# LaTeX2HTML 2002-2-1 (1.71)
# Associate images original text with physical files.


$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{frict.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="284" BORDER="0"
 SRC="|."$dir".q|img659.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{frict.eps}}
\end{figure}">|; 

$key = q/displaystyleI_y,omega_y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1555.png"
 ALT="$\displaystyle I_y \omega_y,$">|; 

$key = q/{displaymath}I=M,frac{int_0^b2pi,sigma^3,dsigma}{int_0^b2pi,sigma,dsigma}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img1355.png"
 ALT="\begin{displaymath}
I = M \frac{\int_0^b 2\pi \sigma^3 d\sigma}{\int_0^b 2\pi \sigma d\sigma}.
\end{displaymath}">|; 

$key = q/{displaymath}(x_2,y_2)=(l_2slash2,l_1).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1709.png"
 ALT="\begin{displaymath}
(x_2, y_2) = (l_2/2, l_1).
\end{displaymath}">|; 

$key = q/displaystylev_0,costheta,,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$\displaystyle v_0 \cos\theta  t,$">|; 

$key = q/omega_1=11,{{rm{rad.slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1606.png"
 ALT="$\omega_1=
11 {\rm rad./s}$">|; 

$key = q/10^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$10^{-1}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{para.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="508" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{para.eps}}
\end{figure}">|; 

$key = q/m_2=3.4,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1705.png"
 ALT="$m_2=3.4 {\rm kg}$">|; 

$key = q/{displaymath}{mitDelta}W=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img654.png"
 ALT="\begin{displaymath}
{\mit\Delta}W = 0.
\end{displaymath}">|; 

$key = q/QP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1237.png"
 ALT="$QP$">|; 

$key = q/225^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$225^\circ$">|; 

$key = q/BC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1686.png"
 ALT="$BC$">|; 

$key = q/|{{bf{s}|,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img613.png"
 ALT="$\vert{\bf s}\vert \cos\theta$">|; 

$key = q/{displaymath}{{bf{p}=m,{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1524.png"
 ALT="\begin{displaymath}
{\bf p} = m {\bf v},
\end{displaymath}">|; 

$key = q/{displaymath}R-M,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1665.png"
 ALT="\begin{displaymath}
R - M g = 0.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{comp.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{comp.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{1}{2},m,v^2-frac{1}{2},m,{v'}^2=m,g,r,(1-costheta),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1147.png"
 ALT="\begin{displaymath}
\frac{1}{2} m v^2 - \frac{1}{2} m {v'}^2 = m g r (1-\cos\theta),
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2.5in{centerline{epsffile{pulw.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img1483.png"
 ALT="\begin{figure*}
\epsfysize =2.5in
\centerline{\epsffile{pulw.eps}}
\end{figure*}">|; 

$key = q/{displaymath}omega=frac{m,v,d}{I+m,d^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1592.png"
 ALT="\begin{displaymath}
\omega = \frac{m v d}{I + m d^2}.
\end{displaymath}">|; 

$key = q/m=10,{{rm{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1220.png"
 ALT="$m=10 {\rm g}$">|; 

$key = q/10^{15};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$10^{15}$">|; 

$key = q/{displaymath}U(r)=-frac{G,M,m}{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1937.png"
 ALT="\begin{displaymath}
U(r) = -\frac{G M m}{r}.
\end{displaymath}">|; 

$key = q/v';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img782.png"
 ALT="$v'$">|; 

$key = q/{displaymath}R=M,g,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1439.png"
 ALT="\begin{displaymath}
R = M g \cos\theta.
\end{displaymath}">|; 

$key = q/BD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="$BD$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{consta1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{consta1.eps}}
\end{figure}">|; 

$key = q/displaystylefrac{M,g,sintheta}{1+M,b^2slashI}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1450.png"
 ALT="$\displaystyle \frac{M g \sin\theta}{1+ M b^2/I}.$">|; 

$key = q/{displaymath}l,ddot{theta}=-g,theta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1824.png"
 ALT="\begin{displaymath}
l \ddot{\theta} = - g \theta,
\end{displaymath}">|; 

$key = q/K_i=K_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1036.png"
 ALT="$K_i=K_f$">|; 

$key = q/{displaymath}x<l,mu,tantheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1671.png"
 ALT="\begin{displaymath}
x&lt;l  \mu \tan\theta.
\end{displaymath}">|; 

$key = q/{displaymath}F=frac{W,(aslashg+1)}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img552.png"
 ALT="\begin{displaymath}
F = \frac{W (a/g+1)}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}I=M,frac{int!int!int(x^2+y^2),,dx,dy,dz}{int!int!int,dx,dy,dz},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1340.png"
 ALT="\begin{displaymath}
I = M \frac{\int\!\int\!\int (x^2+y^2)  dx dy dz}
{\int\!\int\!\int  dx dy dz},
\end{displaymath}">|; 

$key = q/{displaymath}U_B=m,g,h_2=300times9.81times7=2.06times10^4,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="366" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img753.png"
 ALT="\begin{displaymath}
U_B = m g h_2 = 300\times 9.81 \times 7 = 2.06\times 10^4 {\rm J}.
\end{displaymath}">|; 

$key = q/{{bf{f}=(0,-m,g);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img609.png"
 ALT="${\bf f} = (0,-m g)$">|; 

$key = q/{displaymath}z=r,{{rm{e}^{,i,theta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1124.png"
 ALT="\begin{displaymath}
z = r  {\rm e}^{ i \theta}.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{b}+{{bf{a}-mu,{{bf{c}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$\displaystyle {\bf b} + {\bf a} - \mu  {\bf c}.$">|; 

$key = q/displaystylev_{f1}=v_{i2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img936.png"
 ALT="$\displaystyle v_{f1} = v_{i2},$">|; 

$key = q/hat{{bf{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1560.png"
 ALT="$\hat{\bf x}$">|; 

$key = q/{displaymath}x_{cm}=frac{m_1,x_1+m_2,x_2}{m_1+m_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img802.png"
 ALT="\begin{displaymath}
x_{cm} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}.
\end{displaymath}">|; 

$key = q/displaystylefrac{dz}{dt}=3,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$\displaystyle \frac{dz}{dt}=3,$">|; 

$key = q/h=43,{{rm{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img763.png"
 ALT="$h=43 {\rm cm}$">|; 

$key = q/{displaymath}a=v,omega=frac{v^2}{r}=r,omega^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1077.png"
 ALT="\begin{displaymath}
a = v \omega = \frac{v^2}{r} = r \omega^2.
\end{displaymath}">|; 

$key = q/{displaymath}S=frac{(m,gslash2+M,g,xslashl)}{tantheta}=frac{(0.5times40times9.81.81times7slash10)}{tan83.11^circ}=90.09,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="602" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1738.png"
 ALT="\begin{displaymath}
S = \frac{(m g/2 + M g x/l)}{\tan\theta} = \frac{(0.5\tim...
...80\times 9.81\times 7/10)}{\tan 83.11^\circ} = 90.09 {\rm N}.
\end{displaymath}">|; 

$key = q/|{{bf{v}_a|=200,{{rm{mislashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$\vert{\bf v}_a\vert=200 {\rm mi/h}$">|; 

$key = q/omega_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1266.png"
 ALT="$\omega_z$">|; 

$key = q/4.448;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$4.448$">|; 

$key = q/IslashM,b^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1459.png"
 ALT="$I/M b^2$">|; 

$key = q/erightarrow1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2004.png"
 ALT="$e\rightarrow 1$">|; 

$key = q/p_2=m_2,dot{x}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img813.png"
 ALT="$p_2=m_2 \dot{x}_2$">|; 

$key = q/{displaymath}v_f=frac{m_1}{m_1+m_2},v_{i1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img948.png"
 ALT="\begin{displaymath}
v_f = \frac{m_1}{m_1+m_2} v_{i1}.
\end{displaymath}">|; 

$key = q/textstyle<;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1646.png"
 ALT="$\textstyle &lt;$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{rods.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img1675.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{rods.eps}}
\end{figure}">|; 

$key = q/{displaymath}T=frac{2,pi}{omega}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1057.png"
 ALT="\begin{displaymath}
T = \frac{2 \pi}{\omega}
\end{displaymath}">|; 

$key = q/{displaymath}t_{{rm{max}=Tleft(n+frac{phi}{2,pi}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1780.png"
 ALT="\begin{displaymath}
t_{\rm max} = T \left(n + \frac{\phi}{2 \pi}\right).
\end{displaymath}">|; 

$key = q/360^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="$360^\circ$">|; 

$key = q/{displaymath}tau=-k,theta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1811.png"
 ALT="\begin{displaymath}
\tau = -k \theta,
\end{displaymath}">|; 

$key = q/{displaymath}m,g,sintheta>mu,m,g,costheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img513.png"
 ALT="\begin{displaymath}
m g \sin\theta &gt; \mu m g \cos\theta,
\end{displaymath}">|; 

$key = q/{displaymath}omega=sqrt{frac{g}{L}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1833.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{g}{L}},
\end{displaymath}">|; 

$key = q/t=t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img602.png"
 ALT="$t=t$">|; 

$key = q/M=80,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1732.png"
 ALT="$M=80 {\rm kg}$">|; 

$key = q/r_0=0.57,{{rm{AU};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2036.png"
 ALT="$r_0=0.57 {\rm AU}$">|; 

$key = q/(mslashM),v_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img833.png"
 ALT="$(m/M) v_b$">|; 

$key = q/-u_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img896.png"
 ALT="$-u_i$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cylslope.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="316" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img1437.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cylslope.eps}}
\end{figure}">|; 

$key = q/Nrightarrowinfty;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img621.png"
 ALT="$N\rightarrow\infty$">|; 

$key = q/P;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img522.png"
 ALT="$P$">|; 

$key = q/h=r,(1-costheta_0)=aslash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1186.png"
 ALT="$h=r (1-\cos\theta_0) = a/3$">|; 

$key = q/{displaymath}P=mbox{{boldmath{{omega{}cdot{{bf{r}times{{bf{f}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1385.png"
 ALT="\begin{displaymath}
P = \mbox{\boldmath$\omega$}\cdot {\bf r} \times {\bf f}.
\end{displaymath}">|; 

$key = q/B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$B$">|; 

$key = q/Y_1=Y_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1690.png"
 ALT="$Y_1=Y_3$">|; 

$key = q/T({{rm{yr});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1910.png"
 ALT="$T({\rm yr})$">|; 

$key = q/{displaymath}F=frac{34,(3.2slash9.81+1)}{2}=22.55,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img555.png"
 ALT="\begin{displaymath}
F = \frac{34 (3.2/9.81+1)}{2} = 22.55 {\rm N}.
\end{displaymath}">|; 

$key = q/60,{{rm{deg.slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1474.png"
 ALT="$60 {\rm deg./s}$">|; 

$key = q/2times2=4,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$2\times 2=4 {\rm m}$">|; 

$key = q/displaystyleI_z,omega_z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1557.png"
 ALT="$\displaystyle I_z \omega_z,$">|; 

$key = q/{displaymath}deltaA=frac{1}{2}timesr,deltathetatimesr:{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1983.png"
 ALT="\begin{displaymath}
\delta A = \frac{1}{2}\times r \delta\theta \times r:
\end{displaymath}">|; 

$key = q/v_{{rm{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1084.png"
 ALT="$v_{\rm max}$">|; 

$key = q/{displaymath}{v'}^2=5,g,l,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1234.png"
 ALT="\begin{displaymath}
{v'}^2 = 5 g l,
\end{displaymath}">|; 

$key = q/9.79,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1928.png"
 ALT="$9.79 {\rm m/s^2}$">|; 

$key = q/{displaymath}v_{f2}=frac{m_1}{m_2},v_{f1}=frac{5times2.5981}{2.5}=5.1962,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1033.png"
 ALT="\begin{displaymath}
v_{f2} = \frac{m_1}{m_2} v_{f1} = \frac{5\times 2.5981}{2.5} = 5.1962 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{L}=omega,sum_{i=1,N}m_i,{{bf{r}_itimes({{bf{k}times{{bf{r}_i).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1539.png"
 ALT="\begin{displaymath}
{\bf L} = \omega  \sum_{i=1,N} m_i {\bf r}_i\times ({\bf k}\times {\bf r}_i).
\end{displaymath}">|; 

$key = q/{displaymath}{v'}^2=-r,g,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1169.png"
 ALT="\begin{displaymath}
{v'}^2 =- r g \cos\theta.
\end{displaymath}">|; 

$key = q/M+m+dm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img865.png"
 ALT="$M+m+dm$">|; 

$key = q/a=0.2,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1601.png"
 ALT="$a=0.2 {\rm m}$">|; 

$key = q/l=1.3,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1492.png"
 ALT="$l=1.3 {\rm m}$">|; 

$key = q/{{bf{r}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="${\bf r}_2$">|; 

$key = q/{displaymath}tau=f,b.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1443.png"
 ALT="\begin{displaymath}
\tau = f b.
\end{displaymath}">|; 

$key = q/{displaymath}v=r,omega=1.2times314.16=376.99,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1063.png"
 ALT="\begin{displaymath}
v = r \omega = 1.2\times 314.16 =376.99 {\rm m/s}.
\end{displaymath}">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$t$">|; 

$key = q/{displaymath}t_f=C,,sqrt{frac{h}{g}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="\begin{displaymath}
t_f = C  \sqrt{\frac{h}{g}}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{ua589.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="426" HEIGHT="227" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{ua589.eps}}
\end{figure*}">|; 

$key = q/C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$C$">|; 

$key = q/f_1=k_1,x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1867.png"
 ALT="$f_1 = k_1 x_1$">|; 

$key = q/v_{i2}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img939.png"
 ALT="$v_{i2} =0$">|; 

$key = q/X_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1676.png"
 ALT="$X_1$">|; 

$key = q/m=12,{{rm{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1001.png"
 ALT="$m=12 {\rm g}$">|; 

$key = q/{displaymath}K(x)=E-U(x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img693.png"
 ALT="\begin{displaymath}
K(x) = E - U(x).
\end{displaymath}">|; 

$key = q/{displaymath}omega_2=8times11=88,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1616.png"
 ALT="\begin{displaymath}
\omega_2 = 8\times 11 = 88 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/1,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="$1 {\rm N}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{table.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img496.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{table.eps}}
\end{figure}">|; 

$key = q/I=(1slash12),M,(2,b)^2=(1slash3),M,b^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1590.png"
 ALT="$I= (1/12) M (2 b)^2= (1/3) M b^2$">|; 

$key = q/{displaymath}phi=180^circ-tan^{-1}left(frac{1.997times10^2}{2.101times10^2}right)=136.5^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="\begin{displaymath}
\phi = 180^\circ - \tan^{-1} \left(\frac{1.997\times 10^2}{2.101\times 10^2}\right)
= 136.5^\circ.
\end{displaymath}">|; 

$key = q/{displaymath}x(16)-x(0)=4times25=100,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="\begin{displaymath}
x(16)-x(0) = 4\times 25 = 100 {\rm m}.
\end{displaymath}">|; 

$key = q/{displaymath}f=frac{1}{T}=frac{omega}{2,pi}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1058.png"
 ALT="\begin{displaymath}
f = \frac{1}{T} = \frac{\omega}{2 \pi}.
\end{displaymath}">|; 

$key = q/dot{r}=ddot{r}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1140.png"
 ALT="$\dot{r}=\ddot{r}=0$">|; 

$key = q/{displaymath}frac{1}{2},m_1,v_{i1}^{,2}+frac{1}{2},m_2,v_{i2}^{,2}=frac{1}{2},m_1,v_{f1}^{,2}+frac{1}{2},m_2,v_{f2}^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img912.png"
 ALT="\begin{displaymath}
\frac{1}{2} m_1 v_{i1}^{ 2} +\frac{1}{2} m_2 v_{i2}^{ ...
...frac{1}{2} m_1 v_{f1}^{ 2} +\frac{1}{2} m_2 v_{f2}^{ 2},
\end{displaymath}">|; 

$key = q/b,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1486.png"
 ALT="$b \omega$">|; 

$key = q/{{bf{f}_W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="${\bf f}_W$">|; 

$key = q/z=-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$z=-1/2$">|; 

$key = q/{{bf{v}_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="${\bf v}_a$">|; 

$key = q/{displaymath}left.frac{df}{dx}right|_{x_0}<0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img703.png"
 ALT="\begin{displaymath}
\left.\frac{df}{dx}\right\vert _{x_0} &lt;0
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}'=sum_{i=1,N}({{bf{r}_i-{{bf{r}_0)times{{bf{F}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1624.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}' = \sum_{i=1,N} ({\bf r}_i-{\bf r}_0)\times {\bf F}_i.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{beam.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img1642.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{beam.eps}}
\end{figure}">|; 

$key = q/F_1=F_2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img807.png"
 ALT="$F_1=F_2=0$">|; 

$key = q/t=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$t=0$">|; 

$key = q/{displaymath}M,dot{v}=f.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1510.png"
 ALT="\begin{displaymath}
M \dot{v} = f.
\end{displaymath}">|; 

$key = q/{mitDelta}v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="${\mit\Delta} v$">|; 

$key = q/30,{{rm{kmslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1912.png"
 ALT="$30 {\rm km/s}$">|; 

$key = q/m_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img872.png"
 ALT="$m_p$">|; 

$key = q/2pi,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1041.png"
 ALT="$2\pi r$">|; 

$key = q/{displaymath}U(R)=-int_O^R{{bf{f}!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="154" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img661.png"
 ALT="\begin{displaymath}
U(R) = -\int_O^R {\bf f}\!\cdot\!d{\bf r}.
\end{displaymath}">|; 

$key = q/t_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$t_f$">|; 

$key = q/{displaymath}t=frac{L}{v-u}=frac{85}{115+0.115}=0.738,s.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img979.png"
 ALT="\begin{displaymath}
t = \frac{L}{v-u} = \frac{85}{115+0.115} = 0.738 s.
\end{displaymath}">|; 

$key = q/g=9.91,{{rm{m}slash{{rm{s}^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$g=9.91 {\rm m}/{\rm s}^2$">|; 

$key = q/t+deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1049.png"
 ALT="$t+\delta t$">|; 

$key = q/p_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img894.png"
 ALT="$p_f$">|; 

$key = q/v_f=v_{f1}=v_{f2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img946.png"
 ALT="$v_f= v_{f1}=v_{f2}$">|; 

$key = q/displaystylefrac{[0.5times15+36times(0.5slash3)+24times(2.3slash3)]times9.81}{sin40^circ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="386" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1728.png"
 ALT="$\displaystyle \frac{[0.5\times 15 + 36\times(0.5/3)+24\times(2.3/3)]\times 9.81}{\sin 40^\circ}$">|; 

$key = q/v=8,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img992.png"
 ALT="$v=8 {\rm m/s}$">|; 

$key = q/{displaymath}{{bf{v}_0=(v_0,costheta,,0,,v_0,sintheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="\begin{displaymath}
{\bf v}_0 = (v_0 \cos\theta, 0, v_0 \sin\theta).
\end{displaymath}">|; 

$key = q/{displaymath}h_0=frac{2}{3},l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1424.png"
 ALT="\begin{displaymath}
h_0 = \frac{2}{3} l.
\end{displaymath}">|; 

$key = q/displaystylex;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$\displaystyle x$">|; 

$key = q/displaystylez;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$\displaystyle z$">|; 

$key = q/{displaymath}theta(t)=omega,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1038.png"
 ALT="\begin{displaymath}
\theta(t) = \omega t,
\end{displaymath}">|; 

$key = q/sintheta=cos(theta-pislash2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1840.png"
 ALT="$\sin\theta = \cos(\theta-\pi/2)$">|; 

$key = q/s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$s$">|; 

$key = q/lnxsimO(1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img879.png"
 ALT="$\ln x \sim O(1)$">|; 

$key = q/{displaymath}I,{mitDelta}omega=K,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1413.png"
 ALT="\begin{displaymath}
I {\mit\Delta}\omega = K,
\end{displaymath}">|; 

$key = q/z_{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1292.png"
 ALT="$z_{cm}$">|; 

$key = q/f=mu,R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1208.png"
 ALT="$f=\mu R$">|; 

$key = q/T_2,sin30^circ=T_2slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="$T_2 \sin 30^\circ = T_2/2$">|; 

$key = q/{displaymath}tau=r,f,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1377.png"
 ALT="\begin{displaymath}
\tau = r f \sin\theta.
\end{displaymath}">|; 

$key = q/displaystyleMfrac{int!int!int(x^2+y^2),dx,dy,dz}{int!int!int,dx,dy,dz}-2,d,Mfrac{int!int!intx,dx,dy,dz}{int!int!int,dx,dy,dz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="389" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1345.png"
 ALT="$\displaystyle M\frac{\int\!\int\!\int (x^2+y^2) dx dy dz}
{\int\!\int\!\int ...
...z}-2 d M\frac{\int\!\int\!\int x dx dy dz}
{\int\!\int\!\int  dx dy dz}$">|; 

$key = q/-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1110.png"
 ALT="$-1$">|; 

$key = q/{{bf{f}_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="${\bf f}_g$">|; 

$key = q/{displaymath}v'=frac{|{{bf{p}_3|}{M+m}=frac{sqrt{960^2+60^2}}{120+20}=6.87,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img999.png"
 ALT="\begin{displaymath}
v' = \frac{\vert{\bf p}_3\vert}{M+m} = \frac{\sqrt{960^2+60^2}}{120+20} = 6.87 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{rhrule.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="521" HEIGHT="338" BORDER="0"
 SRC="|."$dir".q|img1253.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{rhrule.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{impulse.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="390" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img881.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{impulse.eps}}
\end{figure}">|; 

$key = q/d(sintheta)slashdtheta=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1772.png"
 ALT="$d (\sin\theta)/d\theta = \cos\theta$">|; 

$key = q/dot{{bf{p}={{bf{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1531.png"
 ALT="$\dot{\bf p} = {\bf f}$">|; 

$key = q/{displaymath}{{bf{a}(t)=lim_{{mitDelta}trightarrow0}frac{{{bf{v}(t+{mitDelta}t)-ta}t}=frac{d{{bf{v}}{dt}=frac{d^2{{bf{r}}{dt^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="\begin{displaymath}
{\bf a}(t) = \lim_{{\mit\Delta} t\rightarrow 0}\frac{{\bf v}...
...{\mit\Delta} t} = \frac{d{\bf v}}{dt}=\frac{d^2{\bf r}}{dt^2}.
\end{displaymath}">|; 

$key = q/{displaymath}K=sum_{i=1,N}frac{1}{2},m_i,|{{bf{k}times{{bf{r}_i|^2,omega^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1309.png"
 ALT="\begin{displaymath}
K = \sum_{i=1,N} \frac{1}{2} m_i \vert{\bf k}\times {\bf r}_i\vert^2 \omega^2,
\end{displaymath}">|; 

$key = q/{displaymath}v_{{rm{max}=sqrt{frac{r,T_{{rm{max}}{m}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1083.png"
 ALT="\begin{displaymath}
v_{\rm max} = \sqrt{\frac{r T_{\rm max}}{m}}.
\end{displaymath}">|; 

$key = q/deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1055.png"
 ALT="$\delta t$">|; 

$key = q/{displaymath}I'=I+M,b^2=2,M,b^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1348.png"
 ALT="\begin{displaymath}
I' = I + M b^2 = 2 M b^2.
\end{displaymath}">|; 

$key = q/{{bf{f}_{ba};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="${\bf f}_{ba}$">|; 

$key = q/{displaymath}G=6.6726times10^{-11},{{rm{N,m^2slashkg^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1920.png"
 ALT="\begin{displaymath}
G = 6.6726\times 10^{-11} {\rm N m^2/ kg^2}.
\end{displaymath}">|; 

$key = q/deltathetaslashdeltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1054.png"
 ALT="$\delta\theta /\delta t$">|; 

$key = q/a>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img500.png"
 ALT="$a&gt;0$">|; 

$key = q/m=l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$m=l$">|; 

$key = q/K;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img692.png"
 ALT="$K$">|; 

$key = q/displaystyle486.84,{{rm{N}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1729.png"
 ALT="$\displaystyle 486.84 {\rm N}.$">|; 

$key = q/I=(1slash3),m,l^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1494.png"
 ALT="$I= (1/3) m l^2$">|; 

$key = q/d{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img713.png"
 ALT="$d{\bf r}$">|; 

$key = q/m_2ggm_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img938.png"
 ALT="$m_2\gg m_1$">|; 

$key = q/1,{{rm{ton}=2,240,{{rm{lb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$1 {\rm ton} = 2,240  {\rm lb}$">|; 

$key = q/{displaymath}W=frac{1}{2},m,v_B^{,2}-frac{1}{2},m,v_A^{,2}={mitDelta}K,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img637.png"
 ALT="\begin{displaymath}
W = \frac{1}{2} m v_B^{ 2} -\frac{1}{2} m v_A^{ 2} = {\mit\Delta}K,
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=lim_{deltatrightarrow0}frac{deltambox{{boldphi{}}{deltat}=frac{dmbox{{boldmath{{phi{}}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1257.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}= \lim_{\delta t\rightarrow 0} \frac...
...ldmath$\phi$}}{\delta t} = \frac{d\mbox{\boldmath$\phi$}}{dt}.
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{d^2x}{dt^2}=1-3t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="\begin{displaymath}
a = \frac{d^2 x}{dt^2} = 1 - 3 t^2.
\end{displaymath}">|; 

$key = q/omega_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1264.png"
 ALT="$\omega_x$">|; 

$key = q/{displaymath}frac{m,{v'}^2}{r}=T-m,g,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1150.png"
 ALT="\begin{displaymath}
\frac{m {v'}^2}{r} = T - m g \cos\theta.
\end{displaymath}">|; 

$key = q/{displaymath}x=frac{l}{2},costheta=frac{1.3timescos35^circ}{2}=0.532,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1497.png"
 ALT="\begin{displaymath}
x = \frac{l}{2} \cos\theta = \frac{1.3\times \cos 35^\circ}{2} = 0.532 {\rm m}.
\end{displaymath}">|; 

$key = q/[cdots];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$[\cdots]$">|; 

$key = q/omega_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1265.png"
 ALT="$\omega_y$">|; 

$key = q/9.81,{rmm,s^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$9.81 {\rm m s^{-2}}$">|; 

$key = q/{displaymath}I,ddot{theta}=-M,g,d,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1829.png"
 ALT="\begin{displaymath}
I \ddot{\theta} = - M g d \sin\theta.
\end{displaymath}">|; 

$key = q/-omega,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1786.png"
 ALT="$-\omega a$">|; 

$key = q/l_1=0.5,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1718.png"
 ALT="$l_1=0.5 {\rm m}$">|; 

$key = q/{displaymath}X=-frac{M,g}{2,tantheta}=-frac{M,g}{2,sqrt{3}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1700.png"
 ALT="\begin{displaymath}
X = -\frac{M g}{2 \tan\theta} = -\frac{M g}{2 \sqrt{3}},
\end{displaymath}">|; 

$key = q/-T,l,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1725.png"
 ALT="$-T l \sin\theta$">|; 

$key = q/{displaymath}r_0,u_0=r_1,u_1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2033.png"
 ALT="\begin{displaymath}
r_0 u_0 = r_1 u_1,
\end{displaymath}">|; 

$key = q/displaystyle-v_{i1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img941.png"
 ALT="$\displaystyle - v_{i1},$">|; 

$key = q/{displaymath}f=mu,R=m,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1210.png"
 ALT="\begin{displaymath}
f=\mu R=m g.
\end{displaymath}">|; 

$key = q/1,{{rm{ya}=3,{{rm{ft};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$1 {\rm ya} = 3 {\rm ft}$">|; 

$key = q/{displaymath}f=-k,x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img671.png"
 ALT="\begin{displaymath}
f = -k x,
\end{displaymath}">|; 

$key = q/K_B=(1slash2),m,v_B^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img759.png"
 ALT="$K_B = (1/2) m v_B^{ 2}$">|; 

$key = q/{displaymath}({v'}^2)_0=v^2-4,r,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1158.png"
 ALT="\begin{displaymath}
({v'}^2)_0 = v^2 - 4 r g.
\end{displaymath}">|; 

$key = q/{displaymath}W=sum_iW_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img641.png"
 ALT="\begin{displaymath}
W = \sum_i W_i.
\end{displaymath}">|; 

$key = q/{displaymath}I=M,frac{left[2,pi,sigma^4slash4right]_0^b}{left[2,pi,sigma^2slash2right]_0^b}=frac{1}{2},M,b^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img1356.png"
 ALT="\begin{displaymath}
I = M \frac{\left[2 \pi \sigma^4/4\right]_0^b}{\left[2 \pi \sigma^2/2\right]_0^b}=
\frac{1}{2} M b^2.
\end{displaymath}">|; 

$key = q/v_{f2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img909.png"
 ALT="$v_{f2}$">|; 

$key = q/{displaymath}{{bf{v}=dot{{bf{r}=v_r,{{bf{e}_r+v_theta,{{bf{e}_theta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1102.png"
 ALT="\begin{displaymath}
{\bf v} = \dot{\bf r} = v_r {\bf e}_r + v_\theta {\bf e}_\theta,
\end{displaymath}">|; 

$key = q/8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1607.png"
 ALT="$8$">|; 

$key = q/{displaymath}x=x_0+v,t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="\begin{displaymath}
x = x_0 + v t.
\end{displaymath}">|; 

$key = q/{{bf{a}!cdot!({{bf{b}+{{bf{c})={{bf{a}!cdot!{{bf{b}+{{bf{a}!cdot!{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="${\bf a}\!\cdot\!({\bf b} +{\bf c}) = {\bf a}\!\cdot\!{\bf b}+{\bf a}\!\cdot\!{\bf c}$">|; 

$key = q/displaystyleI_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1323.png"
 ALT="$\displaystyle I_x$">|; 

$key = q/displaystyle{{bf{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$\displaystyle {\bf x}$">|; 

$key = q/{displaymath}omega=frac{2,pi}{T}=frac{2,pi}{0.7}=8.98,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1603.png"
 ALT="\begin{displaymath}
\omega = \frac{2 \pi}{T} = \frac{2 \pi}{0.7}= 8.98 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{complex.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="283" BORDER="0"
 SRC="|."$dir".q|img1114.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{complex.eps}}
\end{figure}">|; 

$key = q/3slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2014.png"
 ALT="$3/2$">|; 

$key = q/{{bf{s}=(x,y,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="${\bf s}=(x,y,z)$">|; 

$key = q/displaystyley;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$\displaystyle y$">|; 

$key = q/{{bf{r}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="${\bf r}_0$">|; 

$key = q/{displaymath}W=int_A^B{{bf{f}({{bf{r})!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img643.png"
 ALT="\begin{displaymath}
W = \int_A^B {\bf f}({\bf r})\!\cdot\!d{\bf r}.
\end{displaymath}">|; 

$key = q/r_1=(3.5+1),R_oplus=4.5,R_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2020.png"
 ALT="$r_1= (3.5+1) R_\oplus= 4.5 R_\oplus$">|; 

$key = q/t_0=v_0,sinthetaslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$t_0=v_0 \sin\theta/g$">|; 

$key = q/{displaymath}m_1,ddot{x}_1+m_2,ddot{x}_2=(m_1+m_2),ddot{x}_{cm}=F_1+F_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img803.png"
 ALT="\begin{displaymath}
m_1 \ddot{x}_1+m_2 \ddot{x}_2 = (m_1+m_2) \ddot{x}_{cm} = F_1 + F_2.
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}=sum_{i=1,N}{{bf{r}_itimesm_i,{{bf{g}=left(sum_iright)times{{bf{g}={{bf{r}_{cm}timesM,{{bf{g},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img1630.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}= \sum_{i=1,N} {\bf r}_i\times m_i {\...
...\bf r}_i\right)\times {\bf g} = {\bf r}_{cm}\times M {\bf g},
\end{displaymath}">|; 

$key = q/m=15,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1749.png"
 ALT="$m=15 {\rm kg}$">|; 

$key = q/displaystyle{{bf{v}_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="$\displaystyle {\bf v}_g$">|; 

$key = q/{displaymath}K=E-frac{1}{2},k,x^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img686.png"
 ALT="\begin{displaymath}
K = E - \frac{1}{2} k x^2.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{s}=lambda,(x,y,z)=(lambda,x,lambda,y,lambda,z).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="\begin{displaymath}
{\bf s} = \lambda (x,y,z) = (\lambda x, \lambda y, \lambda z).
\end{displaymath}">|; 

$key = q/-m,u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img986.png"
 ALT="$-m u$">|; 

$key = q/0.3048,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$0.3048 {\rm m}$">|; 

$key = q/F(x-v,t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$F(x-v t)$">|; 

$key = q/{displaymath}I,alpha=tau,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1496.png"
 ALT="\begin{displaymath}
I \alpha = \tau,
\end{displaymath}">|; 

$key = q/{displaymath}tantheta<3,mu.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1455.png"
 ALT="\begin{displaymath}
\tan\theta &lt; 3 \mu.
\end{displaymath}">|; 

$key = q/dot{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1134.png"
 ALT="$\dot{z}$">|; 

$key = q/m,g,r,(1-costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1146.png"
 ALT="$m g r (1-\cos\theta)$">|; 

$key = q/displaystyle{{bf{0}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1623.png"
 ALT="$\displaystyle {\bf0}.$">|; 

$key = q/{displaymath}M,g,frac{l}{2}-T,l,sintheta=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1655.png"
 ALT="\begin{displaymath}
M g \frac{l}{2} - T l \sin\theta = 0.
\end{displaymath}">|; 

$key = q/r=2.34times10^{11},{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2041.png"
 ALT="$r=2.34\times 10^{11} {\rm m}$">|; 

$key = q/{displaymath}{{bf{L}=I_x,omega_x,hat{{bf{x}+I_y,omega_y,hat{{bf{y}+I_z,omega_z,hat{{bf{z},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1559.png"
 ALT="\begin{displaymath}
{\bf L} = I_x \omega_x \hat{\bf x} + I_y \omega_y \hat{\bf y} + I_z \omega_z \hat{\bf z},
\end{displaymath}">|; 

$key = q/r=|{{bf{r}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="$r=\vert{\bf r}\vert$">|; 

$key = q/+omega,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1787.png"
 ALT="$+\omega a$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{hoop.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img1161.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{hoop.eps}}
\end{figure}">|; 

$key = q/{displaymath}bar{{bf{f}=frac{{{bf{I}}{{mitDelta}t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img904.png"
 ALT="\begin{displaymath}
\bar{\bf f} = \frac{{\bf I}}{{\mit\Delta}t}.
\end{displaymath}">|; 

$key = q/m=5.3,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1491.png"
 ALT="$m=5.3 {\rm kg}$">|; 

$key = q/6.62;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1960.png"
 ALT="$6.62$">|; 

$key = q/T=0.35,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1879.png"
 ALT="$T=0.35 {\rm s}$">|; 

$key = q/{displaymath}f=mu,R=0.25times3times9.81timescos35^circ=6.03,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img767.png"
 ALT="\begin{displaymath}
f = \mu R = 0.25\times 3\times 9.81 \times \cos 35^\circ = 6.03 {\rm N}.
\end{displaymath}">|; 

$key = q/{v'}^2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1165.png"
 ALT="${v'}^2=0$">|; 

$key = q/1,{{rm{yr}=365.26,{{rm{da}=31,558,464,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$1 {\rm yr} = 365.26 {\rm da} = 31,558,464 {\rm s}$">|; 

$key = q/h_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img573.png"
 ALT="$h_1$">|; 

$key = q/{displaymath}v=u,lnleft(frac{M+m_p}{M+m}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img875.png"
 ALT="\begin{displaymath}
v = u \ln\left(\frac{M+m_p}{M+m}\right).
\end{displaymath}">|; 

$key = q/t=16;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$t=16$">|; 

$key = q/{{bf{r}-{{bf{r}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="${\bf r} - {\bf r}_2$">|; 

$key = q/r=85,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1199.png"
 ALT="$r = 85 {\rm m}$">|; 

$key = q/{displaymath}L=I,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1589.png"
 ALT="\begin{displaymath}
L = I \omega,
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{{mitDelta}v}{{mitDelta}t}=frac{v(t=12)-v(t=10)}{12-10}=frac{4-8}{2}=-2,{{rm{m,s^{-2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="\begin{displaymath}
a = \frac{{\mit\Delta}v}{{\mit\Delta} t} = \frac{v(t=12)-v(t=10)}{12-10} =
\frac{4-8}{2} = -2 {\rm m s^{-2}}.
\end{displaymath}">|; 

$key = q/t>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$t&gt;0$">|; 

$key = q/120^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="$120^\circ$">|; 

$key = q/{displaymath}z_{cm}=frac{int_0^h,z,dz!int_{-a,(1-zslashh)slash2}^{+a,(1-zslashh)lash2}^{+a,(1-zslashh)slash2},a,(1-zslashh),dy}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img1301.png"
 ALT="\begin{displaymath}
z_{cm} = \frac{\int_0^h z dz\!\int_{-a (1-z/h)/2}^{+a (1...
...h dz\!\int_{-a (1-z/h)/2}^{+a (1-z/h)/2} 
a (1-z/h) dy}.
\end{displaymath}">|; 

$key = q/f'=105;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img743.png"
 ALT="$f'= 105$">|; 

$key = q/{displaymath}t_f=C,m^x,h^y,g^z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="\begin{displaymath}
t_f = C m^x h^y g^z,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{hooke.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="171" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{hooke.eps}}
\end{figure}">|; 

$key = q/{displaymath}M,{mitDelta}v=-J-J'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1410.png"
 ALT="\begin{displaymath}
M {\mit\Delta} v = -J - J'.
\end{displaymath}">|; 

$key = q/RslashR_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1932.png"
 ALT="$R/R_\oplus$">|; 

$key = q/f=G,Mslashr^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1935.png"
 ALT="$f = G M/r^2$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{multi1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1568.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{multi1.eps}}
\end{figure}">|; 

$key = q/mu_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img517.png"
 ALT="$\mu_k$">|; 

$key = q/u_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img882.png"
 ALT="$u_i$">|; 

$key = q/displaystylev_0,t-frac{1}{2},g,t^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$\displaystyle v_0 t -\frac{1}{2}  g t^2,$">|; 

$key = q/displaystyle-Y_2-Y_3-M,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1688.png"
 ALT="$\displaystyle -Y_2 - Y_3 - M g$">|; 

$key = q/mu=0.15;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1508.png"
 ALT="$\mu=0.15$">|; 

$key = q/x=pma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1808.png"
 ALT="$x=\pm a$">|; 

$key = q/f(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img885.png"
 ALT="$f(t)$">|; 

$key = q/v_z=dzslashdt=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$v_z=dz/dt=0$">|; 

$key = q/v^2>5,r,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1177.png"
 ALT="$v^2 &gt; 5 r g$">|; 

$key = q/displaystyleM,frac{int!int!int[(x^2+y^2)-2,d,x+d^2],dx,dy,dz}{int!int!int,dx,dy,dz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1344.png"
 ALT="$\displaystyle M \frac{\int\!\int\!\int [(x^2+y^2) - 2 d x + d^2] dx dy dz}
{\int\!\int\!\int  dx dy dz}$">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$\phi$">|; 

$key = q/{{bf{s}=lambda,{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="${\bf s} = \lambda {\bf r}$">|; 

$key = q/displaystylesqrt{(-2.101times10^2)^2+(1.997times10^2)^2}=289.9,{{rm{kmslashh}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="$\displaystyle \sqrt{(-2.101\times 10^2)^2 + (1.997\times 10^2)^2} = 289.9 {\rm km/h}.$">|; 

$key = q/x<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1793.png"
 ALT="$x&lt;0$">|; 

$key = q/{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="${\rm km/h}$">|; 

$key = q/displaystyle(-1.500times10^2,-,6.010times10^1,,2.598times10^2,-,6.010times10^1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="$\displaystyle (-1.500\times 10^2 - 6.010\times 10^1, 2.598\times 10^2 - 6.010\times 10^1)$">|; 

$key = q/f=mu,m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img507.png"
 ALT="$f=\mu m g$">|; 

$key = q/dot{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1441.png"
 ALT="$\dot{v}$">|; 

$key = q/60^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$60^\circ$">|; 

$key = q/{displaymath}g_1=g_0left(frac{r_0}{r_1}right)^2=frac{9.81times1}{(4.5)^2}=0.484,{{rm{mslashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img2024.png"
 ALT="\begin{displaymath}
g_1 = g_0\left(\frac{r_0}{r_1}\right)^2 = \frac{9.81\times 1}{(4.5)^2} = 0.484 {\rm m/s^2}.
\end{displaymath}">|; 

$key = q/theta=60^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1701.png"
 ALT="$\theta = 60^\circ$">|; 

$key = q/{displaymath}v_{f1}=x,v_{i1}=-0.5times12=-6,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1024.png"
 ALT="\begin{displaymath}
v_{f1} = x v_{i1} = -0.5\times12 = -6 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{displaymath}bar{f}=frac{I}{{mitDelta}t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img899.png"
 ALT="\begin{displaymath}
\bar{f} = \frac{I}{{\mit\Delta} t}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{string1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{string1.eps}}
\end{figure}">|; 

$key = q/v_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1104.png"
 ALT="$v_\theta$">|; 

$key = q/x_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img819.png"
 ALT="$x_g$">|; 

$key = q/54,{{rm{kmslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2031.png"
 ALT="$54 {\rm km/s}$">|; 

$key = q/U=m,g,(z-z_0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img669.png"
 ALT="$U=m g (z-z_0)$">|; 

$key = q/{displaymath}[T]=[M]^x,[L]^y,left(frac{[L]}{[T^2]}right)^z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="202" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="\begin{displaymath}[T]= [M]^x [L]^y \left(\frac{[L]}{[T^2]}\right)^z.
\end{displaymath}">|; 

$key = q/5000,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img880.png"
 ALT="$5000 {\rm m/s}$">|; 

$key = q/displaystylefrac{m,g}{1+m,b^2slashI}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1488.png"
 ALT="$\displaystyle \frac{m g}{1+ m b^2/I}.$">|; 

$key = q/{displaymath}f=-frac{dU}{dx}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img680.png"
 ALT="\begin{displaymath}
f = -\frac{dU}{dx}.
\end{displaymath}">|; 

$key = q/displaystylea_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="$\displaystyle a_z$">|; 

$key = q/{displaymath}sum_{i=1,N}!m_i,ddot{{bf{r}_i=sum_{i,j=1,N}^{jneqi}!{{bf{f}_{ij}+sum_{i=1,N}!{{bf{F}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img843.png"
 ALT="\begin{displaymath}
\sum_{i=1,N} \!m_i \ddot{\bf r}_i =\sum_{i,j=1,N}^{j\neq i}\! {\bf f}_{ij} + \sum_{i=1,N}\!
{\bf F}_i.
\end{displaymath}">|; 

$key = q/{displaymath}x=x_0+v_0,t+frac{1}{2},g,sintheta,t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="\begin{displaymath}
x = x_0 + v_0 t + \frac{1}{2} g \sin\theta t^2.
\end{displaymath}">|; 

$key = q/x=-a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1775.png"
 ALT="$x=-a$">|; 

$key = q/{displaymath}P=tau,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1393.png"
 ALT="\begin{displaymath}
P = \tau \omega.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}=mbox{{boldmath{{tau{},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1579.png"
 ALT="\begin{displaymath}
\frac{d {\bf L}}{dt} = \mbox{\boldmath$\tau$},
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2},M,v^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1313.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} M v^2,
\end{displaymath}">|; 

$key = q/dot{x}=0.75,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1855.png"
 ALT="$\dot{x} = 0.75 {\rm m/s}$">|; 

$key = q/{displaymath}k_{{rm{eff}=frac{k_1,k_2}{k_1+k_2}=frac{1200times400}{1200+400}=300,{{rm{Nslashm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1875.png"
 ALT="\begin{displaymath}
k_{\rm eff} = \frac{k_1 k_2}{k_1+k_2} = \frac{1200\times 400}{1200+400} = 300 {\rm N/m}.
\end{displaymath}">|; 

$key = q/displaystylefrac{l}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1647.png"
 ALT="$\displaystyle \frac{l}{2},$">|; 

$key = q/{displaymath}alpha=72.54^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="\begin{displaymath}
\alpha = 72.54^\circ.
\end{displaymath}">|; 

$key = q/T,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1087.png"
 ALT="$T \sin\theta$">|; 

$key = q/displaystyle{{bf{c}=(36,0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="$\displaystyle {\bf c} = (36, 0),$">|; 

$key = q/E=m,c^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$E=m c^3$">|; 

$key = q/{displaymath}v=sqrt{2,g,h}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img588.png"
 ALT="\begin{displaymath}
v = \sqrt{2 g h}.
\end{displaymath}">|; 

$key = q/{displaymath}f=G,frac{m,M_oplus}{R_oplus^{,2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1923.png"
 ALT="\begin{displaymath}
f = G  \frac{m M_\oplus}{R_\oplus^{ 2}},
\end{displaymath}">|; 

$key = q/v_{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img810.png"
 ALT="$v_{cm}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{crossp.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1273.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{crossp.eps}}
\end{figure}">|; 

$key = q/a=3.2,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img554.png"
 ALT="$a=3.2 {\rm m/s^2}$">|; 

$key = q/displaystyle-frac{m_2}{m_1+m_2},(v_{i2}-v_{i1}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img917.png"
 ALT="$\displaystyle - \frac{m_2}{m_1+m_2} (v_{i2}-v_{i1}),$">|; 

$key = q/m_1=5.2,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1703.png"
 ALT="$m_1=5.2 {\rm kg}$">|; 

$key = q/{displaymath}m,ddot{x}=-k,x.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1769.png"
 ALT="\begin{displaymath}
m \ddot{x} = - k x.
\end{displaymath}">|; 

$key = q/3.2,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img546.png"
 ALT="$3.2 {\rm m/s^2}$">|; 

$key = q/{displaymath}{{bf{r}_itimes{{bf{f}_{ij}+{{bf{r}_jtimes{{bf{f}_{ji}={{bf{0},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1578.png"
 ALT="\begin{displaymath}
{\bf r}_i\times {\bf f}_{ij}+{\bf r}_j\times {\bf f}_{ji} = {\bf0},
\end{displaymath}">|; 

$key = q/E;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$E$">|; 

$key = q/{displaymath}theta=tan^{-1}!left(frac{|{{bf{p}_2|}{|{{bf{p}_1|}right)=tan^{-1}!left(frac{60}{960}right)=3.58^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="334" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1000.png"
 ALT="\begin{displaymath}
\theta = \tan^{-1}\!\left(\frac{\vert{\bf p}_2\vert}{\vert{\...
...}
\right) =\tan^{-1}\!\left(\frac{60}{960}\right)= 3.58^\circ.
\end{displaymath}">|; 

$key = q/{displaymath}phi(t)=phi_0+omega,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1246.png"
 ALT="\begin{displaymath}
\phi(t) = \phi_0 + \omega t,
\end{displaymath}">|; 

$key = q/{displaymath}Usimeqm,g,z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1944.png"
 ALT="\begin{displaymath}
U \simeq m g z,
\end{displaymath}">|; 

$key = q/m_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img817.png"
 ALT="$m_g$">|; 

$key = q/{{bf{a}cdot{{bf{b}times{{bf{c}={{bf{a}times{{bf{b}cdot{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1541.png"
 ALT="${\bf a}\cdot {\bf b}\times {\bf c} = {\bf a}\times {\bf b} \cdot {\bf c}$">|; 

$key = q/{displaymath}T=frac{2pi}{sqrt{G,M_odot}},,a^{3slash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img2013.png"
 ALT="\begin{displaymath}
T = \frac{2\pi}{\sqrt{G M_\odot}}  a^{3/2}.
\end{displaymath}">|; 

$key = q/{{bf{L}=I,mbox{{boldmath{{omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1562.png"
 ALT="${\bf L} = I \mbox{\boldmath$\omega$}$">|; 

$key = q/{displaymath}dot{v}_{{rm{hollow}=frac{1}{2},g,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1458.png"
 ALT="\begin{displaymath}
\dot{v}_{\rm hollow} = \frac{1}{2} g \sin\theta.
\end{displaymath}">|; 

$key = q/alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1248.png"
 ALT="$\alpha$">|; 

$key = q/t_2-t_1={mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img903.png"
 ALT="$t_2-t_1={\mit\Delta} t$">|; 

$key = q/-dm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img860.png"
 ALT="$-dm$">|; 

$key = q/m_2llm_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img943.png"
 ALT="$m_2\ll m_1$">|; 

$key = q/v-u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img862.png"
 ALT="$v-u$">|; 

$key = q/phi=theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1660.png"
 ALT="$\phi =\theta$">|; 

$key = q/displaystylefrac{g}{1+Islashm,b^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1487.png"
 ALT="$\displaystyle \frac{g}{1+I/m b^2},$">|; 

$key = q/{{bf{r}_{cm}=sum_{i=1,N}m_i,{{bf{r}_islashM;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1632.png"
 ALT="${\bf r}_{cm} = \sum_{i=1,N} m_i {\bf r}_i/M$">|; 

$key = q/Y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1758.png"
 ALT="$Y$">|; 

$key = q/{displaymath}I=frac{2}{3},M,b^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1360.png"
 ALT="\begin{displaymath}
I = \frac{2}{3} M b^2.
\end{displaymath}">|; 

$key = q/{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="${\bf c}$">|; 

$key = q/omega=dthetaslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1133.png"
 ALT="$\omega = d\theta/dt$">|; 

$key = q/{displaymath}m,v=(M+m),v'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1223.png"
 ALT="\begin{displaymath}
m v = (M+m) v'.
\end{displaymath}">|; 

$key = q/1,{{rm{tonne}=1,000,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$1 {\rm tonne} = 1,000 
{\rm kg}$">|; 

$key = q/{displaymath}{mitDelta}x=v_1,{mitDelta}t+frac{1}{2},a,({mitDeltat})^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="\begin{displaymath}
{\mit\Delta} x = v_1 {\mit\Delta} t + \frac{1}{2} a ({\mit\Delta t})^2,
\end{displaymath}">|; 

$key = q/{mitDelta}U=m,g,s=-m,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img585.png"
 ALT="${\mit\Delta}U=m g s=-m g h$">|; 

$key = q/g_0=9.81,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2022.png"
 ALT="$g_0= 9.81 {\rm m/s^2}$">|; 

$key = q/f,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1362.png"
 ALT="$f \cos\theta$">|; 

$key = q/{displaymath}{{bf{f}=-frac{G,M_odot,m}{r^2},{{bf{e}_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1965.png"
 ALT="\begin{displaymath}
{\bf f} = - \frac{G M_\odot m}{r^2} {\bf e}_r.
\end{displaymath}">|; 

$key = q/displaystyle-M,g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1696.png"
 ALT="$\displaystyle -M g.$">|; 

$key = q/{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="${\rm m/s}$">|; 

$key = q/{displaymath}omega=sqrt{frac{g}{h}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1094.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{g}{h}}.
\end{displaymath}">|; 

$key = q/alpha(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1244.png"
 ALT="$\alpha(t)$">|; 

$key = q/0.15;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img728.png"
 ALT="$0.15$">|; 

$key = q/b=0.25,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1506.png"
 ALT="$b=0.25 {\rm m}$">|; 

$key = q/displaystyleT_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$\displaystyle T_1$">|; 

$key = q/{mitDelta}K=(1slash2),m,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img586.png"
 ALT="${\mit\Delta}K=(1/2) m v^2$">|; 

$key = q/{displaymath}W'=-f,x=-69.89times6=-419.3,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img735.png"
 ALT="\begin{displaymath}
W' = - f x = - 69.89\times 6 = -419.3 {\rm J}.
\end{displaymath}">|; 

$key = q/6^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1822.png"
 ALT="$6^\circ$">|; 

$key = q/displaystyleR;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1657.png"
 ALT="$\displaystyle R$">|; 

$key = q/{figure}{epsfysize{=3.5in{centerline{epsffile{attwood.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="548" HEIGHT="401" BORDER="0"
 SRC="|."$dir".q|img502.png"
 ALT="\begin{figure}
\epsfysize =3.5in
\centerline{\epsffile{attwood.eps}}
\end{figure}">|; 

$key = q/{displaymath}a_{{rm{max}=a,omega^2=0.07times418.88times418.88=1.228times10^4,{{rm{mslashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1848.png"
 ALT="\begin{displaymath}
a_{\rm max} = a \omega^2 = 0.07 \times 418.88\times 418.88 = 1.228\times 10^4 {\rm m/s^2}.
\end{displaymath}">|; 

$key = q/theta=theta^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1162.png"
 ALT="$\theta =\theta^\circ$">|; 

$key = q/{{bf{f}({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img644.png"
 ALT="${\bf f}({\bf r})$">|; 

$key = q/displaystylefrac{dy}{dt}=-sint,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="$\displaystyle \frac{dy}{dt}= - \sin t,$">|; 

$key = q/{{bf{P};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img853.png"
 ALT="${\bf P}$">|; 

$key = q/m_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img836.png"
 ALT="$m_i$">|; 

$key = q/0.034,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1913.png"
 ALT="$0.034 {\rm m/s^2}$">|; 

$key = q/I_xneqI_yneqI_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1563.png"
 ALT="$I_x\neq I_y\neq I_z$">|; 

$key = q/{displaymath}t=sqrt{frac{-2,h}{g}}=sqrt{frac{2times41.15}{9.81}}=2.896,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="\begin{displaymath}
t = \sqrt{\frac{-2 h}{g}} = \sqrt{\frac{2\times 41.15}{9.81}} = 2.896 {\rm s}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}W=W_1-W_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img653.png"
 ALT="\begin{displaymath}
{\mit\Delta}W = W_1 - W_2,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{ring.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img1320.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{ring.eps}}
\end{figure}">|; 

$key = q/(M+m),v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1005.png"
 ALT="$(M+m) v$">|; 

$key = q/{displaymath}T_1+T_2-M,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1640.png"
 ALT="\begin{displaymath}
T_1 + T_2 - M g = 0.
\end{displaymath}">|; 

$key = q/m,g,l,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1818.png"
 ALT="$m g l \sin\theta$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{vector.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{vector.eps}}
\end{figure}">|; 

$key = q/{displaymath}I,dot{omega}=tau=f,b,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1444.png"
 ALT="\begin{displaymath}
I \dot{\omega} = \tau=f b,
\end{displaymath}">|; 

$key = q/|{{bf{a}|,|{{bf{b}|,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1275.png"
 ALT="$\vert{\bf a}\vert \vert{\bf b}\vert \sin\theta$">|; 

$key = q/{displaymath}g=frac{(6.673times10^{-11})times(1.08times10^{23})}{(2.403times10^6)^2}=1.25,{{rm{mslashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2018.png"
 ALT="\begin{displaymath}
g = \frac{(6.673\times 10^{-11})\times(1.08\times 10^{23})}{(2.403\times 10^6)^2} = 1.25 {\rm m/s^2}.
\end{displaymath}">|; 

$key = q/m=3,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img762.png"
 ALT="$m=3 {\rm kg}$">|; 

$key = q/{displaymath}r=frac{G,M_oplus}{v^2}=frac{(6.673times10^{-11})times(5.97times10^{24})}{(6000)^2}=1.107times10^7,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="506" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2027.png"
 ALT="\begin{displaymath}
r = \frac{G M_\oplus}{v^2} = \frac{(6.673\times 10^{-11})\times(5.97\times 10^{24})}{(6000)^2} =1.107\times 10^7 {\rm m}.
\end{displaymath}">|; 

$key = q/displaystyle-p_{f1}'=p_{f2}'=mu,(v_{f2}-v_{f1}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img925.png"
 ALT="$\displaystyle -p_{f1}' = p_{f2}' = \mu (v_{f2}-v_{f1}),$">|; 

$key = q/{displaymath}P=f,v.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img717.png"
 ALT="\begin{displaymath}
P = f v.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{pyramid.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="548" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img1294.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{pyramid.eps}}
\end{figure}">|; 

$key = q/{{bf{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="${\bf f}$">|; 

$key = q/{displaymath}T=frac{2,pi}{omega}=1.676,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1862.png"
 ALT="\begin{displaymath}
T = \frac{2 \pi}{\omega} = 1.676 {\rm s}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{ringax.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img1334.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{ringax.eps}}
\end{figure}">|; 

$key = q/8.49,{{rm{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img981.png"
 ALT="$8.49 {\rm cm}$">|; 

$key = q/dot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1785.png"
 ALT="$\dot{x}$">|; 

$key = q/displaystylev_{i2}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img918.png"
 ALT="$\displaystyle v_{i2}'$">|; 

$key = q/{displaymath}alpha=frac{2,phi}{t^2}=frac{2times33.30}{2.3^2}=12.59,{{rm{rad.slashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1465.png"
 ALT="\begin{displaymath}
\alpha = \frac{2 \phi}{t^2} = \frac{2\times 33.30}{2.3^2} = 12.59  {\rm rad./s^2}.
\end{displaymath}">|; 

$key = q/f_{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img788.png"
 ALT="$f_{12}$">|; 

$key = q/2,r,g<v^2<5,r,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1178.png"
 ALT="$2 r g &lt; v^2&lt; 5 r g$">|; 

$key = q/{displaymath}{{bf{f}_{12}=G,frac{{{bf{r}_2-{{bf{r}_1}{|{{bf{r}_2-{{bf{r}_1|^3}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1917.png"
 ALT="\begin{displaymath}
{\bf f}_{12} = G \frac{{\bf r}_2-{\bf r}_1}{\vert{\bf r}_2-{\bf r}_1\vert^3}.
\end{displaymath}">|; 

$key = q/v=115,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img969.png"
 ALT="$v=115 {\rm m/s}$">|; 

$key = q/{displaymath}omega=frac{2,pi}{24times60times60}=7.27times10^{-5},{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1956.png"
 ALT="\begin{displaymath}
\omega = \frac{2 \pi}{24\times 60 \times 60} = 7.27\times 10^{-5} {\rm rad./s}.
\end{displaymath}">|; 

$key = q/theta=90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$\theta =90^\circ$">|; 

$key = q/v^2<2,r,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1176.png"
 ALT="$v^2&lt;2 r g$">|; 

$key = q/l=10,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1731.png"
 ALT="$l=10 {\rm m}$">|; 

$key = q/{displaymath}Wsimeqsum_{i=1}^{N}{{bf{f}_i!cdot!{mitDelta}{{bf{r}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img620.png"
 ALT="\begin{displaymath}
W \simeq \sum_{i=1}^{N} {\bf f}_i\!\cdot\!{\mit\Delta}{\bf r}_i.
\end{displaymath}">|; 

$key = q/{displaymath}a>frac{g}{mu}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img565.png"
 ALT="\begin{displaymath}
a &gt; \frac{g}{\mu}.
\end{displaymath}">|; 

$key = q/p_{i1}'=m_1,v_{i1}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img927.png"
 ALT="$p_{i1}' = m_1 v_{i1}'$">|; 

$key = q/{figure}{epsfysize{=4.in{centerline{epsffile{consta.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="456" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="\begin{figure}
\epsfysize =4.in
\centerline{\epsffile{consta.eps}}
\end{figure}">|; 

$key = q/-m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img596.png"
 ALT="$-m g$">|; 

$key = q/{displaymath}phi=270^circ-alpha=270^circ-72.54^circ=197.46^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="\begin{displaymath}
\phi = 270^\circ -alpha = 270^\circ - 72.54^\circ = 197.46^\circ.
\end{displaymath}">|; 

$key = q/{{bf{L}=(L_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1549.png"
 ALT="${\bf L} = (L_x$">|; 

$key = q/t=uslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$t=u/g$">|; 

$key = q/-W';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img737.png"
 ALT="$-W'$">|; 

$key = q/displaystylev_0,t+frac{1}{2},a,t^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\displaystyle v_0 t + \frac{1}{2}  a t^2,$">|; 

$key = q/r=1.2,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1059.png"
 ALT="$r=1.2 {\rm m}$">|; 

$key = q/displaystylefrac{costheta}{sin(theta+phi)},M,g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1658.png"
 ALT="$\displaystyle \frac{\cos\theta}{\sin(\theta+\phi)} M g.$">|; 

$key = q/{displaymath}m,g,frac{l}{2}+m_1,g,l_1+m_2,g,l_2-T,l,sintheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="324" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1726.png"
 ALT="\begin{displaymath}
m g \frac{l}{2} + m_1 g l_1 + m_2 g l_2 -T l \sin\theta = 0,
\end{displaymath}">|; 

$key = q/{displaymath}h_{{rm{max}=frac{v_0^{,2}}{2,g},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="\begin{displaymath}
h_{\rm max} = \frac{v_0^{ 2}}{2 g},
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{park.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img1207.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{park.eps}}
\end{figure*}">|; 

$key = q/14.59;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$14.59$">|; 

$key = q/V;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$V$">|; 

$key = q/M=5000,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1741.png"
 ALT="$M=5000 {\rm kg}$">|; 

$key = q/{displaymath}I=int_{t_1}^{t_2}f(t),dt.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img889.png"
 ALT="\begin{displaymath}
I = \int_{t_1}^{t_2} f(t) dt.
\end{displaymath}">|; 

$key = q/{displaymath}m_i=rho({{bf{r}_i),V_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1283.png"
 ALT="\begin{displaymath}
m_i = \rho({\bf r}_i) V_i,
\end{displaymath}">|; 

$key = q/x=d,y=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1338.png"
 ALT="$x=d, y=0$">|; 

$key = q/displaystyleY_2-Y_1-M,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1685.png"
 ALT="$\displaystyle Y_2 - Y_1 - M g$">|; 

$key = q/{displaymath}g_oplus=frac{G,M_oplus}{R_oplus^{,2}}=frac{(6.673times10^{-11})time{24})}{(6.378times10^6)^2}=9.79,{{rm{mslashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="482" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1927.png"
 ALT="\begin{displaymath}
g_\oplus = \frac{G  M_\oplus}{R_\oplus^{ 2}} = \frac{ (6.6...
....97\times 10^{24})}{(6.378\times 10^6)^2} = 9.79 {\rm m/s^2}.
\end{displaymath}">|; 

$key = q/{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="${\bf r}$">|; 

$key = q/{displaymath}{{bf{r}_itimesdot{{bf{p}_i=sum_{j=1,N}^{jneqi}{{bf{r}_itimes{{bf{f}_{ij}+{{bf{r}_itimes{{bf{F}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img1570.png"
 ALT="\begin{displaymath}
{\bf r}_i\times \dot{\bf p}_i = \sum_{j=1,N}^{j\neq i} {\bf r}_i\times {\bf f}_{ij} +
{\bf r}_i\times {\bf F}_i.
\end{displaymath}">|; 

$key = q/{displaymath}-m,omega^2,a,cos(omega,t-phi)=-k,a,cos(omega,t-phi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1773.png"
 ALT="\begin{displaymath}
-m \omega^2 a \cos(\omega t-\phi) =- k a \cos(\omega t-\phi).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{bear.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="288" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{bear.eps}}
\end{figure}">|; 

$key = q/{displaymath}1=(1-3,y)^2+3,y^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1021.png"
 ALT="\begin{displaymath}
1 = (1-3 y)^2 + 3 y^2,
\end{displaymath}">|; 

$key = q/v_{g,x}=v_g,cosphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$v_{g x}=v_g \cos\phi$">|; 

$key = q/a_r=v^2slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1201.png"
 ALT="$a_r=v^2/r$">|; 

$key = q/-T=-m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="$-T=-m g$">|; 

$key = q/v(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$v(t)$">|; 

$key = q/{{bf{d}={{bf{b}-{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="${\bf d} = {\bf b} - {\bf a}$">|; 

$key = q/{displaymath}m_1,v_{i1}=m_1,v_{f1}+m_2,v_{f2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1014.png"
 ALT="\begin{displaymath}
m_1 v_{i1} = m_1 v_{f1} + m_2 v_{f2},
\end{displaymath}">|; 

$key = q/displaystyleX_1-X_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1682.png"
 ALT="$\displaystyle X_1 - X_3$">|; 

$key = q/{displaymath}omega=sqrt{frac{k_{{rm{eff}}{m}}=sqrt{frac{300}{3}}=10,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1876.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{k_{\rm eff}}{m}}=\sqrt{\frac{300}{3}} = 10 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/x=x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img701.png"
 ALT="$x=x_1$">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}=sum_{i=1,N}{{bf{r}_itimes{{bf{F}_i{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1580.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}= \sum_{i=1,N} {\bf r}_i\times {\bf F}_i
\end{displaymath}">|; 

$key = q/{displaymath}e=frac{A,h^2}{G,M_odot},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1995.png"
 ALT="\begin{displaymath}
e = \frac{A h^2}{G M_\odot},
\end{displaymath}">|; 

$key = q/displaystylev_0^{,2}-2,g,s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$\displaystyle v_0^{ 2} - 2 g s.$">|; 

$key = q/displaystyles;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\displaystyle s$">|; 

$key = q/t_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img887.png"
 ALT="$t_2$">|; 

$key = q/I_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1544.png"
 ALT="$I_k$">|; 

$key = q/289.9,{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="$289.9 {\rm km/h}$">|; 

$key = q/m=0.35,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img982.png"
 ALT="$m=0.35 {\rm kg}$">|; 

$key = q/{displaymath}v=-g,t=-9.81times2.896=-28.41,{{rm{m,s}^{-1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="343" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="\begin{displaymath}
v =- g t = -9.81\times 2.896 = -28.41 {\rm m s}^{-1}.
\end{displaymath}">|; 

$key = q/{{bf{r}_itimes{{bf{f}_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1574.png"
 ALT="${\bf r}_i\times {\bf f}_{ij}$">|; 

$key = q/{displaymath}omega,d^2={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1584.png"
 ALT="\begin{displaymath}
\omega d^2 = {\rm constant}.
\end{displaymath}">|; 

$key = q/{{bf{v}=(v_x,v_y,v_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="${\bf v}=(v_x,v_y,v_z)$">|; 

$key = q/m,g,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="$m g \sin\theta$">|; 

$key = q/{displaymath}{{bf{r}={{bf{r}_1+{{bf{r}_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img519.png"
 ALT="\begin{displaymath}
{\bf r} = {\bf r}_1 + {\bf r}_2,
\end{displaymath}">|; 

$key = q/{mitDelta}x_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img821.png"
 ALT="${\mit\Delta}x_w$">|; 

$key = q/v_0=v_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img578.png"
 ALT="$v_0=v_1$">|; 

$key = q/3.33times10^5;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1934.png"
 ALT="$3.33\times 10^5$">|; 

$key = q/U(O)=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img664.png"
 ALT="$U(O)=0$">|; 

$key = q/{R_oplus};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1941.png"
 ALT="${R_\oplus}$">|; 

$key = q/dot{~};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img797.png"
 ALT="$\dot{&nbsp;}$">|; 

$key = q/{figure}{epsfysize{=1.8in{centerline{epsffile{equil.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="208" BORDER="0"
 SRC="|."$dir".q|img709.png"
 ALT="\begin{figure}
\epsfysize =1.8in
\centerline{\epsffile{equil.eps}}
\end{figure}">|; 

$key = q/sin0^circ=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1276.png"
 ALT="$\sin 0^\circ=0$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{vadd.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{vadd.eps}}
\end{figure}">|; 

$key = q/p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$p$">|; 

$key = q/K_r=(1slash2),I,omega^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1431.png"
 ALT="$K_r = (1/2) I \omega^2$">|; 

$key = q/m=5,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img539.png"
 ALT="$m= 5 {\rm kg}$">|; 

$key = q/{displaymath}P=frac{1.57times10^5}{746}=210.5,{{rm{hp}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1505.png"
 ALT="\begin{displaymath}
P = \frac{1.57\times 10^5}{746} = 210.5 {\rm hp}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{major.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{major.eps}}
\end{figure*}">|; 

$key = q/{displaymath}6,y,(2,y-1)=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1022.png"
 ALT="\begin{displaymath}
6 y (2 y-1)=0,
\end{displaymath}">|; 

$key = q/y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$y$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{planet.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="414" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1961.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{planet.eps}}
\end{figure}">|; 

$key = q/m=4,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1878.png"
 ALT="$m=4 {\rm kg}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{multi.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="488" HEIGHT="340" BORDER="0"
 SRC="|."$dir".q|img835.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{multi.eps}}
\end{figure}">|; 

$key = q/{displaymath}v=sigma,omega=omega,r,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1279.png"
 ALT="\begin{displaymath}
v=\sigma \omega = \omega r \sin\theta,
\end{displaymath}">|; 

$key = q/5,r,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1172.png"
 ALT="$5 r g$">|; 

$key = q/sigma_i=|{{bf{k}times{{bf{r}_i|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1312.png"
 ALT="$\sigma_i=\vert{\bf k}\times {\bf r}_i\vert$">|; 

$key = q/10^{12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$10^{12}$">|; 

$key = q/costheta=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1153.png"
 ALT="$\cos\theta=-1$">|; 

$key = q/{displaymath}W=int_{x_A}^{x_B}m,frac{d^2x}{dt^2},dx=int_{t_A}^{t_B}m,frac{d^2x}{t[frac{m}{2}!left(frac{dx}{dt}right)^2right],dt,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img634.png"
 ALT="\begin{displaymath}
W = \int_{x_A}^{x_B} m \frac{d^2 x}{dt^2} dx = \int_{t_A}^...
...}\!\left[\frac{m}{2}\!\left(\frac{dx}{dt}\right)^2\right] dt,
\end{displaymath}">|; 

$key = q/displaystylefrac{g,sintheta}{1+IslashM,b^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1448.png"
 ALT="$\displaystyle \frac{g \sin\theta}{1 + I/M b^2},$">|; 

$key = q/omega_0=omega(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1251.png"
 ALT="$\omega_0=\omega(t=0)$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{work3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="310" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img632.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{work3.eps}}
\end{figure}">|; 

$key = q/v_{f1}=v_{f2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img913.png"
 ALT="$v_{f1}=v_{f2}$">|; 

$key = q/displaystylex(t=0)=a,cosphi,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1795.png"
 ALT="$\displaystyle x(t=0) = a \cos\phi,$">|; 

$key = q/M=1200,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img967.png"
 ALT="$M=1200 {\rm kg}$">|; 

$key = q/{displaymath}omega=sqrt{frac{M,g,d}{I}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1831.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{M g d}{I}}.
\end{displaymath}">|; 

$key = q/{displaymath}r=frac{1}{u},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1985.png"
 ALT="\begin{displaymath}
r = \frac{1}{u},
\end{displaymath}">|; 

$key = q/{{bf{e}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="${\bf e}_\theta $">|; 

$key = q/{displaymath}omega=alpha,t=12.59times2.3=28.96,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1466.png"
 ALT="\begin{displaymath}
\omega = \alpha t = 12.59\times 2.3 =28.96 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/displaystyle{{bf{v}_0+{{bf{a},t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="$\displaystyle {\bf v}_0 + {\bf a} t,$">|; 

$key = q/displaystylem_2,ddot{x}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img795.png"
 ALT="$\displaystyle m_2 \ddot{x}_2$">|; 

$key = q/300,{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$300 {\rm km/h}$">|; 

$key = q/{displaymath}I,dot{omega}=-f,b,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1511.png"
 ALT="\begin{displaymath}
I \dot{\omega} = - f b,
\end{displaymath}">|; 

$key = q/-{{bf{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$-{\bf c}$">|; 

$key = q/4,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1842.png"
 ALT="$4 {\rm kg}$">|; 

$key = q/63.6,{{rm{mislashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$63.6 {\rm mi/h}$">|; 

$key = q/m,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img590.png"
 ALT="$m g h$">|; 

$key = q/W=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$W=0$">|; 

$key = q/deltaphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1239.png"
 ALT="$\delta\phi$">|; 

$key = q/{v''}^2slashl;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1232.png"
 ALT="${v''}^2/l$">|; 

$key = q/+omega^2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1790.png"
 ALT="$+\omega^2 a$">|; 

$key = q/{displaymath}frac{dv}{u}=-frac{dm}{M+m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img870.png"
 ALT="\begin{displaymath}
\frac{dv}{u} =- \frac{dm}{M+m}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}_{cm}=frac{1}{V}int!int!int{{bf{r},dV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1288.png"
 ALT="\begin{displaymath}
{\bf r}_{cm} = \frac{1}{V} \int\!\int\!\int {\bf r} dV,
\end{displaymath}">|; 

$key = q/{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="${\bf b}$">|; 

$key = q/{displaymath}f=frac{K_i-K_f}{K_i}=frac{m_1,v_{i1}^{,2}-(m_1+m_2),v_f^{,2}}{m_1,v_{i1}^{,2}}=frac{m_2}{m_1+m_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img949.png"
 ALT="\begin{displaymath}
f = \frac{K_i-K_f}{K_i} = \frac{m_1 v_{i1}^{ 2} - (m_1+m_2) v_f^{ 2}}{m_1 v_{i1}^{ 2}}
= \frac{m_2}{m_1+m_2}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=1.75in{centerline{epsffile{bank.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="201" BORDER="0"
 SRC="|."$dir".q|img1189.png"
 ALT="\begin{figure*}
\epsfysize =1.75in
\centerline{\epsffile{bank.eps}}
\end{figure*}">|; 

$key = q/{displaymath}I=M,frac{int!int!intsigma^2,dV}{int!int!int,dV}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1317.png"
 ALT="\begin{displaymath}
I = M \frac{\int\!\int\!\int \sigma^2 dV}
{\int\!\int\!\int  dV}.
\end{displaymath}">|; 

$key = q/{displaymath}phi=180^circ-tan^{-1}left(frac{v_{g,y}}{|v_{g,x}|}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="\begin{displaymath}
\phi = 180^\circ - \tan^{-1} \left(\frac{v_{g y}}{\vert v_{g x}\vert}\right),
\end{displaymath}">|; 

$key = q/theta=45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$\theta=45^\circ$">|; 

$key = q/displaystyle+frac{m_1}{m_1+m_2},(v_{f2}-v_{f1}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img923.png"
 ALT="$\displaystyle + \frac{m_1}{m_1+m_2} (v_{f2}-v_{f1}).$">|; 

$key = q/m=20,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img993.png"
 ALT="$m=20 {\rm kg}$">|; 

$key = q/X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$X$">|; 

$key = q/displaystylez_1+z_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$\displaystyle z_1 + z_2.$">|; 

$key = q/{displaymath}m_1,v_{i1}=m_1,v_{f1},costheta+m_2,v_{f2},costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1030.png"
 ALT="\begin{displaymath}
m_1 v_{i1} = m_1 v_{f1} \cos\theta + m_2 v_{f2} \cos\theta.
\end{displaymath}">|; 

$key = q/m_1=36,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1716.png"
 ALT="$m_1=36 {\rm kg}$">|; 

$key = q/M_ast;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2043.png"
 ALT="$M_\ast$">|; 

$key = q/{displaymath}omega=30times2,pislash60=3.142,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1469.png"
 ALT="\begin{displaymath}
\omega = 30\times 2 \pi/60 = 3.142 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/displaystyle0.25,cos(phi),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1857.png"
 ALT="$\displaystyle 0.25 \cos(\phi),$">|; 

$key = q/{displaymath}M,ddot{x}_{cm}=F,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img804.png"
 ALT="\begin{displaymath}
M \ddot{x}_{cm} = F,
\end{displaymath}">|; 

$key = q/dm,dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img868.png"
 ALT="$dm dv$">|; 

$key = q/lslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1639.png"
 ALT="$l/2$">|; 

$key = q/displaystylefrac{dv_y}{dt}=frac{d^2y}{dt^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$\displaystyle \frac{dv_y}{dt}=\frac{d^2 y}{dt^2},$">|; 

$key = q/omega=sqrt{gslashl};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1891.png"
 ALT="$\omega = \sqrt{g/l}$">|; 

$key = q/displaystylefrac{dv_y}{dt}=-cost,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$\displaystyle \frac{dv_y}{dt}=- \cos t,$">|; 

$key = q/m=60,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1205.png"
 ALT="$m=60 {\rm kg}$">|; 

$key = q/9.81,{{rm{m,s^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$9.81 {\rm m s^{-2}}$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{vcircle.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1144.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{vcircle.eps}}
\end{figure}">|; 

$key = q/{displaymath}z=z_0+v_0,t-frac{1}{2},g,t^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="\begin{displaymath}
z = z_0 + v_0 t - \frac{1}{2} g t^2,
\end{displaymath}">|; 

$key = q/1,{{rm{da}=86,400,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$1 {\rm da} = 86,400 {\rm s}$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{col.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="519" HEIGHT="172" BORDER="0"
 SRC="|."$dir".q|img905.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{col.eps}}
\end{figure}">|; 

$key = q/M=m_1+m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img805.png"
 ALT="$M=m_1+m_2$">|; 

$key = q/{{bf{a}slashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img529.png"
 ALT="${\bf a}/m$">|; 

$key = q/displaystylex_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1645.png"
 ALT="$\displaystyle x_1$">|; 

$key = q/1,650,763.73;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$1,650,763.73$">|; 

$key = q/{displaymath}E=K+U=K+frac{1}{2},k,x^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img685.png"
 ALT="\begin{displaymath}
E = K + U = K + \frac{1}{2} k x^2.
\end{displaymath}">|; 

$key = q/displaystyle-2,mu,g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1515.png"
 ALT="$\displaystyle - 2 \mu g.$">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}_3=mbox{{boldmath{{omega{}_1+mbox{{boldmath{{omega{}_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1261.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}_3 = \mbox{\boldmath$\omega$}_1 + \mbox{\boldmath$\omega$}_2,
\end{displaymath}">|; 

$key = q/{displaymath}dW=mbox{{boldmath{{tau{}!cdot!dmbox{{boldmath{{phi{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1389.png"
 ALT="\begin{displaymath}
dW = \mbox{\boldmath$\tau$}\!\cdot\!d\mbox{\boldmath$\phi$}.
\end{displaymath}">|; 

$key = q/O;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$O$">|; 

$key = q/I_1slashI_2=8;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1615.png"
 ALT="$I_1/I_2 = 8$">|; 

$key = q/{displaymath}r^2,dot{theta}=h,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1971.png"
 ALT="\begin{displaymath}
r^2 \dot{\theta} = h,
\end{displaymath}">|; 

$key = q/-T_2,cos30^circ=-sqrt{3},T_2slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="$-T_2 \cos 30^\circ = -\sqrt{3} T_2/2$">|; 

$key = q/e=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2003.png"
 ALT="$e=0$">|; 

$key = q/{displaymath}tantheta=frac{omega^2,r}{g}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1092.png"
 ALT="\begin{displaymath}
\tan\theta = \frac{\omega^2 r}{g}.
\end{displaymath}">|; 

$key = q/{displaymath}I=sum_{i=1,N}m_i,|{{bf{k}times{{bf{r}_i|^2=sum_{i=1,N}m_i,sigma_i^{,2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1311.png"
 ALT="\begin{displaymath}
I = \sum_{i=1,N} m_i \vert{\bf k}\times {\bf r}_i\vert^2= \sum_{i=1,N} m_i \sigma_i^{ 2},
\end{displaymath}">|; 

$key = q/displaystyleI_x,omega_x,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1553.png"
 ALT="$\displaystyle I_x \omega_x,$">|; 

$key = q/{displaymath}R=m,r,omega^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1212.png"
 ALT="\begin{displaymath}
R = m r \omega^2.
\end{displaymath}">|; 

$key = q/mu,m,g,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img512.png"
 ALT="$\mu m g \cos\theta$">|; 

$key = q/g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$g$">|; 

$key = q/m=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img876.png"
 ALT="$m=0$">|; 

$key = q/{displaymath}omega=2pi,f=314.16,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1062.png"
 ALT="\begin{displaymath}
\omega = 2\pi f = 314.16 {\rm rad. / s}.
\end{displaymath}">|; 

$key = q/{displaymath}u=A,cos(theta-theta_0)+frac{G,M_odot}{h^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1991.png"
 ALT="\begin{displaymath}
u = A \cos(\theta-\theta_0) + \frac{G M_\odot}{h^2},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{L}=I_z,omega_zhat{{bf{z}=I_z,mbox{{boldmath{{omega{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1565.png"
 ALT="\begin{displaymath}
{\bf L} = I_z \omega_z\hat{\bf z} = I_z \mbox{\boldmath$\omega$}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{p}_2=(0,,m,u)=(0,,20times3)=(0,,60),{{rm{N,s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img997.png"
 ALT="\begin{displaymath}
{\bf p}_2 = (0, m u) = (0, 20\times 3) = (0,  60)  {\rm N s}.
\end{displaymath}">|; 

$key = q/{displaymath}v-b,omega=-(b,omega_0-3,mu,g,t).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1520.png"
 ALT="\begin{displaymath}
v - b \omega = -(b \omega_0 - 3 \mu g t).
\end{displaymath}">|; 

$key = q/{displaymath}b=frac{l}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1421.png"
 ALT="\begin{displaymath}
b = \frac{l}{2}.
\end{displaymath}">|; 

$key = q/displaystylef;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1449.png"
 ALT="$\displaystyle f$">|; 

$key = q/AC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$AC$">|; 

$key = q/underline{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$\underline{r}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{work.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img616.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{work.eps}}
\end{figure}">|; 

$key = q/theta=180^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1152.png"
 ALT="$\theta =180^\circ$">|; 

$key = q/{figure*}{epsfysize{=2.2in{centerline{epsffile{rodpin.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="253" BORDER="0"
 SRC="|."$dir".q|img1721.png"
 ALT="\begin{figure*}
\epsfysize =2.2in
\centerline{\epsffile{rodpin.eps}}
\end{figure*}">|; 

$key = q/{displaymath}L=m,v,d=(2.3times10^4)times210times(1times10^4)=4.83times10^{10},{{rm{kg,m^2slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="525" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1599.png"
 ALT="\begin{displaymath}
L = m v d = (2.3\times 10^4)\times 210\times (1\times 10^4)= 4.83\times 10^{10} {\rm kg m^2/s}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}K=U(O)-U(R)=-{mitDelta}U.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img666.png"
 ALT="\begin{displaymath}
{\mit\Delta} K = U(O) - U(R) = -{\mit\Delta} U.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}times{{bf{f}=mbox{{boldmath{{tau{},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1534.png"
 ALT="\begin{displaymath}
{\bf r} \times {\bf f} = \mbox{\boldmath$\tau$},
\end{displaymath}">|; 

$key = q/deltambox{{boldmath{{phi{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1256.png"
 ALT="$\delta\mbox{\boldmath$\phi$}$">|; 

$key = q/theta>90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img615.png"
 ALT="$\theta&gt;90^\circ$">|; 

$key = q/{{bf{f}_{ba}=-{{bf{f}_{ab};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="${\bf f}_{ba}=-{\bf f}_{ab}$">|; 

$key = q/10^{1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$10^{1}$">|; 

$key = q/{displaymath}x(16)-x(0)=int_0^{16}v(t),dt.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="\begin{displaymath}
x(16)-x(0) = \int_0^{16} v(t) dt.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{P}}{dt}={{bf{F}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img854.png"
 ALT="\begin{displaymath}
\frac{d{\bf P}}{dt} = {\bf F}.
\end{displaymath}">|; 

$key = q/{displaymath}tau=T,b.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1485.png"
 ALT="\begin{displaymath}
\tau = T b.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{rodp.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1651.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{rodp.eps}}
\end{figure}">|; 

$key = q/{displaymath}left[frac{v}{u}right]_{v=0}^{v=v}=-left[ln(M+m)right]_{m=m_p}^{m=m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img874.png"
 ALT="\begin{displaymath}
\left[\frac{v}{u}\right]_{v=0}^{v=v}= -\left[\ln(M+m)\right]_{m=m_p}^{m=m},
\end{displaymath}">|; 

$key = q/{displaymath}I'=frac{3times1.2^2}{3}=1.44,{{rm{kg,m^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="218" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1477.png"
 ALT="\begin{displaymath}
I' = \frac{3\times 1.2^2}{3} = 1.44 {\rm kg m^2}.
\end{displaymath}">|; 

$key = q/{displaymath}I,ddot{theta}=-M,g,d,theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1830.png"
 ALT="\begin{displaymath}
I \ddot{\theta} = - M g d \theta.
\end{displaymath}">|; 

$key = q/|lambda|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$\vert\lambda\vert$">|; 

$key = q/v_0=u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$v_0=u$">|; 

$key = q/x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img785.png"
 ALT="$x_1$">|; 

$key = q/{displaymath}{{bf{v}=dot{r},{{bf{e}_r+r,dot{{bf{e}_r,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1107.png"
 ALT="\begin{displaymath}
{\bf v}=\dot{r} {\bf e}_r + r \dot{\bf e}_r,
\end{displaymath}">|; 

$key = q/r,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1145.png"
 ALT="$r \cos\theta$">|; 

$key = q/{mitDelta}p=m,u_f-m,(-u_i)=m,(u_f+u_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img895.png"
 ALT="${\mit\Delta} p = m u_f -m (-u_i)=
m (u_f+u_i)$">|; 

$key = q/{displaymath}{{bf{v}(t)=frac{d{{bf{r}}{dt}={{bf{v}_0+{{bf{a},t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="\begin{displaymath}
{\bf v}(t) = \frac{d{\bf r}}{dt}= {\bf v}_0 + {\bf a} t.
\end{displaymath}">|; 

$key = q/{{bf{c}={{bf{a}+{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="${\bf c} = {\bf a} + {\bf b}$">|; 

$key = q/{displaymath}t=frac{2,u}{g}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="\begin{displaymath}
t = \frac{2 u}{g}.
\end{displaymath}">|; 

$key = q/lambda=mu=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$\lambda=\mu =1/2$">|; 

$key = q/31.45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="$31.45^\circ$">|; 

$key = q/{displaymath}I=int!int!intrho,sigma^2,dV,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1314.png"
 ALT="\begin{displaymath}
I = \int\!\int\!\int \rho \sigma^2 dV,
\end{displaymath}">|; 

$key = q/-{mitDelta}P=M,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1429.png"
 ALT="$-{\mit\Delta}P = M g h$">|; 

$key = q/{displaymath}I=frac{2}{5},M,a^2=0.4times5times(0.2)^2=0.08,{{rm{kg,m^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1604.png"
 ALT="\begin{displaymath}
I = \frac{2}{5} M a^2 = 0.4\times 5\times(0.2)^2 = 0.08 {\rm kg m^2}.
\end{displaymath}">|; 

$key = q/g=9.81,{{rm{m,s^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$g=9.81 {\rm m s^{-2}}$">|; 

$key = q/{displaymath}tau=-M,g,d,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1828.png"
 ALT="\begin{displaymath}
\tau = - M g d \sin\theta.
\end{displaymath}">|; 

$key = q/+a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1783.png"
 ALT="$+a$">|; 

$key = q/I=(1slash2),M,r^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1896.png"
 ALT="$I= (1/2) M r^2$">|; 

$key = q/m=Wslashg;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img548.png"
 ALT="$m=W/g$">|; 

$key = q/displaystylefrac{g}{1+Mslash2,m}=frac{9.81}{1+6.4slash2times2.6}=4.40,{{rm{mslashs^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1489.png"
 ALT="$\displaystyle \frac{g}{1+M/2 m} = \frac{9.81}{1+6.4/2\times 2.6}=4.40 {\rm m/s^2},$">|; 

$key = q/s>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$s&gt;0$">|; 

$key = q/v=dxslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$v=dx/dt$">|; 

$key = q/{displaymath}-Y_1=Y_2=Y_3=Y,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1755.png"
 ALT="\begin{displaymath}
-Y_1 = Y_2= Y_3 = Y,
\end{displaymath}">|; 

$key = q/theta=pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1999.png"
 ALT="$\theta=\pi$">|; 

$key = q/pi,a,b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2007.png"
 ALT="$\pi a b$">|; 

$key = q/{displaymath}R=m,g,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img766.png"
 ALT="\begin{displaymath}
R = m g \cos\theta.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{equant.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="514" HEIGHT="342" BORDER="0"
 SRC="|."$dir".q|img1900.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{equant.eps}}
\end{figure}">|; 

$key = q/{displaymath}E=K+U={{rm{constant}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img583.png"
 ALT="\begin{displaymath}
E = K+ U= {\rm constant}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{p}=m,{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="\begin{displaymath}
{\bf p} = m {\bf v}.
\end{displaymath}">|; 

$key = q/u(t)equivu(theta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1986.png"
 ALT="$u(t)\equiv u(\theta)$">|; 

$key = q/{displaymath}{{bf{r}(t)={{bf{r}_0+{{bf{v},t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="\begin{displaymath}
{\bf r}(t) = {\bf r}_0 + {\bf v} t,
\end{displaymath}">|; 

$key = q/{displaymath}g=frac{G,M}{R^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2017.png"
 ALT="\begin{displaymath}
g = \frac{G M}{R^2}.
\end{displaymath}">|; 

$key = q/{displaymath}v=C,p^x,rho^y,V^z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="\begin{displaymath}
v = C p^x \rho^y V^z,
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{stunt.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1215.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{stunt.eps}}
\end{figure*}">|; 

$key = q/rho({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1284.png"
 ALT="$\rho({\bf r})$">|; 

$key = q/{displaymath}Y_2-Y_3=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1754.png"
 ALT="\begin{displaymath}
Y_2-Y_3 = 0.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{princ.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="490" HEIGHT="282" BORDER="0"
 SRC="|."$dir".q|img1566.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{princ.eps}}
\end{figure}">|; 

$key = q/y=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$y=1/2$">|; 

$key = q/{displaymath}T,costheta=m,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1088.png"
 ALT="\begin{displaymath}
T \cos\theta = m g.
\end{displaymath}">|; 

$key = q/x-x_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img704.png"
 ALT="$x-x_0&gt;0$">|; 

$key = q/{{bf{b}=(b_x,b_y,b_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1268.png"
 ALT="${\bf b}=(b_x,b_y,b_z)$">|; 

$key = q/{displaymath}S-f=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1664.png"
 ALT="\begin{displaymath}
S - f = 0.
\end{displaymath}">|; 

$key = q/l_2=0.7,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1706.png"
 ALT="$l_2=0.7 {\rm m}$">|; 

$key = q/m_1,v_{i1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img954.png"
 ALT="$m_1 v_{i1}$">|; 

$key = q/{displaymath}m,g,lslash2-X_3,l=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1759.png"
 ALT="\begin{displaymath}
m g l/2 - X_3 l =0,
\end{displaymath}">|; 

$key = q/K_t=(1slash2),M,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1430.png"
 ALT="$K_t = (1/2) M v^2$">|; 

$key = q/f=-f';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img724.png"
 ALT="$f=-f'$">|; 

$key = q/60,{{rm{mislashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="$60 {\rm mi/h}$">|; 

$key = q/gpropto1slashr^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2023.png"
 ALT="$g\propto 1/r^2$">|; 

$key = q/m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$m$">|; 

$key = q/{displaymath}{{bf{f}_{ij}=-{{bf{f}_{ji}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img839.png"
 ALT="\begin{displaymath}
{\bf f}_{ij} = - {\bf f}_{ji}.
\end{displaymath}">|; 

$key = q/displaystylefrac{cosphi}{sin(theta+phi)},M,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1656.png"
 ALT="$\displaystyle \frac{\cos\phi}{\sin(\theta+\phi)} M g,$">|; 

$key = q/{displaymath}r=r_0,frac{1+e}{1+e,costheta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1994.png"
 ALT="\begin{displaymath}
r = r_0 \frac{1+e}{1+ e \cos\theta},
\end{displaymath}">|; 

$key = q/0.25,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1849.png"
 ALT="$0.25 {\rm m}$">|; 

$key = q/v_0,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$v_0 \sin\theta$">|; 

$key = q/{displaymath}l=l'+L,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1591.png"
 ALT="\begin{displaymath}
l = l' + L,
\end{displaymath}">|; 

$key = q/{{bf{f}_{ba}neq-{{bf{f}_{ab};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="${\bf f}_{ba}\neq -{\bf f}_{ab}$">|; 

$key = q/theta=30^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$\theta = 30^\circ$">|; 

$key = q/{mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="${\mit\Delta} x$">|; 

$key = q/s=h_2-h_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img577.png"
 ALT="$s=h_2-h_1$">|; 

$key = q/displaystyle0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$\displaystyle 0$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{bat.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img1409.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{bat.eps}}
\end{figure}">|; 

$key = q/T_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1155.png"
 ALT="$T_0&gt;0$">|; 

$key = q/{mitDelta}omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1414.png"
 ALT="${\mit\Delta}\omega$">|; 

$key = q/{displaymath}v^2=frac{2,g,h}{1+IslashM,b^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1433.png"
 ALT="\begin{displaymath}
v^2 = \frac{2 g h}{1+ I/M b^2}.
\end{displaymath}">|; 

$key = q/{displaymath}omega=sqrt{frac{k}{I}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1815.png"
 ALT="\begin{displaymath}
\omega =\sqrt{\frac{k}{I}}.
\end{displaymath}">|; 

$key = q/1,{{rm{hp}=746,{{rm{W};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1502.png"
 ALT="$1 {\rm hp} = 746 {\rm W}$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{rel.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="169" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{rel.eps}}
\end{figure}">|; 

$key = q/{displaymath}a=frac{F,cos25^circ-m,g,sin25^circ}{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img544.png"
 ALT="\begin{displaymath}
a = \frac{F \cos 25^\circ - m g \sin 25^\circ}{m}.
\end{displaymath}">|; 

$key = q/[L];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$[L]$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$T$">|; 

$key = q/50,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img530.png"
 ALT="$50 {\rm N}$">|; 

$key = q/{displaymath}u=frac{M+m}{m},v=frac{(0.012+5.2)times1.7}{0.012}=738.4,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="399" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1007.png"
 ALT="\begin{displaymath}
u = \frac{M+m}{m} v = \frac{(0.012+5.2)\times 1.7}{0.012} = 738.4 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{displaymath}m,dot{v}=m,g-T.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1484.png"
 ALT="\begin{displaymath}
m  \dot{v} = m g-T.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{l}}{dt}=mbox{{boldmath{{tau{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1535.png"
 ALT="\begin{displaymath}
\frac{d{\bf l}}{dt} = \mbox{\boldmath$\tau$}.
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{{mitDelta}v}{{mitDelta}t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="\begin{displaymath}
a = \frac{{\mit\Delta}v}{{\mit\Delta} t},
\end{displaymath}">|; 

$key = q/{displaymath}2,I_x=I_z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1331.png"
 ALT="\begin{displaymath}
2 I_x = I_z,
\end{displaymath}">|; 

$key = q/{{bf{v}=d{{bf{r}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img716.png"
 ALT="${\bf v} = d{\bf r}/dt$">|; 

$key = q/{displaymath}W=f,h=m,g,h=30times150times9.81=4.415times10^4,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img725.png"
 ALT="\begin{displaymath}
W = f h = m g h = 30\times 150\times 9.81 = 4.415\times 10^4 {\rm J}.
\end{displaymath}">|; 

$key = q/m_2=6,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1013.png"
 ALT="$m_2=6 {\rm kg}$">|; 

$key = q/sigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1243.png"
 ALT="$\sigma$">|; 

$key = q/{displaymath}a_theta=r,dot{omega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1143.png"
 ALT="\begin{displaymath}
a_\theta = r \dot{\omega}.
\end{displaymath}">|; 

$key = q/s=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$s=0$">|; 

$key = q/{displaymath}a=sqrt{x_0^{,2}+(v_0slashomega)^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1800.png"
 ALT="\begin{displaymath}
a = \sqrt{x_0^{ 2} + (v_0/\omega)^2},
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}K=W={{bf{f}!cdot{{bf{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img610.png"
 ALT="\begin{displaymath}
{\mit\Delta} K = W = {\bf f}\!\cdot {\bf s}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}=a_r,{{bf{e}_r+a_theta,{{bf{e}_theta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1962.png"
 ALT="\begin{displaymath}
{\bf a} = a_r {\bf e}_r + a_\theta {\bf e}_\theta,
\end{displaymath}">|; 

$key = q/x_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img627.png"
 ALT="$x_A$">|; 

$key = q/{displaymath}W=frac{1}{2},I,omega^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1470.png"
 ALT="\begin{displaymath}
W = \frac{1}{2} I \omega^2,
\end{displaymath}">|; 

$key = q/{displaymath}theta=tan^{-1}left(frac{(40times1000slash3600)^2}{60times9.81}right)=11.8^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1196.png"
 ALT="\begin{displaymath}
\theta = \tan^{-1}\left(\frac{(40\times 1000/3600)^2}{60\times9.81}\right)= 11.8^\circ.
\end{displaymath}">|; 

$key = q/r_1=35,{{rm{AU};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2037.png"
 ALT="$r_1=35 {\rm AU}$">|; 

$key = q/displaystyle-x-3y+3z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$\displaystyle -x -3 y+ 3z,$">|; 

$key = q/U(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img690.png"
 ALT="$U(x)$">|; 

$key = q/dot{theta}=h,u^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1988.png"
 ALT="$\dot{\theta} = h u^2$">|; 

$key = q/{{bf{s}={{bf{r}-{{bf{r}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="${\bf s} = {\bf r}-{\bf r}_0$">|; 

$key = q/y=-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$y=-1/2$">|; 

$key = q/displaystylev_{f1}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img920.png"
 ALT="$\displaystyle v_{f1}'$">|; 

$key = q/{displaymath}{{bf{r}_{cm}=frac{1}{M}sum_{i=1,N}m_i,{{bf{r}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1282.png"
 ALT="\begin{displaymath}
{\bf r}_{cm} = \frac{1}{M} \sum_{i=1,N} m_i {\bf r}_i.
\end{displaymath}">|; 

$key = q/{displaymath}omega=sqrt{frac{M,g,d}{I'}}=sqrt{frac{3times9.81times0.25}{1.1475}}=2.532,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img1898.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{M g d}{I'}} = \sqrt{\frac{3\times 9.81\times 0.25}{1.1475}} = 2.532 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/{displaymath}K_f=frac{1}{2},m_1,v_{f1}^{,2}+frac{1}{2},m_2,v_{f2}^{,2}=0.5times52.5981^2+0.5times2.5times5.1962^2=50.63,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="583" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1035.png"
 ALT="\begin{displaymath}
K_f= \frac{1}{2} m_1 v_{f1}^{ 2}+ \frac{1}{2} m_2 v_{f2...
...imes 2.5981^2 + 0.5\times 2.5\times 5.1962^2 = 50.63 {\rm J}.
\end{displaymath}">|; 

$key = q/T=4.3times10^7,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2040.png"
 ALT="$T= 4.3\times 10^7 {\rm s}$">|; 

$key = q/displaystyler_{{rm{geo};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1957.png"
 ALT="$\displaystyle r_{\rm geo}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{string.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="408" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{string.eps}}
\end{figure}">|; 

$key = q/f';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img720.png"
 ALT="$f'$">|; 

$key = q/I';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1335.png"
 ALT="$I'$">|; 

$key = q/displaystyle+frac{m_1}{m_1+m_2},(v_{i2}-v_{i1}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img919.png"
 ALT="$\displaystyle + \frac{m_1}{m_1+m_2} (v_{i2}-v_{i1}),$">|; 

$key = q/{displaymath}m_i,ddot{{bf{r}_i=sum_{j=1,N}^{jneqi}!{{bf{f}_{ij}+{{bf{F}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img841.png"
 ALT="\begin{displaymath}
m_i \ddot{\bf r}_i = \sum_{j=1,N}^{j\neq i}\! {\bf f}_{ij} + {\bf F}_i.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{elev.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{elev.eps}}
\end{figure}">|; 

$key = q/x^2+y^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1341.png"
 ALT="$x^2+y^2$">|; 

$key = q/T_4=M,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img532.png"
 ALT="$T_4=M g$">|; 

$key = q/l=b,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1370.png"
 ALT="$l=b \sin\theta$">|; 

$key = q/{{bf{a}=(0,-g);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img600.png"
 ALT="${\bf a}=(0,-g)$">|; 

$key = q/{displaymath}omega=sqrt{frac{g,}{mu,r}}=sqrt{frac{9.81}{0.25times7}}=2.37,{{rm{radslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img1213.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{g }{\mu r}} = \sqrt{\frac{9.81}{0.25\times 7}} = 2.37 {\rm rad/s}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}times{{bf{b}=(a_y,b_z-a_z,b_y,a_z,b_x-a_x,b_z,a_x,b_y-a_y,b_x).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1272.png"
 ALT="\begin{displaymath}
{\bf a}\times{\bf b} = (a_y  b_z-a_z  b_y, a_z  b_x - a_x  b_z, a_x  b_y - a_y  b_x).
\end{displaymath}">|; 

$key = q/[M],([L]slash[T])^2=[M][L^2]slash[T^2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$[M] ([L]/[T])^2= [M][L^2]/[T^2]$">|; 

$key = q/2,T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img550.png"
 ALT="$2 T$">|; 

$key = q/displaystyleY_1=Y_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1695.png"
 ALT="$\displaystyle Y_1=Y_3$">|; 

$key = q/{displaymath}l=m,v,r=m,omega,r^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1536.png"
 ALT="\begin{displaymath}
l = m v r = m \omega r^2.
\end{displaymath}">|; 

$key = q/CP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1902.png"
 ALT="$CP$">|; 

$key = q/{displaymath}x_0=sqrt{frac{2,E}{k}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img687.png"
 ALT="\begin{displaymath}
x_0 = \sqrt{\frac{2 E}{k}}.
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=frac{dmbox{{boldmath{{phi{}}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1388.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}= \frac{d\mbox{\boldmath$\phi$}}{dt}.
\end{displaymath}">|; 

$key = q/D;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$D$">|; 

$key = q/{displaymath}tau=M,g,d=M,g,r,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1635.png"
 ALT="\begin{displaymath}
\tau = M g  d = M g r \sin\theta.
\end{displaymath}">|; 

$key = q/{displaymath}L_k=I_k,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1545.png"
 ALT="\begin{displaymath}
L_k = I_k \omega.
\end{displaymath}">|; 

$key = q/z=x+i,y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1112.png"
 ALT="$z = x + i y$">|; 

$key = q/{displaymath}l'=m,d^2,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1588.png"
 ALT="\begin{displaymath}
l' = m d^2 \omega,
\end{displaymath}">|; 

$key = q/omega^2,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1089.png"
 ALT="$\omega^2 r$">|; 

$key = q/(v_{g,x},v_{g,y});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="$(v_{g x}, v_{g y})$">|; 

$key = q/t=2.3,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1463.png"
 ALT="$t=2.3 {\rm s}$">|; 

$key = q/{displaymath}W_2=int_{ArightarrowB:{{rm{path}2}{{bf{f}!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img648.png"
 ALT="\begin{displaymath}
W_2 = \int_{A\rightarrow B:{\rm path} 2} {\bf f}\!\cdot\!d{\bf r}.
\end{displaymath}">|; 

$key = q/displaystylev_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1131.png"
 ALT="$\displaystyle v_\theta$">|; 

$key = q/CSD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1906.png"
 ALT="$CSD$">|; 

$key = q/10^{-6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$10^{-6}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{cofg.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1634.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{cofg.eps}}
\end{figure}">|; 

$key = q/u_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2034.png"
 ALT="$u_0$">|; 

$key = q/X_1=m,g-X=73.58,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1761.png"
 ALT="$X_1=m g-X = 73.58 {\rm N}$">|; 

$key = q/omega=omega_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1516.png"
 ALT="$\omega=\omega_0$">|; 

$key = q/{displaymath}frac{1}{2},m,v^2=m,g,h,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img587.png"
 ALT="\begin{displaymath}
\frac{1}{2} m v^2 = m g h,
\end{displaymath}">|; 

$key = q/{displaymath}K=int^ttau,dt.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1412.png"
 ALT="\begin{displaymath}
K = \int^t \tau dt.
\end{displaymath}">|; 

$key = q/m,v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img987.png"
 ALT="$m v$">|; 

$key = q/{displaymath}theta=tan^{-1}left(frac{x_{cm}}{y_{cm}}right)=tan^{-1}0.152=8.65^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1712.png"
 ALT="\begin{displaymath}
\theta = \tan^{-1}\left(\frac{x_{cm}}{y_{cm}}\right) = \tan^{-1}0.152= 8.65^\circ.
\end{displaymath}">|; 

$key = q/{displaymath}L=I,omega=2,m,d^2,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1583.png"
 ALT="\begin{displaymath}
L = I \omega = 2 m d^2 \omega.
\end{displaymath}">|; 

$key = q/theta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img952.png"
 ALT="$\theta_1$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{constv.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{constv.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{a}+lambda,({{bf{b}-{{bf{a})={{bf{b}+{{bf{a}-mu,({{bf{a}+{{bf{b}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="\begin{displaymath}
{\bf a} + \lambda ({\bf b} - {\bf a}) = {\bf b} + {\bf a} - \mu  ({\bf a} + {\bf b}).
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=1in{centerline{epsffile{carridge.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="404" HEIGHT="113" BORDER="0"
 SRC="|."$dir".q|img971.png"
 ALT="\begin{figure*}
\epsfysize =1in
\centerline{\epsffile{carridge.eps}}
\end{figure*}">|; 

$key = q/{displaymath}{{bf{L}=sum_{i=1,N}m_i,{{bf{r}_itimes{{bf{v}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1538.png"
 ALT="\begin{displaymath}
{\bf L} = \sum_{i=1,N} m_i {\bf r}_i\times {\bf v}_i.
\end{displaymath}">|; 

$key = q/displaystyle2,v_{i1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img945.png"
 ALT="$\displaystyle 2 v_{i1}.$">|; 

$key = q/h=h_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1425.png"
 ALT="$h=h_0$">|; 

$key = q/{mitDelta}x_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img823.png"
 ALT="${\mit\Delta}x_g$">|; 

$key = q/displaystyle-frac{m_2}{m_1+m_2},(v_{f2}-v_{f1}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img921.png"
 ALT="$\displaystyle - \frac{m_2}{m_1+m_2} (v_{f2}-v_{f1}),$">|; 

$key = q/{displaymath}int!int!int,x,dx,dy,dz=int!int!int,y,dx,dy,dz=int!int!int,z,dx,dy,dz=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="486" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1339.png"
 ALT="\begin{displaymath}
\int\!\int\!\int  x dx dy dz = \int\!\int\!\int y dx dy dz=\int\!\int\!\int  z dx dy dz=0,
\end{displaymath}">|; 

$key = q/displaystylev_0^{,2}+2,{{bf{a}!cdot!{{bf{s}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="$\displaystyle v_0^{ 2} + 2 {\bf a}\!\cdot\!{\bf s}.$">|; 

$key = q/{displaymath}v_f=frac{m_1,v_{i1}+m_2,v_{i2}}{m_1+m_2}=v_{cm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img947.png"
 ALT="\begin{displaymath}
v_f = \frac{m_1 v_{i1} + m_2 v_{i2}}{m_1+m_2}= v_{cm}.
\end{displaymath}">|; 

$key = q/D=4sqrt{A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$D=4\sqrt{A}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{bullet.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="394" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1586.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{bullet.eps}}
\end{figure}">|; 

$key = q/10^{9};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$10^{9}$">|; 

$key = q/{displaymath}v=b,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1521.png"
 ALT="\begin{displaymath}
v = b \omega,
\end{displaymath}">|; 

$key = q/M_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1940.png"
 ALT="$M_\oplus$">|; 

$key = q/{displaymath}v=sigma,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1242.png"
 ALT="\begin{displaymath}
v = \sigma \omega,
\end{displaymath}">|; 

$key = q/PC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1827.png"
 ALT="$PC$">|; 

$key = q/{mitDelta}U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img595.png"
 ALT="${\mit\Delta}U$">|; 

$key = q/{displaymath}{mitDelta}K=int_O^R{{bf{f}!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img665.png"
 ALT="\begin{displaymath}
{\mit\Delta} K = \int_O^R {\bf f}\!\cdot\!d{\bf r}.
\end{displaymath}">|; 

$key = q/displaystyle1-mu.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$\displaystyle 1 -\mu.$">|; 

$key = q/tantheta=sinthetaslashcostheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1803.png"
 ALT="$\tan\theta = \sin\theta/\cos\theta$">|; 

$key = q/{displaymath}t=sqrt{frac{2,h}{g}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="\begin{displaymath}
t = \sqrt{\frac{2 h}{g}}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{q66.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1029.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{q66.eps}}
\end{figure*}">|; 

$key = q/{{bf{a}!cdot{{bf{s}={{bf{s}!cdot{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="${\bf a}\!\cdot{\bf s}= {\bf s}\!\cdot{\bf a}$">|; 

$key = q/{displaymath}{{bf{r}=(x,y,z).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="\begin{displaymath}
{\bf r} = (x, y, z).
\end{displaymath}">|; 

$key = q/pislash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1046.png"
 ALT="$\pi/2$">|; 

$key = q/{mitDelta}p=p_f-p_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img892.png"
 ALT="${\mit\Delta} p = p_f-p_i$">|; 

$key = q/1,{{rm{ton}=2,000,{{rm{lb};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$1 {\rm ton} = 2,000 {\rm lb}$">|; 

$key = q/{displaymath}f=mu,R_n,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img504.png"
 ALT="\begin{displaymath}
f = \mu  R_n,
\end{displaymath}">|; 

$key = q/({v'}^2)_0>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1159.png"
 ALT="$({v'}^2)_0 &gt;0$">|; 

$key = q/t_fproptosqrt{h};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$t_f\propto \sqrt{h}$">|; 

$key = q/m_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img561.png"
 ALT="$m_B$">|; 

$key = q/{displaymath}T_2-T_1,sin40^circ=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img533.png"
 ALT="\begin{displaymath}
T_2 - T_1 \sin 40^\circ = 0.
\end{displaymath}">|; 

$key = q/displaystylef_{21}+F_2.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img796.png"
 ALT="$\displaystyle f_{21} + F_2.$">|; 

$key = q/f,b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1442.png"
 ALT="$f b$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\mu$">|; 

$key = q/{figure*}{epsfysize{=2.5in{centerline{epsffile{rod.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img1493.png"
 ALT="\begin{figure*}
\epsfysize =2.5in
\centerline{\epsffile{rod.eps}}
\end{figure*}">|; 

$key = q/T_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="$T_2$">|; 

$key = q/-Y_1=Y_2=Y_3=127.44,{{rm{N}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1765.png"
 ALT="$-Y_1=Y_2=Y_3=127.44 
{\rm N}.$">|; 

$key = q/m,g,sin25^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img543.png"
 ALT="$m g \sin 25^\circ$">|; 

$key = q/I_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1558.png"
 ALT="$I_x$">|; 

$key = q/{displaymath}deltas=r,deltatheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1052.png"
 ALT="\begin{displaymath}
\delta s = r \delta\theta.
\end{displaymath}">|; 

$key = q/v'';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1230.png"
 ALT="$v''$">|; 

$key = q/{displaymath}tau=f,b,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1368.png"
 ALT="\begin{displaymath}
\tau = f b \sin\theta.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}(t)=[x(t),y(t),z(t)].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="\begin{displaymath}
{\bf r}(t) = [x(t), y(t), z(t)].
\end{displaymath}">|; 

$key = q/{displaymath}M,u+m,v=(M+m),w.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img976.png"
 ALT="\begin{displaymath}
M u + m v= (M+m) w.
\end{displaymath}">|; 

$key = q/d=0.25,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1895.png"
 ALT="$d=0.25 {\rm m}$">|; 

$key = q/I=m,(u_f+u_i);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img897.png"
 ALT="$I = m (u_f+u_i)$">|; 

$key = q/{displaymath}theta({{rm{rad.})=frac{2pi}{360^circ},theta(^circ).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1043.png"
 ALT="\begin{displaymath}
\theta({\rm rad.}) = \frac{2\pi}{360^\circ} \theta(^\circ).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}'=frac{d{{bf{v}'}{dt}={{bf{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img527.png"
 ALT="\begin{displaymath}
{\bf a}' = \frac{ d{\bf v}'}{dt} = {\bf a}.
\end{displaymath}">|; 

$key = q/E=2.5,{{rm{J};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1880.png"
 ALT="$E=2.5 {\rm J}$">|; 

$key = q/displaystyle0.75;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1858.png"
 ALT="$\displaystyle 0.75$">|; 

$key = q/{displaymath}l=m,v,d:{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1587.png"
 ALT="\begin{displaymath}
l = m v d:
\end{displaymath}">|; 

$key = q/G,M_astslashr^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2042.png"
 ALT="$G M_\ast/r^2$">|; 

$key = q/{displaymath}I,{mitDelta}omega=-J,h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1418.png"
 ALT="\begin{displaymath}
I {\mit\Delta}\omega = -J h.
\end{displaymath}">|; 

$key = q/R=60times3times0.3048=54.86,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="$R= 60\times 3\times 0.3048 = 54.86 {\rm m}$">|; 

$key = q/{displaymath}W=|{{bf{f}|,,|{{bf{s}|,costheta.,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img611.png"
 ALT="\begin{displaymath}
W = \vert{\bf f}\vert  \vert{\bf s}\vert \cos\theta. 
\end{displaymath}">|; 

$key = q/r_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2001.png"
 ALT="$r_1$">|; 

$key = q/1,{{rm{min}=60,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$1 {\rm min}
= 60 {\rm s}$">|; 

$key = q/W=34,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img553.png"
 ALT="$W=34 {\rm N}$">|; 

$key = q/2,pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1044.png"
 ALT="$2 \pi$">|; 

$key = q/{displaymath}l=m,r,v_theta=m,r^2,dot{theta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1975.png"
 ALT="\begin{displaymath}
l = m r v_\theta = m r^2 \dot{\theta},
\end{displaymath}">|; 

$key = q/{displaymath}M=frac{50}{9.81times0.8391}=6.074,{{rm{kg}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img537.png"
 ALT="\begin{displaymath}
M = \frac{50}{9.81\times 0.8391} = 6.074 {\rm kg}.
\end{displaymath}">|; 

$key = q/{displaymath}E=frac{1}{2},m,v_{{rm{esc}^{,2}-frac{G,M,m}{R}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1951.png"
 ALT="\begin{displaymath}
E = \frac{1}{2} m v_{\rm esc}^{ 2} - \frac{G M m}{R} = 0.
\end{displaymath}">|; 

$key = q/{displaymath}U(x)=-int_0^xf(x'),dx'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img679.png"
 ALT="\begin{displaymath}
U(x) = - \int_0^x f(x') dx'.
\end{displaymath}">|; 

$key = q/v_{g,x}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$v_{g x}&lt; 0$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{cirshm.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img1834.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{cirshm.eps}}
\end{figure}">|; 

$key = q/r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$r$">|; 

$key = q/{displaymath}y=frac{K}{f'}=frac{8.13}{7.36}=1.10,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img776.png"
 ALT="\begin{displaymath}
y = \frac{K}{f'} = \frac{8.13}{7.36} = 1.10 {\rm m}.
\end{displaymath}">|; 

$key = q/{{bf{a}times{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1270.png"
 ALT="${\bf a}\times{\bf b}$">|; 

$key = q/a=0.25,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1852.png"
 ALT="$a=0.25 {\rm m}$">|; 

$key = q/10^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$10^{2}$">|; 

$key = q/{displaymath}I,frac{domega}{dt}=tau,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1411.png"
 ALT="\begin{displaymath}
I \frac{d\omega}{dt} = \tau,
\end{displaymath}">|; 

$key = q/{displaymath}1=x^2+3,y^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1020.png"
 ALT="\begin{displaymath}
1 = x^2 + 3 y^2.
\end{displaymath}">|; 

$key = q/{displaymath}h=r-R_oplus=1.107times10^7-6.378times10^6=4.69times10^6,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="456" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img2028.png"
 ALT="\begin{displaymath}
h = r - R_\oplus = 1.107\times 10^7 - 6.378\times 10^6 = 4.69\times 10^6 {\rm m}.
\end{displaymath}">|; 

$key = q/{displaymath}m_1,v_{i1}+m_2,v_{i2}=m_1,v_{f1}+m_2,v_{f2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img911.png"
 ALT="\begin{displaymath}
m_1 v_{i1} + m_2 v_{i2} = m_1 v_{f1} + m_2 v_{f2}.
\end{displaymath}">|; 

$key = q/displaystylecost,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$\displaystyle \cos t,$">|; 

$key = q/{displaymath}P=mbox{{boldmath{{tau{}!cdot!mbox{{boldmath{{omega{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1387.png"
 ALT="\begin{displaymath}
P = \mbox{\boldmath$\tau$}\!\cdot\!\mbox{\boldmath$\omega$}.
\end{displaymath}">|; 

$key = q/{displaymath}T_1,cos40^circ-T_4=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img534.png"
 ALT="\begin{displaymath}
T_1 \cos 40^\circ - T_4 = 0.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{p}_1=(M,v,,0)=(120times8,,0)=(960,,0),{{rm{N,s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img996.png"
 ALT="\begin{displaymath}
{\bf p}_1 = (M v, 0) = (120\times 8, 0 ) = (960,  0) {\rm N s}.
\end{displaymath}">|; 

$key = q/displaystylefrac{dy}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="$\displaystyle \frac{dy}{dt},$">|; 

$key = q/p_1=m_1,dot{x}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img812.png"
 ALT="$p_1=m_1 \dot{x}_1$">|; 

$key = q/{displaymath}{{bf{r}={{bf{r}_0+{{bf{v},t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="\begin{displaymath}
{\bf r} = {\bf r}_0 + {\bf v} t,
\end{displaymath}">|; 

$key = q/m=40,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1730.png"
 ALT="$m = 40 {\rm kg}$">|; 

$key = q/SP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1978.png"
 ALT="$SP$">|; 

$key = q/T_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="$T_1$">|; 

$key = q/{{bf{v}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="${\bf v}={\bf0}$">|; 

$key = q/{{bf{f}_ipropto-{mitDelta}{{bf{r}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img656.png"
 ALT="${\bf f}_i\propto
-{\mit\Delta}{\bf r}_i$">|; 

$key = q/f'=-f=k,x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img742.png"
 ALT="$f' = -f = k x$">|; 

$key = q/J,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1416.png"
 ALT="$J h$">|; 

$key = q/{displaymath}M_ast=frac{4,pi^2,r^3}{G,T^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2045.png"
 ALT="\begin{displaymath}
M_\ast = \frac{4 \pi^2 r^3}{G T^2}.
\end{displaymath}">|; 

$key = q/{displaymath}h=frac{u^2}{2,g}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="\begin{displaymath}
h = \frac{u^2}{2 g}.
\end{displaymath}">|; 

$key = q/pm7,{{rm{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1841.png"
 ALT="$\pm 7 {\rm cm}$">|; 

$key = q/{{bf{F}=sum_{i=1}^N{{bf{F}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img846.png"
 ALT="${\bf F} = \sum_{i=1}^N {\bf F}_i$">|; 

$key = q/mbox{{boldmath{{omega{}=(omega_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1547.png"
 ALT="$\mbox{\boldmath$\omega$}= (\omega_x$">|; 

$key = q/m_2=2.5,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1028.png"
 ALT="$m_2 =2.5 {\rm kg}$">|; 

$key = q/theta_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1992.png"
 ALT="$\theta_0$">|; 

$key = q/a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$a$">|; 

$key = q/{displaymath}W_1=int_{ArightarrowB:{{rm{path}1}{{bf{f}!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img646.png"
 ALT="\begin{displaymath}
W_1 = \int_{A\rightarrow B: {\rm path} 1} {\bf f}\!\cdot\!d{\bf r}.
\end{displaymath}">|; 

$key = q/{displaymath}m,frac{v^2}{r}=m,g-R.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1217.png"
 ALT="\begin{displaymath}
m \frac{v^2}{r} = m g- R.
\end{displaymath}">|; 

$key = q/m_2>m_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img501.png"
 ALT="$m_2&gt;m_1$">|; 

$key = q/m_2,v_{f2},sintheta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img959.png"
 ALT="$m_2 v_{f2} \sin\theta_2$">|; 

$key = q/M_oplus=5.97times10^{24},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1921.png"
 ALT="$M_\oplus =5.97\times 10^{24} {\rm kg}$">|; 

$key = q/{mitDelta}K;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img756.png"
 ALT="${\mit\Delta} K$">|; 

$key = q/m,v^2slashrneqm,g,tantheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1197.png"
 ALT="$m v^2/r\neq m g \tan\theta$">|; 

$key = q/{displaymath}I,frac{dmbox{{boldmath{{omega{}}{dt}=I,mbox{{boldmath{{alpha{}=mbox{{boldmath{{tau{},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1374.png"
 ALT="\begin{displaymath}
I \frac{d\mbox{\boldmath$\omega$}}{dt} = I \mbox{\boldmath$\alpha$}= \mbox{\boldmath$\tau$},
\end{displaymath}">|; 

$key = q/{displaymath}theta=cos^{-1}(dslashl)=cos^{-1}(1.2slash10)=83.11^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1736.png"
 ALT="\begin{displaymath}
\theta = \cos^{-1}(d/l)= \cos^{-1}(1.2/10) = 83.11^\circ.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}W=sum_i{{bf{f}_i!cdot!{mitDelta}{{bf{r}_i=-sum_i|{{bf{f}_i|,|{mitDelta}{{bf{r}_i|<0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="309" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img658.png"
 ALT="\begin{displaymath}
{\mit\Delta}W = \sum_i {\bf f}_i\!\cdot\!{\mit\Delta}{\bf r}...
...m_i \vert{\bf f}_i\vert  \vert{\mit\Delta}{\bf r}_i\vert &lt; 0.
\end{displaymath}">|; 

$key = q/X_1=X_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1689.png"
 ALT="$X_1 = X_3$">|; 

$key = q/{displaymath}tau=-m,g,l,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1819.png"
 ALT="\begin{displaymath}
\tau = - m g l \sin\theta.
\end{displaymath}">|; 

$key = q/displaystyle0.15;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1856.png"
 ALT="$\displaystyle 0.15$">|; 

$key = q/displaystylefrac{[mslash2+m_1,(l_1slashl)+m_2,(l_2slashl)],g}{sintheta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="253" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1727.png"
 ALT="$\displaystyle \frac{ [m/2 + m_1 (l_1/l) + m_2 (l_2/l)] g }{\sin\theta}$">|; 

$key = q/v_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img760.png"
 ALT="$v_B$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{work1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="530" HEIGHT="227" BORDER="0"
 SRC="|."$dir".q|img624.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{work1.eps}}
\end{figure}">|; 

$key = q/gslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="$g/2$">|; 

$key = q/{displaymath}T=frac{m,v^2}{r}+m,g,(3,costheta-2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1151.png"
 ALT="\begin{displaymath}
T = \frac{m v^2}{r} + m g (3 \cos\theta -2).
\end{displaymath}">|; 

$key = q/displaystyleL_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1552.png"
 ALT="$\displaystyle L_x$">|; 

$key = q/{displaymath}l,ddot{theta}=-g,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1820.png"
 ALT="\begin{displaymath}
l \ddot{\theta} = - g \sin\theta.
\end{displaymath}">|; 

$key = q/m_2slashm_1=3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1018.png"
 ALT="$m_2/m_1=3$">|; 

$key = q/{{bf{p}times{{bf{p}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1533.png"
 ALT="${\bf p} \times {\bf p} = {\bf0}$">|; 

$key = q/{displaymath}a=frac{T}{m_1}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="\begin{displaymath}
a = \frac{T}{m_1}.
\end{displaymath}">|; 

$key = q/{displaymath}x=frac{1}{2},mu,g,t^2=0.5times0.15times9.81times1.98^2=2.88,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1523.png"
 ALT="\begin{displaymath}
x = \frac{1}{2} \mu g t^2 = 0.5\times 0.15 \times 9.81\times 1.98^2=2.88 {\rm m}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{e51.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="171" BORDER="0"
 SRC="|."$dir".q|img531.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{e51.eps}}
\end{figure*}">|; 

$key = q/{displaymath}left|{{rm{e}^{,i,theta}right|=sqrt{cos^2theta+sin^2theta}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img1117.png"
 ALT="\begin{displaymath}
\left\vert{\rm e}^{ i \theta}\right\vert = \sqrt{\cos^2\theta + \sin^2\theta } = 1.
\end{displaymath}">|; 

$key = q/Y_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1681.png"
 ALT="$Y_3$">|; 

$key = q/M,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1438.png"
 ALT="$M g$">|; 

$key = q/a=-g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$a=-g$">|; 

$key = q/{displaymath}{mitDelta}x_g=-frac{m_w}{m_g},{mitDelta}x_w.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img825.png"
 ALT="\begin{displaymath}
{\mit\Delta}x_g =- \frac{m_w}{m_g}  {\mit\Delta}x_w.
\end{displaymath}">|; 

$key = q/displaystylev_0^{,2}+2,a,s.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$\displaystyle v_0^{ 2} + 2 a s.$">|; 

$key = q/{displaymath}Y_3=Y=frac{X}{tantheta}=frac{73.58}{tan30^circ}=127.44,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1763.png"
 ALT="\begin{displaymath}
Y_3 = Y = \frac{X}{\tan \theta} = \frac{73.58}{\tan 30^\circ}
=127.44 {\rm N}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{f}=m,{{bf{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="\begin{displaymath}
{\bf f} = m {\bf a}.
\end{displaymath}">|; 

$key = q/OC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1633.png"
 ALT="$OC$">|; 

$key = q/{displaymath}x_1,T_1+x_2,T_2-frac{l}{2},M,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1641.png"
 ALT="\begin{displaymath}
x_1 T_1 + x_2 T_2 - \frac{l}{2} M g = 0.
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{alpha{}=dmbox{{boldmath{{omega{}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1375.png"
 ALT="$\mbox{\boldmath$\alpha$}= d\mbox{\boldmath$\omega$}/dt$">|; 

$key = q/1,{{rm{mi}=5,280,{{rm{ft};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$1 {\rm mi} = 5,280 {\rm ft}$">|; 

$key = q/{displaymath}M_ast=frac{4timespi^2times(2.34times10^{11})^3}{(6.673times10^{-11}imes(4.3times10^7)^2}=4.01times10^{30},{{rm{kg}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2046.png"
 ALT="\begin{displaymath}
M_\ast = \frac{4\times\pi^2\times (2.34\times 10^{11})^3}{(6...
...11})\times
(4.3\times 10^7)^2} = 4.01\times 10^{30} {\rm kg}.
\end{displaymath}">|; 

$key = q/displaystylefrac{m_2-m_1}{m_1+m_2},g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img499.png"
 ALT="$\displaystyle \frac{m_2-m_1}{m_1+m_2} g.$">|; 

$key = q/{{bf{r}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img837.png"
 ALT="${\bf r}_i$">|; 

$key = q/{displaymath}W=int_{x_A}^{x_B}f(x),dx.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img631.png"
 ALT="\begin{displaymath}
W = \int_{x_A}^{x_B} f(x) dx.
\end{displaymath}">|; 

$key = q/displaystyle-frac{G,M_odot}{r^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1968.png"
 ALT="$\displaystyle - \frac{G M_\odot}{r^2},$">|; 

$key = q/{figure}{epsfysize{=0.8in{centerline{epsffile{track.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="430" HEIGHT="92" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="\begin{figure}
\epsfysize =0.8in
\centerline{\epsffile{track.eps}}
\end{figure}">|; 

$key = q/ddot{~};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img799.png"
 ALT="$\ddot{&nbsp;}$">|; 

$key = q/displaystyleX,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1694.png"
 ALT="$\displaystyle X,$">|; 

$key = q/{displaymath}U=-frac{G,M_oplus,m}{R_oplus}=frac{(6.673times10^{-11})times(5.97tis120}{(6.378times10^6)}=-7.495times10^9,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="597" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2048.png"
 ALT="\begin{displaymath}
U = - \frac{G M_\oplus m}{R_\oplus} = \frac{(6.673\times 1...
...times 120}
{( 6.378\times 10^6)} = -7.495\times 10^9 {\rm J}.
\end{displaymath}">|; 

$key = q/1,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="$1 {\rm kg}$">|; 

$key = q/mu,R(=mu,M,g,costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1453.png"
 ALT="$\mu R (=\mu M g \cos\theta)$">|; 

$key = q/v=210,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1595.png"
 ALT="$v= 210 {\rm m/s}$">|; 

$key = q/x,y=+a,(1-zslashh)slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1297.png"
 ALT="$x,y=+a (1-z/h)/2$">|; 

$key = q/u_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img883.png"
 ALT="$u_f$">|; 

$key = q/{displaymath}m,ddot{{bf{r}=-G,frac{m,M_oplus}{R_oplus^{,2}},hat{{bf{z},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1924.png"
 ALT="\begin{displaymath}
m \ddot{\bf r} = - G  \frac{m M_\oplus}{R_\oplus^{ 2}} \hat{\bf z},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{cv.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="361" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{cv.eps}}
\end{figure}">|; 

$key = q/ASB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1905.png"
 ALT="$ASB$">|; 

$key = q/x_2>x_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1638.png"
 ALT="$x_2&gt;x_1$">|; 

$key = q/{{bf{l};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1527.png"
 ALT="${\bf l}$">|; 

$key = q/{figure}{epsfysize{=1.6in{centerline{epsffile{block.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="556" HEIGHT="182" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="\begin{figure}
\epsfysize =1.6in
\centerline{\epsffile{block.eps}}
\end{figure}">|; 

$key = q/ddot{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1788.png"
 ALT="$\ddot{x}$">|; 

$key = q/0.3048;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$0.3048$">|; 

$key = q/{displaymath}I=frac{1}{12},M,l^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1357.png"
 ALT="\begin{displaymath}
I = \frac{1}{12} M l^2.
\end{displaymath}">|; 

$key = q/v_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img575.png"
 ALT="$v_1$">|; 

$key = q/{displaymath}z_{cm}=frac{int_0^h,a^2,z,(1-zslashh)^2,dz}{int_0^h,a^2,(1-zslashh)^2,dz}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img1302.png"
 ALT="\begin{displaymath}
z_{cm} = \frac{\int_0^h a^2 z (1-z/h)^2 dz}{\int_0^h a^2 (1-z/h)^2 dz}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{2},m,v^2=m,g,r,(1-costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1181.png"
 ALT="\begin{displaymath}
\frac{1}{2} m v^2 = m g r  (1-\cos\theta).
\end{displaymath}">|; 

$key = q/mu=0.25;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img765.png"
 ALT="$\mu=0.25$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{oblique.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="288" BORDER="0"
 SRC="|."$dir".q|img951.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{oblique.eps}}
\end{figure}">|; 

$key = q/L_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1550.png"
 ALT="$L_y$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{axle.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img1581.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{axle.eps}}
\end{figure}">|; 

$key = q/v_{g,x}geq0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="$v_{g x}\geq 0$">|; 

$key = q/v_{i2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img907.png"
 ALT="$v_{i2}$">|; 

$key = q/{displaymath}I,ddot{theta}=-k,theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1813.png"
 ALT="\begin{displaymath}
I \ddot{\theta} = - k \theta.
\end{displaymath}">|; 

$key = q/dm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img861.png"
 ALT="$dm$">|; 

$key = q/CA;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$CA$">|; 

$key = q/{displaymath}U=m,g,h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img582.png"
 ALT="\begin{displaymath}
U = m g h.
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{x_2-lslash2}{x_2-x_1}right)M,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1643.png"
 ALT="$\displaystyle \left(\frac{x_2-l/2}{x_2-x_1}\right)M g,$">|; 

$key = q/Y_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1680.png"
 ALT="$Y_2$">|; 

$key = q/displaystylefrac{2,m_1}{m_1+m_2},v_{i1}-frac{(m_1-m_2)}{m_1+m_2},v_{i2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img934.png"
 ALT="$\displaystyle \frac{2 m_1}{m_1+m_2} v_{i1} - \frac{(m_1-m_2)}{m_1+m_2} v_{i2}.$">|; 

$key = q/{{bf{f}=M,{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1399.png"
 ALT="${\bf f} = M {\bf a}$">|; 

$key = q/phi=cos^{-1}(0.15slash0.25)=53.13^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1860.png"
 ALT="$\phi = \cos^{-1}(0.15/0.25) = 53.13^\circ$">|; 

$key = q/f<mu,R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1454.png"
 ALT="$f&lt; \mu R$">|; 

$key = q/{displaymath}ddot{r}=-h,frac{d}{dt}!left(frac{du}{dtheta}right)=-h,dot{theta},frc{d^2u}{dtheta^2}=-h^2,u^2,frac{d^2u}{dtheta^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1989.png"
 ALT="\begin{displaymath}
\ddot{r} =- h \frac{d}{dt}\!\left(\frac{du}{d\theta}\right)...
...frac{d^2 u}{d \theta^2}
= - h^2 u^2 \frac{d^2 u}{d\theta^2}.
\end{displaymath}">|; 

$key = q/displaystyledot{x}(t=0)=a,omega,sinphi.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1797.png"
 ALT="$\displaystyle \dot{x}(t=0) =a \omega \sin\phi.$">|; 

$key = q/S;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$S$">|; 

$key = q/{displaymath}M,b,{mitDelta}omega=-J-J'.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1419.png"
 ALT="\begin{displaymath}
M b {\mit\Delta} \omega = -J - J'.
\end{displaymath}">|; 

$key = q/displaystylelambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$\displaystyle \lambda$">|; 

$key = q/{displaymath}P=frac{dW}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img712.png"
 ALT="\begin{displaymath}
P = \frac{dW}{dt}.
\end{displaymath}">|; 

$key = q/{displaymath}m,frac{d^2x}{dt^2}=m,g,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="\begin{displaymath}
m \frac{d^2 x}{dt^2} = m g \sin\theta,
\end{displaymath}">|; 

$key = q/{displaymath}P=f,v{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img778.png"
 ALT="\begin{displaymath}
P = f v
\end{displaymath}">|; 

$key = q/{displaymath}J'=-left(1-frac{M,b,h}{I}right)J.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1420.png"
 ALT="\begin{displaymath}
J' = -\left(1 - \frac{M b h}{I}\right) J.
\end{displaymath}">|; 

$key = q/v_x=dxslashdt=v_0,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="$v_x =dx/dt = v_0 \cos\theta$">|; 

$key = q/{displaymath}K=J,x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1415.png"
 ALT="\begin{displaymath}
K = J x,
\end{displaymath}">|; 

$key = q/(1slash2),(M+m),{v'}^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1225.png"
 ALT="$(1/2) (M+m) {v'}^2$">|; 

$key = q/L_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1551.png"
 ALT="$L_z)$">|; 

$key = q/{displaymath}X_1=m,g-X.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1757.png"
 ALT="\begin{displaymath}
X_1 = m g -X.
\end{displaymath}">|; 

$key = q/x=7,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1733.png"
 ALT="$x=7 {\rm m}$">|; 

$key = q/j=i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img842.png"
 ALT="$j=i$">|; 

$key = q/{displaymath}v=frac{dx}{dt}=1+t-t^3.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="\begin{displaymath}
v = \frac{dx}{dt} = 1 + t - t^3.
\end{displaymath}">|; 

$key = q/g=9.81,{{rm{m},{{rm{s}^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="$g=9.81 {\rm m} {\rm s}^{-1}$">|; 

$key = q/{{bf{e}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="${\bf e}_r$">|; 

$key = q/v_A=(dxslashdt)_{t_A};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="126" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img638.png"
 ALT="$v_A = (dx/dt)_{t_A}$">|; 

$key = q/{{bf{f}_R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="${\bf f}_R$">|; 

$key = q/lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$\lambda$">|; 

$key = q/{displaymath}v_r=-frac{m}{M},v_b.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img832.png"
 ALT="\begin{displaymath}
v_r = - \frac{m}{M} v_b.
\end{displaymath}">|; 

$key = q/-90^circ<theta<90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1168.png"
 ALT="$-90^\circ&lt;\theta&lt;90^\circ$">|; 

$key = q/i^2=-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1111.png"
 ALT="$i^2=-1$">|; 

$key = q/v=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$v=0$">|; 

$key = q/{displaymath}W=-f,x,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img769.png"
 ALT="\begin{displaymath}
W = -f x,
\end{displaymath}">|; 

$key = q/v^2slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img589.png"
 ALT="$v^2/2$">|; 

$key = q/displaystylefrac{dx}{dt}=cost,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$\displaystyle \frac{dx}{dt}=\cos t,$">|; 

$key = q/(m_wslashm_g),{mitDelta}x_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img826.png"
 ALT="$(m_w/m_g)  {\mit\Delta}x_w$">|; 

$key = q/{displaymath}{{bf{a}!cdot!{{bf{s}=a_x,s_x+a_y,s_y+a_z,s_z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="\begin{displaymath}
{\bf a}\!\cdot\!{\bf s} = a_x s_x+ a_y s_y+a_z s_z.
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{alpha{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1376.png"
 ALT="$\mbox{\boldmath$\alpha$}$">|; 

$key = q/{{bf{F}=mbox{{boldmath{{tau{}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1626.png"
 ALT="${\bf F} = \mbox{\boldmath$\tau$}={\bf0}$">|; 

$key = q/v^2slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1079.png"
 ALT="$v^2/r$">|; 

$key = q/x,y=-a,(1-zslashh)slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1296.png"
 ALT="$x,y=-a (1-z/h)/2$">|; 

$key = q/x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img786.png"
 ALT="$x_2$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$u$">|; 

$key = q/v_{f1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img908.png"
 ALT="$v_{f1}$">|; 

$key = q/f,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1363.png"
 ALT="$f \sin\theta$">|; 

$key = q/{displaymath}v_{{rm{max}=a,omega=0.07times418.88=29.32,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1847.png"
 ALT="\begin{displaymath}
v_{\rm max} = a \omega = 0.07\times 418.88 = 29.32 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{{rm{e}^{,i,theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1115.png"
 ALT="${\rm e}^{ i \theta}$">|; 

$key = q/displaystyleI';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1343.png"
 ALT="$\displaystyle I'$">|; 

$key = q/a({{rm{AU});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1909.png"
 ALT="$a({\rm AU})$">|; 

$key = q/{displaymath}dot{r}=-frac{dot{u}}{u^2}=-frac{dot{theta}}{u^2}frac{du}{dtheta}=-h,frac{du}{dtheta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="239" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1987.png"
 ALT="\begin{displaymath}
\dot{r} =- \frac{\dot{u}}{u^2} = - \frac{\dot{\theta}}{u^2}\frac{du}{d\theta} =- h \frac{du}{d\theta}.
\end{displaymath}">|; 

$key = q/F=F_1+F_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img806.png"
 ALT="$F=F_1+F_2$">|; 

$key = q/T,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1086.png"
 ALT="$T \cos\theta$">|; 

$key = q/sigma=|{{bf{k}times{{bf{r}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1315.png"
 ALT="$\sigma = \vert{\bf k}\times {\bf r}\vert$">|; 

$key = q/{displaymath}t=frac{b,omega_0}{3,mu,g}=frac{0.25times35}{3times0.15times9.81}=1.98,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="297" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1522.png"
 ALT="\begin{displaymath}
t = \frac{b \omega_0}{3 \mu g}=\frac{0.25\times 35}{3\times 0.15\times 9.81} = 1.98 {\rm s}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{I}=int_{t_1}^{t_2}{{bf{f}(t),dt.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img901.png"
 ALT="\begin{displaymath}
{\bf I} = \int_{t_1}^{t_2} {\bf f}(t) dt.
\end{displaymath}">|; 

$key = q/phi_0=phi(t=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1247.png"
 ALT="$\phi_0 = \phi(t=0)$">|; 

$key = q/100,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$100 {\rm m}$">|; 

$key = q/v_t=-b,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1426.png"
 ALT="$v_t=-b \omega$">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{bridge.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="171" BORDER="0"
 SRC="|."$dir".q|img1744.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{bridge.eps}}
\end{figure*}">|; 

$key = q/{displaymath}l=r,p,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1526.png"
 ALT="\begin{displaymath}
l = r p \sin\theta,
\end{displaymath}">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img672.png"
 ALT="$k$">|; 

$key = q/displaystyle3,t.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$\displaystyle 3 t.$">|; 

$key = q/P';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1238.png"
 ALT="$P'$">|; 

$key = q/K_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img757.png"
 ALT="$K_B$">|; 

$key = q/M=1.08times10^{23},{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2015.png"
 ALT="$M= 1.08\times 10^{23}  {\rm kg}$">|; 

$key = q/r_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1998.png"
 ALT="$r_0$">|; 

$key = q/{displaymath}cosalpha=frac{|{{bf{u}|}{|{{bf{v}_a|}=frac{60}{200}=0.3.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="\begin{displaymath}
\cos\alpha = \frac{\vert{\bf u}\vert}{\vert{\bf v}_a\vert} = \frac{60}{200} =0.3.
\end{displaymath}">|; 

$key = q/x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$x_0$">|; 

$key = q/{displaymath}E=K+U,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1946.png"
 ALT="\begin{displaymath}
E = K + U,
\end{displaymath}">|; 

$key = q/{displaymath}W=m,(g+a).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="\begin{displaymath}
W = m (g+a).
\end{displaymath}">|; 

$key = q/x=x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img702.png"
 ALT="$x=x_2$">|; 

$key = q/s=x-x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$s=x-x_0$">|; 

$key = q/40^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1720.png"
 ALT="$40^\circ$">|; 

$key = q/{displaymath}{{bf{v}_g={{bf{v}_a+{{bf{u}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="\begin{displaymath}
{\bf v}_g = {\bf v}_a + {\bf u}.
\end{displaymath}">|; 

$key = q/displaystyleM,frac{int,int,x^2,dx,dy}{int,int,dx,dy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1326.png"
 ALT="$\displaystyle M \frac{\int \int  x^2 dx dy}{\int \int dx dy},$">|; 

$key = q/t_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img886.png"
 ALT="$t_1$">|; 

$key = q/{displaymath}s=frac{2pi}{360^circ},r,theta(^circ).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1042.png"
 ALT="\begin{displaymath}
s = \frac{2\pi}{360^\circ}  r \theta(^\circ).
\end{displaymath}">|; 

$key = q/{displaymath}v=r,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1056.png"
 ALT="\begin{displaymath}
v = r \omega.
\end{displaymath}">|; 

$key = q/{{bf{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="${\bf s}$">|; 

$key = q/{displaymath}{{bf{r}_itimesdot{{bf{p}_i=frac{d({{bf{r}_itimes{{bf{p}_i)}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1571.png"
 ALT="\begin{displaymath}
{\bf r}_i\times \dot{\bf p}_i = \frac{d ({\bf r}_i\times {\bf p}_i)}{dt}.
\end{displaymath}">|; 

$key = q/M=sum_{i=1,N}m_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1631.png"
 ALT="$M=\sum_{i=1,N}m_i$">|; 

$key = q/M=sum_{i=1}^Nm_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img845.png"
 ALT="$M = \sum_{i=1}^N m_i$">|; 

$key = q/{displaymath}{{bf{r}_{cm}=frac{sum_{i=1}^Nm_i,{{bf{r}_i}{sum_{i=1}^Nm_i}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img848.png"
 ALT="\begin{displaymath}
{\bf r}_{cm} = \frac{\sum_{i=1}^N m_i {\bf r}_i}{\sum_{i=1}^N m_i}.
\end{displaymath}">|; 

$key = q/v=6000,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2025.png"
 ALT="$v=6000 {\rm m/s}$">|; 

$key = q/x,y=pmaslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1298.png"
 ALT="$x,y=\pm a/2$">|; 

$key = q/{displaymath}r_1=r_0,frac{1+e}{1-e}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2000.png"
 ALT="\begin{displaymath}
r_1 = r_0 \frac{1+e}{1-e}.
\end{displaymath}">|; 

$key = q/[M];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$[M]$">|; 

$key = q/sigma=b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1318.png"
 ALT="$\sigma=b$">|; 

$key = q/{displaymath}{mitDelta}U=U_B-U_A=2.06times10^4-7.36times10^4=-5.30times10^4,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img754.png"
 ALT="\begin{displaymath}
{\mit\Delta} U = U_B-U_A = 2.06\times 10^4-7.36\times 10^4 = -5.30\times 10^4 {\rm J}.
\end{displaymath}">|; 

$key = q/{displaymath}P=mbox{{boldmath{{omega{}times{{bf{r}cdot{{bf{f}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1382.png"
 ALT="\begin{displaymath}
P = \mbox{\boldmath$\omega$}\times {\bf r} \cdot {\bf f}
\end{displaymath}">|; 

$key = q/u=5,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$u=5 {\rm m/s}$">|; 

$key = q/dV;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1316.png"
 ALT="$dV$">|; 

$key = q/3000,{{rm{rev.slashmin.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1501.png"
 ALT="$3000 {\rm rev./min.}$">|; 

$key = q/m_2,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$m_2 g$">|; 

$key = q/W';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img722.png"
 ALT="$W'$">|; 

$key = q/{displaymath}a=frac{r_0+r_1}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="44" BORDER="0"
 SRC="|."$dir".q|img2009.png"
 ALT="\begin{displaymath}
a = \frac{r_0+r_1}{2}.
\end{displaymath}">|; 

$key = q/omega_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1612.png"
 ALT="$\omega_2$">|; 

$key = q/v_1=110,{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$v_1= 110 {\rm km/h}$">|; 

$key = q/{displaymath}R_{{rm{max}=frac{v_0^{,2}}{g},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="\begin{displaymath}
R_{\rm max} = \frac{v_0^{ 2}}{g},
\end{displaymath}">|; 

$key = q/{displaymath}h=z(t_0)=frac{v_0^{,2}}{2,g},sin^2theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="\begin{displaymath}
h = z(t_0)= \frac{v_0^{ 2}}{2 g} \sin^2\theta.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}={{bf{r}_1+{{bf{r}_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="\begin{displaymath}
{\bf r} = {\bf r}_1 + {\bf r}_2.
\end{displaymath}">|; 

$key = q/{displaymath}M,ddot{{bf{r}_{cm}={{bf{F},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img844.png"
 ALT="\begin{displaymath}
M \ddot{\bf r}_{cm}= {\bf F},
\end{displaymath}">|; 

$key = q/displaystyleddot{r}-r,dot{theta}^2=ddot{r}-r,omega^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1137.png"
 ALT="$\displaystyle \ddot{r}-r \dot{\theta}^2 = \ddot{r}-r \omega^2,$">|; 

$key = q/{displaymath}ddot{{bf{r}=-g_oplus,hat{{bf{z},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1926.png"
 ALT="\begin{displaymath}
\ddot{\bf r} = - g_\oplus \hat{\bf z},
\end{displaymath}">|; 

$key = q/R>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1166.png"
 ALT="$R&gt;0$">|; 

$key = q/displaystyleb,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1518.png"
 ALT="$\displaystyle b \omega$">|; 

$key = q/{displaymath}f=-k,x.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1766.png"
 ALT="\begin{displaymath}
f = -k x.
\end{displaymath}">|; 

$key = q/(lslash2),costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1698.png"
 ALT="$(l/2) \cos\theta$">|; 

$key = q/10^{-15};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$10^{-15}$">|; 

$key = q/{displaymath}I_x=frac{I_z}{2}=frac{1}{2},M,b^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1332.png"
 ALT="\begin{displaymath}
I_x = \frac{I_z}{2} = \frac{1}{2} M b^2.
\end{displaymath}">|; 

$key = q/I_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1611.png"
 ALT="$I_2$">|; 

$key = q/{displaymath}P={{bf{f}!cdot!{{bf{v},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img715.png"
 ALT="\begin{displaymath}
P = {\bf f}\!\cdot\!{\bf v},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}=dot{{bf{v}=a_r,{{bf{e}_r+a_theta,{{bf{e}_theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1103.png"
 ALT="\begin{displaymath}
{\bf a} = \dot{\bf v} = a_r {\bf e}_r + a_\theta {\bf e}_\theta.
\end{displaymath}">|; 

$key = q/{displaymath}h=frac{l}{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1976.png"
 ALT="\begin{displaymath}
h = \frac{l}{m}.
\end{displaymath}">|; 

$key = q/{displaymath}alpha=frac{domega}{dt}=frac{d^2phi}{dt^2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1245.png"
 ALT="\begin{displaymath}
\alpha = \frac{d\omega}{dt}= \frac{d^2\phi}{dt^2},
\end{displaymath}">|; 

$key = q/{displaymath}k_1,x_1=k_2,x_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1870.png"
 ALT="\begin{displaymath}
k_1  x_1 = k_2 x_2.
\end{displaymath}">|; 

$key = q/110,{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$110 {\rm km/h}$">|; 

$key = q/d=1.2,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1734.png"
 ALT="$d=1.2 {\rm m}$">|; 

$key = q/b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="$b$">|; 

$key = q/dx,dy;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1354.png"
 ALT="$dx dy$">|; 

$key = q/{{bf{a}!cdot!{{bf{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="${\bf a}\!\cdot\!{\bf s}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{cable.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img1078.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{cable.eps}}
\end{figure}">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$x$">|; 

$key = q/x=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1779.png"
 ALT="$x=a$">|; 

$key = q/displaystyler,ddot{theta}+2,dot{r},dot{theta}=r,dot{omega}+2,dot{r},omega.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1139.png"
 ALT="$\displaystyle r \ddot{\theta}+2 \dot{r} 
\dot{\theta} = r \dot{\omega} + 2 \dot{r} \omega.$">|; 

$key = q/{displaymath}{{bf{f}_{ij}propto({{bf{r}_i-{{bf{r}_j).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1567.png"
 ALT="\begin{displaymath}
{\bf f}_{ij} \propto ({\bf r}_i - {\bf r}_j).
\end{displaymath}">|; 

$key = q/displaystylex_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1648.png"
 ALT="$\displaystyle x_2$">|; 

$key = q/|W|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="$\vert W\vert$">|; 

$key = q/{displaymath}sin(theta+phi)=2,sintheta,cosphi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1659.png"
 ALT="\begin{displaymath}
\sin(\theta+\phi) = 2 \sin\theta \cos\phi.
\end{displaymath}">|; 

$key = q/displaystyleT;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$\displaystyle T$">|; 

$key = q/v_b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img828.png"
 ALT="$v_b$">|; 

$key = q/{displaymath}dW={{bf{f}!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img714.png"
 ALT="\begin{displaymath}
d W = {\bf f}\!\cdot\!d{\bf r}.
\end{displaymath}">|; 

$key = q/{displaymath}U(x)=frac{1}{2},k,x^{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img676.png"
 ALT="\begin{displaymath}
U(x) = \frac{1}{2} k x^{2}.
\end{displaymath}">|; 

$key = q/I=(1slash2),M,b^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1434.png"
 ALT="$I=(1/2) M b^2$">|; 

$key = q/{displaymath}v_{{rm{esc}=sqrt{frac{2,G,M_oplus}{R_oplus}}=sqrt{frac{2times(6.673s10^{24})}{6.378times10^6}}=11.2,{{rm{kmslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="569" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img1953.png"
 ALT="\begin{displaymath}
v_{\rm esc} = \sqrt{\frac{2 G M_\oplus}{R_\oplus}} =
\sqr...
...mes(5.97\times 10^{24})}{6.378\times 10^6}}=
11.2 {\rm km/s}.
\end{displaymath}">|; 

$key = q/{displaymath}k=m,omega^2=4times17.95times17.95=1289.1,{{rm{Nslashm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1883.png"
 ALT="\begin{displaymath}
k = m \omega^2 = 4 \times 17.95\times 17.95 = 1289.1 {\rm N/m}.
\end{displaymath}">|; 

$key = q/{{bf{r}_jtimes{{bf{f}_{ji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1575.png"
 ALT="${\bf r}_j\times {\bf f}_{ji}$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{cannon.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="171" BORDER="0"
 SRC="|."$dir".q|img827.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{cannon.eps}}
\end{figure}">|; 

$key = q/2pi,sigma,dsigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1353.png"
 ALT="$2\pi \sigma d\sigma$">|; 

$key = q/displaystylev^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle v^2$">|; 

$key = q/{{bf{s}=({mitDelta}x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img603.png"
 ALT="${\bf s} = ({\mit\Delta}x$">|; 

$key = q/{displaymath}theta=tan^{-1}left(frac{v^2}{r,g}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1195.png"
 ALT="\begin{displaymath}
\theta =\tan^{-1}\left(\frac{v^2}{r g}\right).
\end{displaymath}">|; 

$key = q/{{bf{a}!cdot!{{bf{b};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1267.png"
 ALT="${\bf a}\!\cdot\!{\bf b}$">|; 

$key = q/{displaymath}X_2=X_3=X,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1756.png"
 ALT="\begin{displaymath}
X_2 = X_3 = X,
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{omega{}=omega_z,hat{{bf{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1564.png"
 ALT="$\mbox{\boldmath$\omega$}= \omega_z \hat{\bf z}$">|; 

$key = q/D=4l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$D=4l$">|; 

$key = q/int_A^B{{bf{f}({{bf{r})!cdot!d{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img623.png"
 ALT="$\int_A^B {\bf f}({\bf r})\!\cdot\!d{\bf r}$">|; 

$key = q/{displaymath}m_1,v_{i1}=m_1,v_{f1},costheta_1+m_2,v_{f2},costheta_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img957.png"
 ALT="\begin{displaymath}
m_1 v_{i1} = m_1 v_{f1} \cos\theta_1+ m_2 v_{f2} \cos\theta_2.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}=(0,0,-g).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="\begin{displaymath}
{\bf a} = (0,0,-g).
\end{displaymath}">|; 

$key = q/displaystyle{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$\displaystyle {\bf v}$">|; 

$key = q/M=3,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1472.png"
 ALT="$M=3 {\rm kg}$">|; 

$key = q/displaystyleddot{r}-r,dot{theta}^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1963.png"
 ALT="$\displaystyle \ddot{r} - r \dot{\theta}^2,$">|; 

$key = q/displaystylev_{f2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img933.png"
 ALT="$\displaystyle v_{f2}$">|; 

$key = q/d=10,000,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1594.png"
 ALT="$d=10,000 {\rm m}$">|; 

$key = q/{displaymath}{mitDelta}{{bf{p}={{bf{I}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img902.png"
 ALT="\begin{displaymath}
{\mit\Delta} {\bf p} = {\bf I}.
\end{displaymath}">|; 

$key = q/{displaymath}L=I,omega=0.08times8.98=0.718,{{rm{kg,m^2slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1605.png"
 ALT="\begin{displaymath}
L = I \omega = 0.08\times 8.98 = 0.718 {\rm kg m^2/s}.
\end{displaymath}">|; 

$key = q/{{bf{r}={{bf{r}_1+{{bf{r}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="${\bf r}= {\bf r}_1 + {\bf r}_2$">|; 

$key = q/{displaymath}T-m,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="\begin{displaymath}
T - m g = 0.
\end{displaymath}">|; 

$key = q/M+m;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img858.png"
 ALT="$M+m$">|; 

$key = q/5,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$5 {\rm m/s}$">|; 

$key = q/h=-135times0.3048=-41.15,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="$h = -135\times 0.3048 = -41.15  {\rm m}$">|; 

$key = q/j;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$j$">|; 

$key = q/{displaymath}|{{bf{a}times{{bf{b}|=|{{bf{a}|,|{{bf{b}|,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1271.png"
 ALT="\begin{displaymath}
\vert{\bf a}\times{\bf b} \vert
=\vert{\bf a}\vert \vert{\bf b}\vert \sin\theta.
\end{displaymath}">|; 

$key = q/{displaymath}R-m,g-M,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1739.png"
 ALT="\begin{displaymath}
R - m g - M g = 0.
\end{displaymath}">|; 

$key = q/{displaymath}tau=m,g,x.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1498.png"
 ALT="\begin{displaymath}
\tau = m g x.
\end{displaymath}">|; 

$key = q/{displaymath}dot{{bf{p}_i=sum_{j=1,N}^{jneqi}{{bf{f}_{ij}+{{bf{F}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img1569.png"
 ALT="\begin{displaymath}
\dot{\bf p}_i = \sum_{j=1,N}^{j\neq i} {\bf f}_{ij} + {\bf F}_i.
\end{displaymath}">|; 

$key = q/{displaymath}f=frac{4000}{60}=66.6666,{{rm{Hz}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1844.png"
 ALT="\begin{displaymath}
f = \frac{4000}{60} = 66.6666 {\rm Hz}.
\end{displaymath}">|; 

$key = q/{displaymath}I,dot{omega}=tau,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1367.png"
 ALT="\begin{displaymath}
I  \dot{\omega} = \tau,
\end{displaymath}">|; 

$key = q/displaystylex,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$\displaystyle x,$">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{roller.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="415" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img751.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{roller.eps}}
\end{figure*}">|; 

$key = q/0^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$0^\circ$">|; 

$key = q/{displaymath}I'=I+M,left(frac{L}{2}right)^2=frac{1}{3},M,L^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1476.png"
 ALT="\begin{displaymath}
I' = I + M \left(\frac{L}{2}\right)^2 = \frac{1}{3} M L^2,
\end{displaymath}">|; 

$key = q/m=2.6,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1480.png"
 ALT="$m=2.6 {\rm kg}$">|; 

$key = q/a_theta=0.6,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1200.png"
 ALT="$a_\theta = 0.6 {\rm m/s}$">|; 

$key = q/{displaymath}{mitDelta}t=frac{{mitDelta}x}{bar{v}}=frac{120}{85times(1000slash3600)}=5.082,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="\begin{displaymath}
{\mit\Delta}t = \frac{{\mit\Delta}x}{\bar{v}} = \frac{120}{85\times (1000/3600)} = 5.082 {\rm s}.
\end{displaymath}">|; 

$key = q/9,192,631,770;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$9,192,631,770$">|; 

$key = q/{displaymath}t=frac{v_0+sqrt{v_0^{,2}+2,g,z_0}}{g}=20.88,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="\begin{displaymath}
t = \frac{v_0 +\sqrt{v_0^{ 2}+2 g z_0}}{g} = 20.88 {\rm s}.
\end{displaymath}">|; 

$key = q/{displaymath}tau=f,l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1371.png"
 ALT="\begin{displaymath}
\tau = f l.
\end{displaymath}">|; 

$key = q/hat{{bf{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1925.png"
 ALT="$\hat{\bf z}$">|; 

$key = q/MslashM_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1931.png"
 ALT="$M/M_\oplus$">|; 

$key = q/{displaymath}dot{z}=dot{r},{{rm{e}^{,i,theta}+r,dot{theta},i,{{rm{e}^{,i,theta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1128.png"
 ALT="\begin{displaymath}
\dot{z} = \dot{r} {\rm e}^{ i \theta} + r \dot{\theta} i {\rm e}^{ i \theta}.
\end{displaymath}">|; 

$key = q/displaystyler,ddot{theta}+2,dot{r},dot{theta}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1964.png"
 ALT="$\displaystyle r \ddot{\theta} + 2 \dot{r} \dot{\theta}.$">|; 

$key = q/m_1,v_{f1},costheta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img955.png"
 ALT="$m_1 v_{f1} \cos\theta_1$">|; 

$key = q/e;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2005.png"
 ALT="$e$">|; 

$key = q/30,{{rm{rev.slashmin.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1467.png"
 ALT="$30  {\rm rev./min.}$">|; 

$key = q/mathaccent"7016relax{F}(k);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$\mathaccent ''7016\relax {F}(k)$">|; 

$key = q/sin^2theta+cos^2theta=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1802.png"
 ALT="$\sin^2\theta+\cos^2\theta =1$">|; 

$key = q/t+dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img859.png"
 ALT="$t+dt$">|; 

$key = q/F>f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img509.png"
 ALT="$F&gt;f$">|; 

$key = q/mbox{{boldmath{{omega{}_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1262.png"
 ALT="$\mbox{\boldmath$\omega$}_3$">|; 

$key = q/displaystylex_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1794.png"
 ALT="$\displaystyle x_0$">|; 

$key = q/m_B=15,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img570.png"
 ALT="$m_B = 15 {\rm kg}$">|; 

$key = q/{displaymath}{{bf{f}=(0,0,-m,g).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img684.png"
 ALT="\begin{displaymath}
{\bf f} = (0,0,-m g).
\end{displaymath}">|; 

$key = q/h_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img574.png"
 ALT="$h_2$">|; 

$key = q/displaystyle{{bf{b}=(0,12),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="$\displaystyle {\bf b} = (0,12),$">|; 

$key = q/h=l,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1095.png"
 ALT="$h=l \cos\theta$">|; 

$key = q/{displaymath}phi=frac{1}{2},alpha,t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1464.png"
 ALT="\begin{displaymath}
\phi = \frac{1}{2} \alpha t^2.
\end{displaymath}">|; 

$key = q/{{bf{f}_{ab};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="${\bf f}_{ab}$">|; 

$key = q/{{bf{a}=(a_x,a_y,a_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="${\bf a}=(a_x,a_y,a_z)$">|; 

$key = q/l,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1654.png"
 ALT="$l \sin\theta$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{perpax.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1328.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{perpax.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{bicycle.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img1365.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{bicycle.eps}}
\end{figure}">|; 

$key = q/{displaymath}R=(m+M),g=(40+80)times9.81=1177.2,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="380" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1740.png"
 ALT="\begin{displaymath}
R = (m+M) g = (40+80)\times 9.81 = 1177.2 {\rm N}.
\end{displaymath}">|; 

$key = q/displaystylefrac{dv_x}{dt}=frac{d^2x}{dt^2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$\displaystyle \frac{dv_x}{dt}=\frac{d^2 x}{dt^2},$">|; 

$key = q/G;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1919.png"
 ALT="$G$">|; 

$key = q/T=F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img549.png"
 ALT="$T=F$">|; 

$key = q/P=mbox{{boldmath{{tau{}!cdot!mbox{{boldmath{{omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1404.png"
 ALT="$P = \mbox{\boldmath$\tau$}\!\cdot\!\mbox{\boldmath$\omega$}$">|; 

$key = q/theta=0^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1037.png"
 ALT="$\theta=0^\circ$">|; 

$key = q/{displaymath}v_theta=r,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1141.png"
 ALT="\begin{displaymath}
v_\theta = r \omega.
\end{displaymath}">|; 

$key = q/{displaymath}R+S-M,g-m,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1745.png"
 ALT="\begin{displaymath}
R+S-M g - m g = 0.
\end{displaymath}">|; 

$key = q/{{bf{v}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1304.png"
 ALT="${\bf v}_i$">|; 

$key = q/{displaymath}deltavsimeqv,deltatheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1073.png"
 ALT="\begin{displaymath}
\delta v\simeq v \delta\theta.
\end{displaymath}">|; 

$key = q/F_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img790.png"
 ALT="$F_1$">|; 

$key = q/F,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img730.png"
 ALT="$F \cos\theta$">|; 

$key = q/dW;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img710.png"
 ALT="$dW$">|; 

$key = q/{displaymath}u=-frac{m}{M},v=-frac{1.2times115}{1200}=-0.115,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img973.png"
 ALT="\begin{displaymath}
u = - \frac{m}{M} v = -\frac{1.2\times 115}{1200} = -0.115 {\rm m/s}.
\end{displaymath}">|; 

$key = q/displaystylefrac{dv_z}{dt}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="$\displaystyle \frac{dv_z}{dt}=0.$">|; 

$key = q/costheta<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1170.png"
 ALT="$\cos\theta&lt;0$">|; 

$key = q/{displaymath}K=sum_{i=1,N}frac{1}{2},m_i,v_i^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1305.png"
 ALT="\begin{displaymath}
K = \sum_{i=1,N} \frac{1}{2} m_i v_i^{ 2}.
\end{displaymath}">|; 

$key = q/{displaymath}omega^2,r=frac{G,M_oplus}{r^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1955.png"
 ALT="\begin{displaymath}
\omega^2 r = \frac{G M_\oplus}{r^2}.
\end{displaymath}">|; 

$key = q/displaystylefrac{dv_z}{dt}=frac{d^2z}{dt^2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="$\displaystyle \frac{dv_z}{dt}=\frac{d^2 z}{dt^2}.$">|; 

$key = q/V_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1285.png"
 ALT="$V_i$">|; 

$key = q/200,{{rm{mislashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$200 {\rm mi/h}$">|; 

$key = q/R=m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img506.png"
 ALT="$R=m g$">|; 

$key = q/{displaymath}X_1+X_3-m,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1751.png"
 ALT="\begin{displaymath}
X_1 + X_3 - m g = 0.
\end{displaymath}">|; 

$key = q/{displaymath}v^2=2,g,h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1436.png"
 ALT="\begin{displaymath}
v^2 = 2 g h.
\end{displaymath}">|; 

$key = q/M=5.2,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1002.png"
 ALT="$M=5.2 {\rm kg}$">|; 

$key = q/{displaymath}k=frac{105}{0.13}=807.7,{{rm{Nslashm}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img745.png"
 ALT="\begin{displaymath}
k = \frac{105}{0.13} = 807.7 {\rm N/m}.
\end{displaymath}">|; 

$key = q/{{bf{f}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="${\bf f}_1$">|; 

$key = q/K=I,omega^2slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1406.png"
 ALT="$K = I \omega^2/2$">|; 

$key = q/f'=W,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img780.png"
 ALT="$f'= W \sin\theta$">|; 

$key = q/{{bf{r}_{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img847.png"
 ALT="${\bf r}_{cm}$">|; 

$key = q/sigma=sqrt{x^2+y^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1351.png"
 ALT="$\sigma=\sqrt{x^2+y^2}$">|; 

$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img975.png"
 ALT="$w$">|; 

$key = q/{displaymath}frac{dP}{dt}=F.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img815.png"
 ALT="\begin{displaymath}
\frac{dP}{dt} = F.
\end{displaymath}">|; 

$key = q/{displaymath}X_2-X_3=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1753.png"
 ALT="\begin{displaymath}
X_2-X_3=0.
\end{displaymath}">|; 

$key = q/x=v_{f1}slashv_{i1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1016.png"
 ALT="$x=v_{f1}/v_{i1}$">|; 

$key = q/displaystyle{{bf{a}+lambda,{{bf{d},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$\displaystyle {\bf a} + \lambda {\bf d},$">|; 

$key = q/57.296^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1047.png"
 ALT="$57.296^\circ$">|; 

$key = q/mbox{{boldmath{{phi{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1252.png"
 ALT="$\mbox{\boldmath$\phi$}$">|; 

$key = q/{displaymath}{{bf{r}=r,{{bf{e}_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1101.png"
 ALT="\begin{displaymath}
{\bf r} = r {\bf e}_r.
\end{displaymath}">|; 

$key = q/{displaymath}frac{g}{g_oplus}=frac{MslashM_oplus}{(RslashR_oplus)^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1930.png"
 ALT="\begin{displaymath}
\frac{g}{g_\oplus} = \frac{M/M_\oplus}{(R/R_\oplus)^2}.
\end{displaymath}">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$z$">|; 

$key = q/{displaymath}{{bf{L}=sum_{i=1,N}{{bf{r}_itimes{{bf{p}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1572.png"
 ALT="\begin{displaymath}
{\bf L} = \sum_{i=1,N} {\bf r}_i\times {\bf p}_i.
\end{displaymath}">|; 

$key = q/dot{z}=dzslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1127.png"
 ALT="$\dot{z} = dz/dt$">|; 

$key = q/Y_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1679.png"
 ALT="$Y_1$">|; 

$key = q/-u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img863.png"
 ALT="$-u$">|; 

$key = q/N;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img617.png"
 ALT="$N$">|; 

$key = q/{displaymath}theta=omega,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1835.png"
 ALT="\begin{displaymath}
\theta = \omega t,
\end{displaymath}">|; 

$key = q/displaystyle{{bf{a}=(-9,0),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$\displaystyle {\bf a} = (-9,0),$">|; 

$key = q/mu=0.45;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img558.png"
 ALT="$\mu = 0.45$">|; 

$key = q/{displaymath}ddot{z}=(ddot{r}-r,dot{theta}^2),{{rm{e}^{,i,theta}+(r,ddot{theta}+2,dot{r},dot{theta}),i,{{rm{e}^{,i,theta}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="295" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1135.png"
 ALT="\begin{displaymath}
\ddot{z} = (\ddot{r}-r \dot{\theta}^2) {\rm e}^{ i \thet...
...{\theta}+2 \dot{r} 
\dot{\theta}) i {\rm e}^{ i \theta}.
\end{displaymath}">|; 

$key = q/{{bf{p}_i=m_i,dot{{bf{r}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img851.png"
 ALT="${\bf p}_i = m_i \dot{\bf r}_i$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{tinc.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img962.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{tinc.eps}}
\end{figure}">|; 

$key = q/displaystyle{{bf{F};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1620.png"
 ALT="$\displaystyle {\bf F}$">|; 

$key = q/v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1171.png"
 ALT="$v^2$">|; 

$key = q/{displaymath}alpha=frac{tau}{I}=frac{m,g,x}{I}=frac{5.3times9.81times0.532}{2.986}=9.26,{{rm{rad.slashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1499.png"
 ALT="\begin{displaymath}
\alpha = \frac{\tau}{I} = \frac{m g x}{I} = \frac{5.3\times 9.81\times 0.532}{2.986} = 9.26 {\rm
rad./s^2}.
\end{displaymath}">|; 

$key = q/mu_s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img516.png"
 ALT="$\mu_s$">|; 

$key = q/a<-g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="$a&lt;-g$">|; 

$key = q/{displaymath}tantheta=frac{v^2}{r,g},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1194.png"
 ALT="\begin{displaymath}
\tan\theta = \frac{v^2}{r g},
\end{displaymath}">|; 

$key = q/displaystyle+d^2,Mfrac{int!int!intdx,dy,dz}{int!int!int,dx,dy,dz}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1346.png"
 ALT="$\displaystyle + d^2 M \frac{\int\!\int\!\int dx dy dz}
{\int\!\int\!\int  dx dy dz}.$">|; 

$key = q/U=m,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img597.png"
 ALT="$U=m g h$">|; 

$key = q/9.81,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1929.png"
 ALT="$9.81 {\rm m/s^2}$">|; 

$key = q/{{bf{f}=(f,0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img629.png"
 ALT="${\bf f} = (f,0,0)$">|; 

$key = q/x=x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img700.png"
 ALT="$x=x_0$">|; 

$key = q/{displaymath}D=4timessqrt{1.619times10^5}=1.609times10^3,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="\begin{displaymath}
D = 4\times \sqrt{1.619\times 10^5} = 1.609\times 10^3  {\rm m}.
\end{displaymath}">|; 

$key = q/displaystylemu,g,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1517.png"
 ALT="$\displaystyle \mu g t,$">|; 

$key = q/{displaymath}M=frac{T_2}{g,tan40^circ}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img535.png"
 ALT="\begin{displaymath}
M = \frac{T_2}{g \tan 40^\circ}.
\end{displaymath}">|; 

$key = q/{{bf{f}_{ji}=-{{bf{f}_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="${\bf f}_{ji}=-{\bf f}_{ij}$">|; 

$key = q/L;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$L$">|; 

$key = q/{displaymath}omega=lim_{deltatrightarrow0}frac{deltaphi}{deltat}=frac{dphi}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1241.png"
 ALT="\begin{displaymath}
\omega = \lim_{\delta t\rightarrow 0}\frac{\delta\phi}{\delta t}=\frac{d\phi}{dt}.
\end{displaymath}">|; 

$key = q/{displaymath}W-m,g=m,a.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="\begin{displaymath}
W -m g = m a.
\end{displaymath}">|; 

$key = q/T_2=50,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img536.png"
 ALT="$T_2 =50 {\rm N}$">|; 

$key = q/{displaymath}omega=60timesfrac{pi}{180}=1.047,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1478.png"
 ALT="\begin{displaymath}
\omega = 60 \times \frac{\pi}{180} = 1.047 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{kepler2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="462" HEIGHT="172" BORDER="0"
 SRC="|."$dir".q|img1977.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{kepler2.eps}}
\end{figure}">|; 

$key = q/{displaymath}E=K+U={{rm{constant}:{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img667.png"
 ALT="\begin{displaymath}
E = K + U = {\rm constant}:
\end{displaymath}">|; 

$key = q/{displaymath}L=frac{I}{M,d}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1832.png"
 ALT="\begin{displaymath}
L = \frac{I}{M d}.
\end{displaymath}">|; 

$key = q/|{{bf{a}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="$\vert{\bf a}\vert$">|; 

$key = q/F=27,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img540.png"
 ALT="$F= 27 {\rm N}$">|; 

$key = q/{displaymath}f<mu,R.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1670.png"
 ALT="\begin{displaymath}
f &lt; \mu R.
\end{displaymath}">|; 

$key = q/Mggm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img834.png"
 ALT="$M\gg m$">|; 

$key = q/t=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$t=10$">|; 

$key = q/{displaymath}f=frac{m,v^2}{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1080.png"
 ALT="\begin{displaymath}
f = \frac{m v^2}{r}.
\end{displaymath}">|; 

$key = q/I_z=M,b^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1330.png"
 ALT="$I_z=M b^2$">|; 

$key = q/{displaymath}L_2=I_2,omega_2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1610.png"
 ALT="\begin{displaymath}
L_2 = I_2 \omega_2,
\end{displaymath}">|; 

$key = q/x,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1666.png"
 ALT="$x \cos\theta$">|; 

$key = q/R_n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img503.png"
 ALT="$R_n$">|; 

$key = q/f=-k,x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img741.png"
 ALT="$f=-k x$">|; 

$key = q/-a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1784.png"
 ALT="$-a$">|; 

$key = q/{displaymath}f=frac{omega}{2,pi}=1.20,{{rm{Hz}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1099.png"
 ALT="\begin{displaymath}
f = \frac{\omega}{2 \pi} = 1.20 {\rm Hz}.
\end{displaymath}">|; 

$key = q/{displaymath}R,sintheta=m,frac{v^2}{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1193.png"
 ALT="\begin{displaymath}
R \sin\theta = m \frac{v^2}{r},
\end{displaymath}">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$\theta$">|; 

$key = q/a_r=a_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1202.png"
 ALT="$a_r = a_\theta$">|; 

$key = q/a^3slashT^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1908.png"
 ALT="$a^3/T^2$">|; 

$key = q/displaystylev_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="$\displaystyle v_z$">|; 

$key = q/v=200,{{rm{m}slash{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="$v=200 {\rm m}/{\rm s}$">|; 

$key = q/{displaymath}a=frac{F}{m_A+m_B},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img566.png"
 ALT="\begin{displaymath}
a = \frac{F}{m_A+m_B},
\end{displaymath}">|; 

$key = q/AB;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="$AB$">|; 

$key = q/28times(12)^2=4032;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$28\times (12)^2 = 4032$">|; 

$key = q/theta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img953.png"
 ALT="$\theta_2$">|; 

$key = q/tantheta>1slashmu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1673.png"
 ALT="$\tan\theta &gt; 1/\mu$">|; 

$key = q/deltas;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1051.png"
 ALT="$\delta s$">|; 

$key = q/{displaymath}dot{{bf{r}_{cm}=frac{sum_{i=1}^Nm_i,dot{{bf{r}_i}{sum_{i=1}^Nm_i},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img850.png"
 ALT="\begin{displaymath}
\dot{\bf r}_{cm} = \frac{\sum_{i=1}^N m_i \dot{\bf r}_i}{\sum_{i=1}^N m_i},
\end{displaymath}">|; 

$key = q/{displaymath}(M+m),v=(M+m+dm),(v+dv)+(-dm),(v-u).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img867.png"
 ALT="\begin{displaymath}
(M+m) v = (M+m+dm) (v+dv) + (-dm) (v-u).
\end{displaymath}">|; 

$key = q/sin(-theta)=-sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1799.png"
 ALT="$\sin(-\theta)
=-\sin\theta$">|; 

$key = q/displaystyle3,M,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1691.png"
 ALT="$\displaystyle 3 M g,$">|; 

$key = q/v_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$v_0$">|; 

$key = q/m(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img857.png"
 ALT="$m(t)$">|; 

$key = q/{displaymath}K=W+W'=550.6-419.3=131.3,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img738.png"
 ALT="\begin{displaymath}
K = W + W' = 550.6 - 419.3 = 131.3 {\rm J}.
\end{displaymath}">|; 

$key = q/{displaymath}v=frac{{mitDelta}x}{{mitDelta}t},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="\begin{displaymath}
v = \frac{{\mit\Delta}x}{{\mit\Delta} t},
\end{displaymath}">|; 

$key = q/{displaymath}z=x+i,y,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1109.png"
 ALT="\begin{displaymath}
z = x + i y,
\end{displaymath}">|; 

$key = q/2.3,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1461.png"
 ALT="$2.3 {\rm s}$">|; 

$key = q/m=1000,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1742.png"
 ALT="$m=1000 {\rm kg}$">|; 

$key = q/J';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1408.png"
 ALT="$J'$">|; 

$key = q/G,M_oplusslashr^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1954.png"
 ALT="$G M_\oplus/r^2$">|; 

$key = q/displaystylev_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="$\displaystyle v_g$">|; 

$key = q/58.56^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$58.56^\circ$">|; 

$key = q/{displaymath}l=m,r,u.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img2032.png"
 ALT="\begin{displaymath}
l = m r u.
\end{displaymath}">|; 

$key = q/dxslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$dx/dt$">|; 

$key = q/E_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img695.png"
 ALT="$E_1$">|; 

$key = q/{displaymath}f=frac{P}{v}=frac{160times10^3}{35}=4.57times10^3,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img779.png"
 ALT="\begin{displaymath}
f = \frac{P}{v} = \frac{160\times 10^3}{35} = 4.57\times 10^3 {\rm N}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{coord.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="444" HEIGHT="174" BORDER="0"
 SRC="|."$dir".q|img608.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{coord.eps}}
\end{figure}">|; 

$key = q/displaystyle0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img942.png"
 ALT="$\displaystyle 0.$">|; 

$key = q/T=m,v^2slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1081.png"
 ALT="$T=m v^2/r$">|; 

$key = q/r=60,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1187.png"
 ALT="$r=60 {\rm m}$">|; 

$key = q/{displaymath}a=frac{h^2}{2,G,M_odot}left(frac{1}{1+e}+frac{1}{1-e}right)=frac{h^2}{G,M_odot,(1-e^2)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="384" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img2012.png"
 ALT="\begin{displaymath}
a = \frac{h^2}{2 G M_\odot}\left(\frac{1}{1+e}+\frac{1}{1-e}\right) = \frac{h^2}{G M_\odot (1-e^2)}.
\end{displaymath}">|; 

$key = q/l=100,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1743.png"
 ALT="$l=100 {\rm m}$">|; 

$key = q/m_2,g,l_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1724.png"
 ALT="$m_2 g l_2$">|; 

$key = q/{displaymath}{{bf{r}_itimes{{bf{f}_{ij}+{{bf{r}_jtimes{{bf{f}_{ji}=({{bf{r}_i-{{bf{r}_j)times{{bf{f}_{ij}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1576.png"
 ALT="\begin{displaymath}
{\bf r}_i\times {\bf f}_{ij}+{\bf r}_j\times {\bf f}_{ji} =
({\bf r}_i-{\bf r}_j)\times {\bf f}_{ij}.
\end{displaymath}">|; 

$key = q/m=120,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2047.png"
 ALT="$m=120 {\rm kg}$">|; 

$key = q/{displaymath}{{bf{r}(t)={{bf{r}_0+{{bf{v}_0,t+frac{1}{2},{{bf{a},t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="\begin{displaymath}
{\bf r}(t) = {\bf r}_0 + {\bf v}_0 t + \frac{1}{2} {\bf a} t^2.
\end{displaymath}">|; 

$key = q/displaystyleX_3-X_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1687.png"
 ALT="$\displaystyle X_3 - X_2$">|; 

$key = q/x_Arightarrow0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img678.png"
 ALT="$x_A\rightarrow 0$">|; 

$key = q/f_2=k_2,x_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1868.png"
 ALT="$f_2=k_2 x_2$">|; 

$key = q/m_1=2,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1011.png"
 ALT="$m_1=2 {\rm kg}$">|; 

$key = q/displaystyle1-mu,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$\displaystyle 1 - \mu,$">|; 

$key = q/P={{bf{f}!cdot!{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1403.png"
 ALT="$P = {\bf f}\!\cdot\!{\bf v}$">|; 

$key = q/T_{{rm{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1082.png"
 ALT="$T_{\rm max}$">|; 

$key = q/mbox{{boldmath{{tau{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1373.png"
 ALT="$\mbox{\boldmath$\tau$}$">|; 

$key = q/theta=35^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img764.png"
 ALT="$\theta=35^\circ$">|; 

$key = q/{displaymath}v=Csqrt{frac{p}{rho}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="\begin{displaymath}
v = C \sqrt{\frac{p}{\rho}}.
\end{displaymath}">|; 

$key = q/O';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img521.png"
 ALT="$O'$">|; 

$key = q/a=0.07,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1843.png"
 ALT="$a=0.07 {\rm m}$">|; 

$key = q/R=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1218.png"
 ALT="$R=0$">|; 

$key = q/{displaymath}T=R=frac{M,g}{2,sintheta},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1662.png"
 ALT="\begin{displaymath}
T = R = \frac{M g}{2 \sin\theta},
\end{displaymath}">|; 

$key = q/{displaymath}f_{12}=-f_{21}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img789.png"
 ALT="\begin{displaymath}
f_{12} = - f_{21}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2.5in{centerline{epsffile{strut.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="383" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img1750.png"
 ALT="\begin{figure*}
\epsfysize =2.5in
\centerline{\epsffile{strut.eps}}
\end{figure*}">|; 

$key = q/{displaymath}T=frac{2,pi,r}{v}=frac{2timespitimes(1.107times10^7)}{6000}=3.22,{{rm{hours}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2029.png"
 ALT="\begin{displaymath}
T = \frac{2 \pi r}{v} = \frac{2\times\pi\times (1.107\times 10^7)}{6000} = 3.22 {\rm hours}.
\end{displaymath}">|; 

$key = q/10^{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$10^{3}$">|; 

$key = q/{mitDelta}phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1394.png"
 ALT="${\mit\Delta}\phi$">|; 

$key = q/displaystyleX_2-X_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1684.png"
 ALT="$\displaystyle X_2 - X_1$">|; 

$key = q/{displaymath}f=k_{{rm{eff},x=k_1,x_1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1872.png"
 ALT="\begin{displaymath}
f = k_{\rm eff}  x = k_1 x_1.
\end{displaymath}">|; 

$key = q/[M][L^2]slash[T^2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$[M][L^2]/[T^2]$">|; 

$key = q/m_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img818.png"
 ALT="$m_w$">|; 

$key = q/{displaymath}W'=-f',y.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img774.png"
 ALT="\begin{displaymath}
W' = -f' y.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{torque.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="338" BORDER="0"
 SRC="|."$dir".q|img1372.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{torque.eps}}
\end{figure}">|; 

$key = q/{displaymath}a=sqrt{frac{2,E}{k}}=sqrt{frac{2times2.5}{1289.1}}=0.06228,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="292" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1885.png"
 ALT="\begin{displaymath}
a = \sqrt{\frac{2 E}{k}} = \sqrt{\frac{2\times 2.5}{1289.1}} = 0.06228 {\rm m}.
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{tau{}equiv{{bf{r}times{{bf{f}=I,mbox{{boldmath{{alpha{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1400.png"
 ALT="$\mbox{\boldmath$\tau$}\equiv{\bf r}\times {\bf f}= I \mbox{\boldmath$\alpha$}$">|; 

$key = q/{displaymath}M,dot{v}=M,g,sintheta-f,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1440.png"
 ALT="\begin{displaymath}
M \dot{v} = M g \sin\theta - f,
\end{displaymath}">|; 

$key = q/{displaymath}R,costheta=m,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1190.png"
 ALT="\begin{displaymath}
R \cos\theta = m g.
\end{displaymath}">|; 

$key = q/{displaymath}P={{bf{f}!cdot!{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1380.png"
 ALT="\begin{displaymath}
P = {\bf f}\!\cdot\!{\bf v}.
\end{displaymath}">|; 

$key = q/displaystyler,ddot{theta}+2,dot{r},dot{theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1969.png"
 ALT="$\displaystyle r \ddot{\theta} + 2 \dot{r} \dot{\theta}$">|; 

$key = q/{displaymath}P=4032times(4.448)slash(0.3048)^2=1.93times10^5,{{rm{N}{{rm{m}^{-2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="\begin{displaymath}
P = 4032\times (4.448) / (0.3048)^2 = 1.93\times 10^5 {\rm N}{\rm m}^{-2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{b}{a}=sqrt{1-e^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2010.png"
 ALT="\begin{displaymath}
\frac{b}{a} = \sqrt{1-e^2}.
\end{displaymath}">|; 

$key = q/k_1=1200,{{rm{Nslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1863.png"
 ALT="$k_1= 1200 {\rm N/m}$">|; 

$key = q/f<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img705.png"
 ALT="$f &lt;0$">|; 

$key = q/{displaymath}T=frac{pi,a,b}{hslash2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img2008.png"
 ALT="\begin{displaymath}
T = \frac{\pi a b}{h/2}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{rigid.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="519" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img1236.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{rigid.eps}}
\end{figure}">|; 

$key = q/f(x);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img626.png"
 ALT="$f(x)$">|; 

$key = q/28.41,{{rm{m,s}^{-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$28.41 {\rm m s}^{-1}$">|; 

$key = q/displaystylev_{f1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img931.png"
 ALT="$\displaystyle v_{f1}$">|; 

$key = q/v=1.7,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1003.png"
 ALT="$v=1.7 {\rm m/s}$">|; 

$key = q/{displaymath}K_i=frac{1}{2},m_1,v_{i1}^{,2}=0.5times5times4.5^2=50.63,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1034.png"
 ALT="\begin{displaymath}
K_i = \frac{1}{2} m_1 v_{i1}^{ 2} = 0.5\times 5\times 4.5^2 = 50.63 {\rm J}.
\end{displaymath}">|; 

$key = q/{displaymath}M,g,x,costheta-S,l,sintheta=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1667.png"
 ALT="\begin{displaymath}
M g x \cos\theta - S l \sin\theta = 0.
\end{displaymath}">|; 

$key = q/mbox{{boldmath{{omega{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1258.png"
 ALT="$\mbox{\boldmath$\omega$}$">|; 

$key = q/{displaymath}omega=sqrt{frac{g}{l,costheta}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1096.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{g}{l \cos\theta}}.
\end{displaymath}">|; 

$key = q/F,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img731.png"
 ALT="$F \sin\theta$">|; 

$key = q/displaystyledot{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1447.png"
 ALT="$\displaystyle \dot{v}$">|; 

$key = q/120,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$120 {\rm m}$">|; 

$key = q/{figure}{epsfysize{=5.5in{centerline{epsffile{solarsystem.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="631" HEIGHT="631" BORDER="0"
 SRC="|."$dir".q|img1903.png"
 ALT="\begin{figure}
\epsfysize =5.5in
\centerline{\epsffile{solarsystem.eps}}
\end{figure}">|; 

$key = q/-(1slash2),m,v^2=-m,g,h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img593.png"
 ALT="$-(1/2) m v^2 = -m g h$">|; 

$key = q/{displaymath}T_0=frac{m,v^2}{r}-5,m,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1154.png"
 ALT="\begin{displaymath}
T_0 = \frac{m v^2}{r} -5 m g.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{gal.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img528.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{gal.eps}}
\end{figure}">|; 

$key = q/displaystylev_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1129.png"
 ALT="$\displaystyle v_r$">|; 

$key = q/W=int{{bf{f}!cdot!d{{bf{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1401.png"
 ALT="$W = \int {\bf f}\!\cdot\!d{\bf r}$">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{missile.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="546" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img1596.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{missile.eps}}
\end{figure*}">|; 

$key = q/{displaymath}I'=I+M,d^2=frac{3times0.8times0.8}{2}+3times0.25times0.25=1.1475,{{rm{kg,m^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="520" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1897.png"
 ALT="\begin{displaymath}
I' = I + M d^2 = \frac{3\times 0.8\times 0.8}{2} + 3\times 0.25\times 0.25 = 1.1475 {\rm kg m^2}.
\end{displaymath}">|; 

$key = q/displaystylesint,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$\displaystyle \sin t,$">|; 

$key = q/dmbox{{boldmath{{phi{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1391.png"
 ALT="$d\mbox{\boldmath$\phi$}$">|; 

$key = q/{displaymath}I=frac{2,W}{omega^2}=frac{2times5500}{3.142^2}=1114.6,{{rm{kg,m^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1471.png"
 ALT="\begin{displaymath}
I = \frac{2 W}{\omega^2} = \frac{2\times 5500}{3.142^2}= 1114.6 {\rm kg m^2}.
\end{displaymath}">|; 

$key = q/displaystylex+y,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$\displaystyle x + y,$">|; 

$key = q/g=G,M_oplusslashR_oplus^{,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1945.png"
 ALT="$g = G M_\oplus/R_\oplus^{ 2}$">|; 

$key = q/displaystylefrac{sqrt{3},m,g}{2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$\displaystyle \frac{\sqrt{3} m g}{2},$">|; 

$key = q/{displaymath}E=m,c^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="\begin{displaymath}
E = m c^2.
\end{displaymath}">|; 

$key = q/omega(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1240.png"
 ALT="$\omega(t)$">|; 

$key = q/displaystyle1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\displaystyle 1$">|; 

$key = q/displaystylefrac{(m_1-m_2)}{m_1+m_2},v_{i1}+frac{2,m_2}{m_1+m_2},v_{i2},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="246" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img932.png"
 ALT="$\displaystyle \frac{(m_1-m_2)}{m_1+m_2} v_{i1} + \frac{2 m_2}{m_1+m_2} v_{i2},$">|; 

$key = q/=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$=$">|; 

$key = q/{displaymath}{{bf{v}'=frac{d{{bf{r}'}{dt}={{bf{v}-{{bf{v}_0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img525.png"
 ALT="\begin{displaymath}
{\bf v}' = \frac{ d{\bf r}'}{dt} = {\bf v} - {\bf v}_0.
\end{displaymath}">|; 

$key = q/{displaymath}dot{v}=frac{2}{3},g,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1451.png"
 ALT="\begin{displaymath}
\dot{v} = \frac{2}{3} g \sin\theta,
\end{displaymath}">|; 

$key = q/(x,y,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$(x,y,z)$">|; 

$key = q/dot{{bf{e}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1108.png"
 ALT="$\dot{\bf e}_r$">|; 

$key = q/{displaymath}{v'}^2=4,g,l,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1228.png"
 ALT="\begin{displaymath}
{v'}^2 = 4 g l,
\end{displaymath}">|; 

$key = q/displaystyleb,omega_0-2,mu,g,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1519.png"
 ALT="$\displaystyle b \omega_0 - 2 \mu g t,$">|; 

$key = q/{figure*}{epsfysize{=2.5in{centerline{epsffile{tworod.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1707.png"
 ALT="\begin{figure*}
\epsfysize =2.5in
\centerline{\epsffile{tworod.eps}}
\end{figure*}">|; 

$key = q/{displaymath}bar{v}=frac{{mitDelta}x}{{mitDelta}t}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="\begin{displaymath}
\bar{v} = \frac{{\mit\Delta}x}{{\mit\Delta} t}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{2},m,v_1^{,2}+m,g,h_1=frac{1}{2},m,v_2^{,2}+m,g,h_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img580.png"
 ALT="\begin{displaymath}
\frac{1}{2} m v_1^{ 2} + m g h_1 = \frac{1}{2} m v_2^{ 2} + m g h_2.
\end{displaymath}">|; 

$key = q/{displaymath}0=z_0+v_0,t-frac{1}{2},g,t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="\begin{displaymath}
0 = z_0 + v_0 t - \frac{1}{2} g t^2.
\end{displaymath}">|; 

$key = q/F_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img791.png"
 ALT="$F_2$">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}=sum_{i,j=1,N}^{ineqj}{{bf{r}_itimes{{bf{f}_{ij}+sum_{i=1,N}{{bf{r}_itimes{{bf{F}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="62" BORDER="0"
 SRC="|."$dir".q|img1573.png"
 ALT="\begin{displaymath}
\frac{d {\bf L}}{dt} = \sum_{i,j = 1,N}^{i\neq j} {\bf r}_i\times {\bf f}_{ij} +
\sum_{i=1,N} {\bf r}_i\times {\bf F}_i.
\end{displaymath}">|; 

$key = q/displaystylev_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1796.png"
 ALT="$\displaystyle v_0$">|; 

$key = q/{displaymath}|z|=sqrt{x^2+y^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img1113.png"
 ALT="\begin{displaymath}
\vert z\vert = \sqrt{x^2 + y^2}.
\end{displaymath}">|; 

$key = q/displaystyle-2,z.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle -2 z.$">|; 

$key = q/{displaymath}{mitDelta}U=-m,g,h=-3times9.81times0.43=-12.65,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="374" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img768.png"
 ALT="\begin{displaymath}
{\mit\Delta}U = -m g h = - 3\times 9.81 \times 0.43 = -12.65 {\rm J}.
\end{displaymath}">|; 

$key = q/{displaymath}omega=2,pi,f=2timespitimes1.833=5.341,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1890.png"
 ALT="\begin{displaymath}
\omega = 2 \pi f = 2\times \pi\times 1.833 = 5.341 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2},I',omega^2=0.5times1.44times1.047^2=0.789,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="359" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1479.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} I' \omega^2 = 0.5\times 1.44\times 1.047^2 = 0.789 {\rm J}.
\end{displaymath}">|; 

$key = q/10^{-9};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$10^{-9}$">|; 

$key = q/{displaymath}frac{d{{bf{l}}{dt}=frac{{{bf{p}times{{bf{p}}{m}+{{bf{r}times{{bf{f}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1532.png"
 ALT="\begin{displaymath}
\frac{d{\bf l}}{dt} = \frac{{\bf p} \times {\bf p}}{m} + {\bf r} \times {\bf f}.
\end{displaymath}">|; 

$key = q/{displaymath}x=x_1+x_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1866.png"
 ALT="\begin{displaymath}
x = x_1 + x_2.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}=(x,,y)=(r,cosphi,,r,sinphi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="\begin{displaymath}
{\bf r} = (x, y) = (r \cos\phi,  r \sin\phi).
\end{displaymath}">|; 

$key = q/W'=-K;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img775.png"
 ALT="$W'=-K$">|; 

$key = q/x(t_B)=x_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img636.png"
 ALT="$x(t_B)=x_B$">|; 

$key = q/{displaymath}r=frac{1}{A,cos(theta-theta_0)+G,M_odotslashh^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1993.png"
 ALT="\begin{displaymath}
r = \frac{1}{A \cos(\theta-\theta_0) + G M_\odot/h^2}.
\end{displaymath}">|; 

$key = q/displaystylefrac{dx}{dt},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$\displaystyle \frac{dx}{dt},$">|; 

$key = q/x=(1-3,y)=-1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1023.png"
 ALT="$x=
(1-3 y)=-1/2$">|; 

$key = q/{{bf{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="${\bf u}$">|; 

$key = q/sin2,theta=2,sintheta,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1661.png"
 ALT="$\sin 2 \theta
= 2 \sin\theta \cos\theta$">|; 

$key = q/x_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img820.png"
 ALT="$x_w$">|; 

$key = q/{{bf{F}={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img849.png"
 ALT="${\bf F} = {\bf0}$">|; 

$key = q/displaystyleL_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1554.png"
 ALT="$\displaystyle L_y$">|; 

$key = q/X_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1677.png"
 ALT="$X_2$">|; 

$key = q/d(costheta)slashdtheta=-sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1771.png"
 ALT="$d(\cos\theta)/d\theta = -\sin\theta$">|; 

$key = q/E=a^2,kslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1884.png"
 ALT="$E= a^2 k/2$">|; 

$key = q/U=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1949.png"
 ALT="$U=0$">|; 

$key = q/10^{-3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$10^{-3}$">|; 

$key = q/R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$R$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{rotn.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="345" BORDER="0"
 SRC="|."$dir".q|img1281.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{rotn.eps}}
\end{figure}">|; 

$key = q/M=1.3,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1221.png"
 ALT="$M=1.3 {\rm kg}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{v.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="266" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{v.eps}}
\end{figure}">|; 

$key = q/{v'}^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="21" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1157.png"
 ALT="${v'}^2$">|; 

$key = q/{displaymath}v^2=v_0^{,2}+2,{{bf{a}!cdot!{{bf{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img601.png"
 ALT="\begin{displaymath}
v^2 = v_0^{ 2} + 2 {\bf a}\!\cdot\!{\bf s}.
\end{displaymath}">|; 

$key = q/u_0=54,{{rm{kmslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2038.png"
 ALT="$u_0 = 54 {\rm km/s}$">|; 

$key = q/{mitDelta}K=-{mitDelta}U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img755.png"
 ALT="${\mit\Delta}K = -{\mit\Delta} U$">|; 

$key = q/{{bf{f}_i!cdot!{mitDelta}{{bf{r}_i=-|{{bf{f}_i|,|{mitDelta}{{bf{r}_i|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img657.png"
 ALT="${\bf f}_i\!\cdot \! {\mit\Delta}{\bf r}_i
= - \vert{\bf f}_i\vert \vert{\mit\Delta}{\bf r}_i\vert$">|; 

$key = q/{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="${\bf a}$">|; 

$key = q/W;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$W$">|; 

$key = q/{displaymath}T=frac{2,pi}{omega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1777.png"
 ALT="\begin{displaymath}
T = \frac{2 \pi}{\omega}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{twospring.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="467" HEIGHT="171" BORDER="0"
 SRC="|."$dir".q|img1865.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{twospring.eps}}
\end{figure*}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{a.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="294" HEIGHT="271" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{a.eps}}
\end{figure}">|; 

$key = q/{{bf{f}={{bf{f}_{ab}+{{bf{f}_{ba};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="${\bf f} = {\bf f}_{ab}+{\bf f}_{ba}$">|; 

$key = q/|{{bf{s}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="$\vert{\bf s}\vert$">|; 

$key = q/{displaymath}{{bf{v}=mbox{{boldmath{{omega{}times{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1381.png"
 ALT="\begin{displaymath}
{\bf v} = \mbox{\boldmath$\omega$}\times {\bf r},
\end{displaymath}">|; 

$key = q/displaystylex_1+x_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$\displaystyle x_1+x_2,$">|; 

$key = q/{displaymath}frac{v^2}{r}=a_theta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1203.png"
 ALT="\begin{displaymath}
\frac{v^2}{r} = a_\theta,
\end{displaymath}">|; 

$key = q/theta_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img963.png"
 ALT="$\theta_i$">|; 

$key = q/1,{{rm{ft}=12,{{rm{in};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$1 {\rm ft} = 12 {\rm in}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{polar.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1100.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{polar.eps}}
\end{figure}">|; 

$key = q/theta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1809.png"
 ALT="$\theta=0$">|; 

$key = q/EP;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1901.png"
 ALT="$EP$">|; 

$key = q/t=12;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$t=12$">|; 

$key = q/{displaymath}omega=2,pi,f=418.88,{{rm{rad.slashsec}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1845.png"
 ALT="\begin{displaymath}
\omega = 2 \pi f = 418.88 {\rm rad./sec}.
\end{displaymath}">|; 

$key = q/A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$A$">|; 

$key = q/{displaymath}a=frac{deltav}{deltat}=v,frac{deltatheta}{deltat}=v,omega,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1074.png"
 ALT="\begin{displaymath}
a = \frac{\delta v}{\delta t} = v \frac{\delta\theta}{\delta t} = v \omega,
\end{displaymath}">|; 

$key = q/{displaymath}tantheta=frac{r}{h}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1093.png"
 ALT="\begin{displaymath}
\tan\theta = \frac{r}{h}.
\end{displaymath}">|; 

$key = q/i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$i$">|; 

$key = q/m,v^2slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1192.png"
 ALT="$m v^2/r$">|; 

$key = q/{displaymath}z=x,tantheta-frac{1}{2}frac{g,x^2}{v_0^{,2}},sec^2theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="\begin{displaymath}
z =x  \tan\theta -\frac{1}{2}\frac{g x^2}{v_0^{ 2}} \sec^2\theta.
\end{displaymath}">|; 

$key = q/stackrel{displaystylerightarrow}{ZX};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1069.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{ZX}$">|; 

$key = q/{displaymath}v_B=sqrt{frac{2,K_B}{m}}=sqrt{frac{2times5.30times10^4}{300}}=18.8,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img761.png"
 ALT="\begin{displaymath}
v_B = \sqrt{\frac{2 K_B}{m}} = \sqrt{\frac{2\times 5.30\times 10^4}{300}} = 18.8 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{slope1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="496" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img515.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{slope1.eps}}
\end{figure}">|; 

$key = q/v'=v-v_{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img915.png"
 ALT="$v'=v-v_{cm}$">|; 

$key = q/displaystyler,dot{theta}=r,omega,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1132.png"
 ALT="$\displaystyle r \dot{\theta}=r \omega,$">|; 

$key = q/1.93;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$1.93$">|; 

$key = q/displaystyley_1+y_2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$\displaystyle y_1 + y_2,$">|; 

$key = q/displaystylef_{12}+F_1,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img794.png"
 ALT="$\displaystyle f_{12} + F_1,$">|; 

$key = q/W_1neqW_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img650.png"
 ALT="$W_1\neq W_2$">|; 

$key = q/cos(-theta)=costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1798.png"
 ALT="$\cos(-\theta) =\cos\theta$">|; 

$key = q/stackrel{displaystylerightarrow}{PX};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1066.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{PX}$">|; 

$key = q/{displaymath}f=frac{omega}{2,pi}=frac{2.37}{2times3.1415}=0.38,{{rm{Hz}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1214.png"
 ALT="\begin{displaymath}
f = \frac{\omega}{2 \pi} = \frac{2.37}{2\times 3.1415} = 0.38 {\rm Hz}.
\end{displaymath}">|; 

$key = q/n;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1781.png"
 ALT="$n$">|; 

$key = q/{displaymath}{{bf{v}=frac{d{{bf{r}}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img524.png"
 ALT="\begin{displaymath}
{\bf v} = \frac{d{\bf r}}{dt}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{2},(M+m),{v''}^2=frac{1}{2},(M+m),{v'}^2-(M+m),2,g,l.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="393" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1231.png"
 ALT="\begin{displaymath}
\frac{1}{2} (M+m) {v''}^2 = \frac{1}{2} (M+m) {v'}^2 - (M+m) 2 g l.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2},m,dot{x}^2=frac{m,a^2,omega^2,sin^2(omega,t-phi)}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1804.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} m \dot{x}^2 = \frac{m a^2 \omega^2 \sin^2(\omega t-\phi)}{2}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{e51a.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="170" BORDER="0"
 SRC="|."$dir".q|img538.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{e51a.eps}}
\end{figure*}">|; 

$key = q/{figure}{epsfysize{=1.4in{centerline{epsffile{rh.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="160" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="\begin{figure}
\epsfysize =1.4in
\centerline{\epsffile{rh.eps}}
\end{figure}">|; 

$key = q/tau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1369.png"
 ALT="$\tau$">|; 

$key = q/{{bf{v}_g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="${\bf v}_g$">|; 

$key = q/{displaymath}x=x_0+v_0,t+frac{1}{2},a,t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="\begin{displaymath}
x = x_0 + v_0 t + \frac{1}{2} a t^2.
\end{displaymath}">|; 

$key = q/displaystylefrac{dz}{dt}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="$\displaystyle \frac{dz}{dt}.$">|; 

$key = q/L=1.2,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1473.png"
 ALT="$L=1.2 {\rm m}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{angmom.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="452" HEIGHT="287" BORDER="0"
 SRC="|."$dir".q|img1528.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{angmom.eps}}
\end{figure}">|; 

$key = q/stackrel{displaystylerightarrow}{QY};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1067.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{QY}$">|; 

$key = q/R=2403,{{rm{km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2016.png"
 ALT="$R= 2403  {\rm km}$">|; 

$key = q/dV=dx,dy,dz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1287.png"
 ALT="$dV = dx dy dz$">|; 

$key = q/{displaymath}{{bf{r}=(x,y)=(-9+0+36,,0+12+0)=(27,12).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="400" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="\begin{displaymath}
{\bf r} = (x,y)= (-9+0+36, 0+12+0) = (27, 12).
\end{displaymath}">|; 

$key = q/displaystyle{{bf{u};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="$\displaystyle {\bf u}$">|; 

$key = q/thetarightarrow90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1674.png"
 ALT="$\theta\rightarrow 90^\circ$">|; 

$key = q/displaystylea_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$\displaystyle a_x$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{spring.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="563" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img673.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{spring.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{r}'={{bf{r}-{{bf{v}_0,t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img523.png"
 ALT="\begin{displaymath}
{\bf r}' = {\bf r} - {\bf v}_0 t.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{krypton.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="517" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img855.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{krypton.eps}}
\end{figure}">|; 

$key = q/{displaymath}sinthetasimeqtheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1823.png"
 ALT="\begin{displaymath}
\sin\theta \simeq \theta.
\end{displaymath}">|; 

$key = q/{displaymath}phi(t)=phi_0+omega_0,t+frac{1}{2},alpha,t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1250.png"
 ALT="\begin{displaymath}
\phi(t) = \phi_0 + \omega_0 t + \frac{1}{2} \alpha t^2.
\end{displaymath}">|; 

$key = q/{displaymath}m,u=(M+m),v,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1006.png"
 ALT="\begin{displaymath}
m u = (M+m) v,
\end{displaymath}">|; 

$key = q/{displaymath}r=frac{v^2}{g}=frac{(500times1000slash3600)^2}{9.81}=1.97,{{rm{km}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="333" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1219.png"
 ALT="\begin{displaymath}
r = \frac{v^2}{g} = \frac{(500\times 1000/3600)^2}{9.81} = 1.97 {\rm km}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{r}_{cm}=frac{1}{M}int!int!intrho,{{bf{r},dV.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1286.png"
 ALT="\begin{displaymath}
{\bf r}_{cm} = \frac{1}{M} \int\!\int\!\int \rho {\bf r} dV.
\end{displaymath}">|; 

$key = q/{displaymath}dot{v}_{{rm{solid}=frac{2}{3},g,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1457.png"
 ALT="\begin{displaymath}
\dot{v}_{\rm solid} = \frac{2}{3} g \sin\theta.
\end{displaymath}">|; 

$key = q/{displaymath}dot{x}=-a,omega,sin(omega,t-phi).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1853.png"
 ALT="\begin{displaymath}
\dot{x} =-a \omega \sin(\omega t -\phi).
\end{displaymath}">|; 

$key = q/{displaymath}I'=M,frac{int!int!int[(x-d)^2+y^2],,dx,dy,dz}{int!int!int,dx,dy,dz}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1342.png"
 ALT="\begin{displaymath}
I' = M \frac{\int\!\int\!\int [(x-d)^2+y^2]  dx dy dz}
{\int\!\int\!\int  dx dy dz}.
\end{displaymath}">|; 

$key = q/sintheta=1slash20;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img777.png"
 ALT="$\sin\theta = 1/20$">|; 

$key = q/v=v_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img579.png"
 ALT="$v=v_2$">|; 

$key = q/x=hslashsintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img770.png"
 ALT="$x= h/\sin\theta$">|; 

$key = q/x=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$x=0$">|; 

$key = q/|x|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$\vert x\vert$">|; 

$key = q/7.495times10^9,{{rm{J};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2049.png"
 ALT="$7.495\times 10^9 {\rm J}$">|; 

$key = q/5.3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1460.png"
 ALT="$5.3$">|; 

$key = q/{bfe}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1122.png"
 ALT="${\bf e}_\theta$">|; 

$key = q/0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1417.png"
 ALT="$0$">|; 

$key = q/stackrel{displaystylerightarrow}{PZ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1068.png"
 ALT="$\stackrel{\displaystyle \rightarrow}{PZ}$">|; 

$key = q/{displaymath}W=f,x.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img599.png"
 ALT="\begin{displaymath}
W = f x.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{F}=sum_{i=1,N}{{bf{F}_i{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1617.png"
 ALT="\begin{displaymath}
{\bf F} = \sum_{i=1,N} {\bf F}_i
\end{displaymath}">|; 

$key = q/displaystylev_{f2}=v_{i1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img937.png"
 ALT="$\displaystyle v_{f2} = v_{i1}.$">|; 

$key = q/{displaymath}P=m,v_b+M,v_r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img830.png"
 ALT="\begin{displaymath}
P = m v_b + M v_r.
\end{displaymath}">|; 

$key = q/X_2=X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1697.png"
 ALT="$X_2=X$">|; 

$key = q/hslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2006.png"
 ALT="$h/2$">|; 

$key = q/{displaymath}W=F,costheta,x=95timescos15^circtimes6=550.6,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img732.png"
 ALT="\begin{displaymath}
W = F \cos\theta x = 95\times \cos 15^\circ\times 6 = 550.6 {\rm J}.
\end{displaymath}">|; 

$key = q/F=F_1+F_2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img808.png"
 ALT="$F=F_1+F_2=0$">|; 

$key = q/dsigma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1352.png"
 ALT="$d\sigma$">|; 

$key = q/136.5^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="$136.5^\circ$">|; 

$key = q/{displaymath}P=omega,tau=314.12times500=1.57times10^5,{{rm{W}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1504.png"
 ALT="\begin{displaymath}
P = \omega \tau = 314.12\times 500 = 1.57\times 10^5 {\rm W}.
\end{displaymath}">|; 

$key = q/{{bf{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="${\bf p}$">|; 

$key = q/(x_2,y_2,z_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="$(x_2, y_2, z_2)$">|; 

$key = q/{displaymath}W=int{{bf{f}!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1392.png"
 ALT="\begin{displaymath}
W = \int {\bf f}\!\cdot\!d {\bf r}.
\end{displaymath}">|; 

$key = q/v=40,{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1188.png"
 ALT="$v=40 {\rm km/h}$">|; 

$key = q/f_{12}=-f_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img910.png"
 ALT="$f_{12}=-f_{21}$">|; 

$key = q/[L]slash[T];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$[L]/[T]$">|; 

$key = q/d{{bf{r}=(dx,0,0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img630.png"
 ALT="$d{\bf r}=(dx,0,0)$">|; 

$key = q/W=m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="$W=m g$">|; 

$key = q/{displaymath}deltav=2,v,sin(deltathetaslash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="150" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1071.png"
 ALT="\begin{displaymath}
\delta v = 2 v \sin(\delta\theta/2).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{ladder.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="459" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1663.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{ladder.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{P}=sum_{i=1}^Nm_i,dot{{bf{r}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img852.png"
 ALT="\begin{displaymath}
{\bf P} = \sum_{i=1}^N m_i \dot{\bf r}_i.
\end{displaymath}">|; 

$key = q/x_B;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img628.png"
 ALT="$x_B$">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{e52.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="170" BORDER="0"
 SRC="|."$dir".q|img541.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{e52.eps}}
\end{figure*}">|; 

$key = q/displaystylefrac{l}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1650.png"
 ALT="$\displaystyle \frac{l}{2}.$">|; 

$key = q/{displaymath}{{bf{L}=I,mbox{{boldmath{{omega{}?{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1546.png"
 ALT="\begin{displaymath}
{\bf L} = I  \mbox{\boldmath$\omega$}?
\end{displaymath}">|; 

$key = q/{displaymath}(x_1,y_1)=(0,l_1slash2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1708.png"
 ALT="\begin{displaymath}
(x_1, y_1) = (0, l_1/2).
\end{displaymath}">|; 

$key = q/{displaymath}L=m,v,r,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1597.png"
 ALT="\begin{displaymath}
L = m v r \sin\theta,
\end{displaymath}">|; 

$key = q/{displaymath}L_k={{bf{L}cdot{{bf{k}=omega,sum_{i=1,N}m_i,{{bf{k}cdot{{bf{r}_itimes({{bf{k}times{{bf{r}_i).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1540.png"
 ALT="\begin{displaymath}
L_k = {\bf L}\cdot {\bf k} = \omega  \sum_{i=1,N} m_i 
{\bf k}\cdot {\bf r}_i\times ({\bf k}\times {\bf r}_i).
\end{displaymath}">|; 

$key = q/{displaymath}v^2=frac{4}{3},g,h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1435.png"
 ALT="\begin{displaymath}
v^2 = \frac{4}{3}  g h.
\end{displaymath}">|; 

$key = q/{displaymath}x=a,cos(omega,t-phi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1770.png"
 ALT="\begin{displaymath}
x = a \cos(\omega t-\phi),
\end{displaymath}">|; 

$key = q/W_1=W_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img649.png"
 ALT="$W_1=W_2$">|; 

$key = q/X_1=X_2=X_3=73.58,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1764.png"
 ALT="$X_1=X_2=X_3=
73.58 {\rm N}$">|; 

$key = q/{displaymath}T=frac{1}{f}=0.02,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1061.png"
 ALT="\begin{displaymath}
T = \frac{1}{f} = 0.02 {\rm s}.
\end{displaymath}">|; 

$key = q/sintheta=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1636.png"
 ALT="$\sin\theta =0$">|; 

$key = q/{displaymath}r,sintheta=d,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1598.png"
 ALT="\begin{displaymath}
r \sin\theta = d,
\end{displaymath}">|; 

$key = q/{displaymath}e=frac{r_1-r_0}{r_1+r_0}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img2002.png"
 ALT="\begin{displaymath}
e = \frac{r_1-r_0}{r_1+r_0}
\end{displaymath}">|; 

$key = q/{displaymath}F>frac{(75+15)times9.81}{0.45}=1.962times10^3,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="313" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img571.png"
 ALT="\begin{displaymath}
F &gt; \frac{(75+15)\times 9.81}{0.45} = 1.962\times 10^3 {\rm N}.
\end{displaymath}">|; 

$key = q/{displaymath}theta=frac{1}{2},sin^{-1}left(frac{R,g}{v_0^{,2}}right)=frac{1}{2},rac{54.86times9.81}{(24.59)^2}right)=31.45^circ.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="\begin{displaymath}
\theta = \frac{1}{2} \sin^{-1}\left(\frac{R g}{v_0^{ 2}}\...
...\left(\frac{54.86\times 9.81}{(24.59)^2}\right) =
31.45^\circ.
\end{displaymath}">|; 

$key = q/{displaymath}L_1=I_1,omega_1,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1608.png"
 ALT="\begin{displaymath}
L_1 = I_1 \omega_1,
\end{displaymath}">|; 

$key = q/F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img505.png"
 ALT="$F$">|; 

$key = q/{displaymath}L_k=omega,sum_{i=1,N}m_i,({{bf{k}times{{bf{r}_i)cdot({{bf{k}times{{=omega,sum_{i=1,N}m_i,|{{bf{k}times{{bf{r}_i|^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="451" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1542.png"
 ALT="\begin{displaymath}
L_k = \omega  \sum_{i=1,N} m_i 
({\bf k}\times {\bf r}_i)\...
...mega  \sum_{i=1,N} m_i 
\vert{\bf k}\times {\bf r}_i\vert^2.
\end{displaymath}">|; 

$key = q/dvslashdt=a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$dv/dt=a$">|; 

$key = q/-omega^2,a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1789.png"
 ALT="$-\omega^2 a$">|; 

$key = q/f>m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1209.png"
 ALT="$f&gt;m g$">|; 

$key = q/sqrt{100slash1}=10;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\sqrt{100/1}=10$">|; 

$key = q/v_{i1}=12,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="105" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1012.png"
 ALT="$v_{i1}= 12 {\rm m/s}$">|; 

$key = q/K=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img689.png"
 ALT="$K=0$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{balloon.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="344" BORDER="0"
 SRC="|."$dir".q|img816.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{balloon.eps}}
\end{figure}">|; 

$key = q/M=6.4,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1481.png"
 ALT="$M=6.4 {\rm kg}$">|; 

$key = q/|u|=0.115,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img974.png"
 ALT="$\vert u\vert= 0.115 {\rm m/s}$">|; 

$key = q/z_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img670.png"
 ALT="$z_0$">|; 

$key = q/displaystyle4.22times10^7,{{rm{m}=6.62,R_oplus.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1959.png"
 ALT="$\displaystyle 4.22\times 10^7 {\rm m} = 6.62 R_\oplus.$">|; 

$key = q/{displaymath}{v'}^2=v^2-2,r,g,(1-costheta).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1148.png"
 ALT="\begin{displaymath}
{v'}^2 = v^2 - 2 r g (1-\cos\theta).
\end{displaymath}">|; 

$key = q/{displaymath}i,{{rm{e}^{,i,theta}=-sintheta+i,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1119.png"
 ALT="\begin{displaymath}
i {\rm e}^{ i \theta} = -\sin\theta + i \cos\theta.
\end{displaymath}">|; 

$key = q/displaystylev;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$\displaystyle v$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{paraax.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="569" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img1349.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{paraax.eps}}
\end{figure}">|; 

$key = q/a=d^2xslashdt^2=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$a=d^2 x/dt^2=0$">|; 

$key = q/{{bf{r}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="${\bf r}_1$">|; 

$key = q/omega=deltathetaslashdeltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1075.png"
 ALT="$\omega = \delta \theta/\delta t$">|; 

$key = q/{displaymath}lim_{deltatrightarrow0}frac{deltaA}{deltat}=frac{1}{2},r^2,lim_{deleta}{deltat}=frac{r^2,dot{theta}}{2}=frac{h}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1984.png"
 ALT="\begin{displaymath}
\lim_{\delta t\rightarrow 0}\frac{\delta A}{\delta t} = \fra...
...\theta}{\delta t}
= \frac{r^2 \dot{\theta}}{2} = \frac{h}{2}.
\end{displaymath}">|; 

$key = q/R_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2019.png"
 ALT="$R_\oplus$">|; 

$key = q/{displaymath}R=frac{v_0^{,2}}{g},sin2theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="\begin{displaymath}
R= \frac{v_0^{ 2}}{g} \sin 2\theta.
\end{displaymath}">|; 

$key = q/displaystyle(300,cos120^circ,,300,sin120^circ)=(-1.500times10^2,,2.598times10^2),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$\displaystyle (300 \cos 120^\circ, 300 \sin 120^\circ) = (-1.500\times 10^2,  2.598\times 10^2),$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{potential.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img691.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{potential.eps}}
\end{figure}">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}={{bf{r}times{{bf{f}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1378.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}= {\bf r}\times {\bf f}.
\end{displaymath}">|; 

$key = q/displaystylefrac{m_2}{m_1+m_2},g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="$\displaystyle \frac{m_2}{m_1+m_2} g.$">|; 

$key = q/displaystyleT_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$\displaystyle T_2$">|; 

$key = q/h=100,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$h=100 {\rm m}$">|; 

$key = q/d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1337.png"
 ALT="$d$">|; 

$key = q/PSP';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1982.png"
 ALT="$PSP'$">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}'=sum_{i=1,N}{{bf{r}_itimes{{bf{F}_i-{{bf{r}_0ht)=mbox{{boldmath{{tau{}+{{bf{r}_0times{{bf{F}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="407" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img1625.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}' = \sum_{i=1,N} {\bf r}_i\times {\bf ...
...F}_i\right)
=\mbox{\boldmath$\tau$}+ {\bf r}_0\times {\bf F} .
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{2,({mitDelta}x-v_1,{mitDelta}t)}{({mitDelta}t)^2}=frac{2,(1200)times5.082)}{(5.082)^2}=-2.73,{{rm{m,s^{-2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="605" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="\begin{displaymath}
a = \frac{2 ({\mit\Delta}x - v_1 {\mit\Delta} t)}{({\mit\D...
...1000/3600)\times 5.082)}{(5.082)^2} = -2.73 
{\rm m s^{-2}}.
\end{displaymath}">|; 

$key = q/{displaymath}a_r=-r,omega^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1142.png"
 ALT="\begin{displaymath}
a_r = -r \omega^2.
\end{displaymath}">|; 

$key = q/{displaymath}omega=3000times2,pislash60=314.12,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1503.png"
 ALT="\begin{displaymath}
\omega = 3000\times 2 \pi/ 60= 314.12 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/M,u+m,v=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img977.png"
 ALT="$M u + m v=0$">|; 

$key = q/U=-G,M,mslashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1947.png"
 ALT="$U= - G M m/r$">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{e54.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="172" BORDER="0"
 SRC="|."$dir".q|img559.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{e54.eps}}
\end{figure*}">|; 

$key = q/displaystyleddot{r}-r,dot{theta}^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1967.png"
 ALT="$\displaystyle \ddot{r} - r \dot{\theta}^2$">|; 

$key = q/{displaymath}I={mitDelta}p=11.55,{{rm{N,s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img989.png"
 ALT="\begin{displaymath}
I = {\mit\Delta} p = 11.55 {\rm N s}.
\end{displaymath}">|; 

$key = q/{{bf{a}!cdot!{{bf{b}=(a_x,b_x+a_y,b_y+a_z,b_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img605.png"
 ALT="${\bf a}\!\cdot\!{\bf b}= (a_x b_x+ a_y b_y+a_z b_z)$">|; 

$key = q/{displaymath}a=frac{T}{m_1}-g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img497.png"
 ALT="\begin{displaymath}
a = \frac{T}{m_1} - g.
\end{displaymath}">|; 

$key = q/{{bf{b}times{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1277.png"
 ALT="${\bf b}\times {\bf a}$">|; 

$key = q/displaystylev_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="$\displaystyle v_x$">|; 

$key = q/W=5500,{{rm{J};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1468.png"
 ALT="$W=5500 {\rm J}$">|; 

$key = q/{displaymath}a=frac{{mitDelta}v}{{mitDelta}t}=frac{v(t=2)-v(t=0)}{2-0}=frac{8-0}{2}=4,{{rm{m,s^{-2}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="\begin{displaymath}
a = \frac{{\mit\Delta}v}{{\mit\Delta} t} = \frac{v(t=2)-v(t=0)}{2-0} =
\frac{8-0}{2} = 4 {\rm m s^{-2}}.
\end{displaymath}">|; 

$key = q/{displaymath}E=K+U=frac{a^2,k}{2},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1806.png"
 ALT="\begin{displaymath}
E = K + U = \frac{a^2 k}{2},
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=2.5in{centerline{epsffile{ladder1.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="502" HEIGHT="288" BORDER="0"
 SRC="|."$dir".q|img1735.png"
 ALT="\begin{figure*}
\epsfysize =2.5in
\centerline{\epsffile{ladder1.eps}}
\end{figure*}">|; 

$key = q/f_1=f_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1869.png"
 ALT="$f_1 = f_2$">|; 

$key = q/h_1=25,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img749.png"
 ALT="$h_1=25 {\rm m}$">|; 

$key = q/U=m,g,z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img668.png"
 ALT="$U=m g z$">|; 

$key = q/M=5,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1600.png"
 ALT="$M=5 {\rm kg}$">|; 

$key = q/{mitDelta}xslash{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="${\mit\Delta}x/{\mit\Delta} t$">|; 

$key = q/gslashg_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1933.png"
 ALT="$g/g_\oplus$">|; 

$key = q/{figure}{epsfysize{=1.7in{centerline{epsffile{circle.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="194" BORDER="0"
 SRC="|."$dir".q|img1040.png"
 ALT="\begin{figure}
\epsfysize =1.7in
\centerline{\epsffile{circle.eps}}
\end{figure}">|; 

$key = q/f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="$f$">|; 

$key = q/{displaymath}U(infty)-U(r)=-int_{r}^infty[-f(r)],dr.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="267" HEIGHT="42" BORDER="0"
 SRC="|."$dir".q|img1936.png"
 ALT="\begin{displaymath}
U(\infty) - U(r) = -\int_{r}^\infty [-f(r)] dr.
\end{displaymath}">|; 

$key = q/{displaymath}r_0=frac{h^2}{G,M_odot,(1+e)}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img1996.png"
 ALT="\begin{displaymath}
r_0 = \frac{h^2}{G M_\odot (1+e)}.
\end{displaymath}">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{wedge.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="169" BORDER="0"
 SRC="|."$dir".q|img784.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{wedge.eps}}
\end{figure*}">|; 

$key = q/costheta_0=2slash3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1184.png"
 ALT="$\cos\theta_0 = 2/3$">|; 

$key = q/10^{-12};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$10^{-12}$">|; 

$key = q/{{bf{a}!cdot!(lambda{{bf{s})=lambda({{bf{a}!cdot{{bf{s});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="${\bf a}\!\cdot\!(\lambda {\bf s}) = \lambda({\bf a}\!\cdot{\bf s})$">|; 

$key = q/m=2.3times10^4,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1593.png"
 ALT="$m=2.3\times 10^4 {\rm kg}$">|; 

$key = q/{displaymath}R=M,g+m,g-S=(5000+1000)times9.81-2.13times10^4=3.76times10^4,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="553" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1748.png"
 ALT="\begin{displaymath}
R = M g + m g - S = (5000+1000)\times 9.81- 2.13\times 10^4 = 3.76\times 10^4 {\rm N}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1in{centerline{epsffile{twomass.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="478" HEIGHT="115" BORDER="0"
 SRC="|."$dir".q|img792.png"
 ALT="\begin{figure}
\epsfysize =1in
\centerline{\epsffile{twomass.eps}}
\end{figure}">|; 

$key = q/10^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$10^{-2}$">|; 

$key = q/v_B=(dxslashdt)_{t_B};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img639.png"
 ALT="$v_B = (dx/dt)_{t_B}$">|; 

$key = q/(1slash2),m,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img592.png"
 ALT="$(1/2) m v^2$">|; 

$key = q/F,cos25^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img542.png"
 ALT="$F \cos 25^\circ$">|; 

$key = q/c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$c$">|; 

$key = q/textstyle>;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1649.png"
 ALT="$\textstyle &gt;$">|; 

$key = q/w=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img978.png"
 ALT="$w=0$">|; 

$key = q/displaystyledot{r},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1130.png"
 ALT="$\displaystyle \dot{r},$">|; 

$key = q/15,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img557.png"
 ALT="$15 {\rm kg}$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{rhobus.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="172" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{rhobus.eps}}
\end{figure}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{torsion.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img1810.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{torsion.eps}}
\end{figure}">|; 

$key = q/0.6,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1888.png"
 ALT="$0.6 {\rm m}$">|; 

$key = q/{displaymath}v=b,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1366.png"
 ALT="\begin{displaymath}
v = b \omega.
\end{displaymath}">|; 

$key = q/{{bf{v}=dot{r},{{bf{e}_r+r,dot{theta},{{bf{e}_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1974.png"
 ALT="${\bf v} = \dot{r} {\bf e}_r+r \dot{\theta} 
{\bf e}_\theta$">|; 

$key = q/{displaymath}{{bf{f}=m,{{bf{a},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img520.png"
 ALT="\begin{displaymath}
{\bf f} = m {\bf a},
\end{displaymath}">|; 

$key = q/r=0.8,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1894.png"
 ALT="$r = 0.8 {\rm m}$">|; 

$key = q/m,omega^2=k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1807.png"
 ALT="$m \omega^2 = k$">|; 

$key = q/dot{omega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1445.png"
 ALT="$\dot{\omega}$">|; 

$key = q/v_0=v,sintheta=200timessin30^circ=100,{{rm{m}slash{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="315" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$v_0 = v \sin\theta = 200\times\sin 30^\circ = 100 {\rm m}/{\rm s}$">|; 

$key = q/85,{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$85 {\rm km/h}$">|; 

$key = q/|{{bf{u}|=60,{{rm{mislashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$\vert{\bf u}\vert = 60 {\rm mi/h}$">|; 

$key = q/{displaymath}v_{f1}=frac{v_{i1}}{2,costheta}=frac{4.5}{2timescos30^circ}=2.5981,{{rm{mslashs},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1032.png"
 ALT="\begin{displaymath}
v_{f1} = \frac{v_{i1}}{2 \cos\theta} = \frac{4.5}{2\times\cos 30^\circ} = 2.5981 {\rm m/s},
\end{displaymath}">|; 

$key = q/textstylesimeq;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img940.png"
 ALT="$\textstyle \simeq$">|; 

$key = q/displaystylea,cos(omega,t),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1838.png"
 ALT="$\displaystyle a \cos(\omega t),$">|; 

$key = q/{{bf{f}(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img900.png"
 ALT="${\bf f}(t)$">|; 

$key = q/{displaymath}v=frac{(M+m),sqrt{5,g,l}}{m}=frac{1.31timessqrt{5times9.81times0.6}}{0.01}=710.7,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1235.png"
 ALT="\begin{displaymath}
v = \frac{(M+m) \sqrt{5  g l} }{ m } =
\frac{1.31\times\sqrt{5\times 9.81\times 0.6}}{0.01}
=710.7 {\rm m/s}.
\end{displaymath}">|; 

$key = q/displaystyleL_z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1556.png"
 ALT="$\displaystyle L_z$">|; 

$key = q/v+dv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img864.png"
 ALT="$v+dv$">|; 

$key = q/theta_0=48.19^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1185.png"
 ALT="$\theta_0 = 48.19^\circ$">|; 

$key = q/{displaymath}frac{T_1}{2}-frac{sqrt{3},T_2}{2}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="\begin{displaymath}
\frac{T_1}{2} -\frac{\sqrt{3} T_2}{2} = 0.
\end{displaymath}">|; 

$key = q/2,r,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1173.png"
 ALT="$2 r g$">|; 

$key = q/{displaymath}frac{d{{bf{p}}{dt}={{bf{f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="\begin{displaymath}
\frac{d{\bf p}}{dt} = {\bf f},
\end{displaymath}">|; 

$key = q/{displaymath}f=frac{51}{60}=0.85,{{rm{Hz}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1889.png"
 ALT="\begin{displaymath}
f = \frac{51}{60} = 0.85  {\rm Hz}.
\end{displaymath}">|; 

$key = q/{{bf{f}_{ji};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img838.png"
 ALT="${\bf f}_{ji}$">|; 

$key = q/{displaymath}g=omega^2,l=5.341times5.341times0.6=17.11,{{rm{mslashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1892.png"
 ALT="\begin{displaymath}
g = \omega^2 l = 5.341\times 5.341\times 0.6 = 17.11 {\rm m/s^2}.
\end{displaymath}">|; 

$key = q/x>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1792.png"
 ALT="$x&gt;0$">|; 

$key = q/s<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$s&lt;0$">|; 

$key = q/mbox{{boldmath{{omega{}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1259.png"
 ALT="$\mbox{\boldmath$\omega$}_1$">|; 

$key = q/displaystylefrac{dv_x}{dt}=-sint,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$\displaystyle \frac{dv_x}{dt}=-\sin t,$">|; 

$key = q/0.57,{{rm{AU};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2030.png"
 ALT="$0.57 {\rm AU}$">|; 

$key = q/mu=m_1,m_2slash(m_1+m_2);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img926.png"
 ALT="$\mu = m_1 m_2/(m_1+m_2)$">|; 

$key = q/W_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img647.png"
 ALT="$W_2$">|; 

$key = q/180^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$180^\circ$">|; 

$key = q/{displaymath}I,ddot{theta}=tau,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1812.png"
 ALT="\begin{displaymath}
I \ddot{\theta} = \tau,
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2U}{dx^2}>0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img707.png"
 ALT="\begin{displaymath}
\frac{d^2 U}{dx^2}&gt;0.
\end{displaymath}">|; 

$key = q/v_0=55times5280times0.3048slash3600=24.59,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="$v_0 = 55\times 5280\times 0.3048/3600 = 24.59 {\rm m/s}$">|; 

$key = q/x,y=pm0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1299.png"
 ALT="$x,y=\pm 0$">|; 

$key = q/displaystylea_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1136.png"
 ALT="$\displaystyle a_r$">|; 

$key = q/({{bf{r}_i-{{bf{r}_j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1577.png"
 ALT="$({\bf r}_i-{\bf r}_j)$">|; 

$key = q/d^2slashdt^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img800.png"
 ALT="$d^2/dt^2$">|; 

$key = q/{displaymath}frac{dU}{dx}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img699.png"
 ALT="\begin{displaymath}
\frac{dU}{dx} = 0.
\end{displaymath}">|; 

$key = q/{displaymath}dot{v}=b,dot{omega}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1446.png"
 ALT="\begin{displaymath}
\dot{v} = b \dot{\omega}.
\end{displaymath}">|; 

$key = q/f=f_1=f_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1871.png"
 ALT="$f = f_1 = f_2$">|; 

$key = q/0.15,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1850.png"
 ALT="$0.15 {\rm m}$">|; 

$key = q/displaystylev_0+a,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\displaystyle v_0 + a t,$">|; 

$key = q/{displaymath}omega=frac{0.75}{0.25timessin(53.13^circ)}=3.75,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1861.png"
 ALT="\begin{displaymath}
\omega = \frac{0.75}{0.25\times\sin(53.13^\circ)} = 3.75 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{F-f}{m}=frac{F}{m}-mu,g,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img508.png"
 ALT="\begin{displaymath}
a = \frac{F - f}{m} = \frac{F}{m}-\mu g,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{l}={{bf{r}times{{bf{p}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1525.png"
 ALT="\begin{displaymath}
{\bf l} = {\bf r}\times {\bf p}.
\end{displaymath}">|; 

$key = q/m_2,v_{f2},costheta_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img956.png"
 ALT="$m_2 v_{f2} \cos\theta_2$">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{q21.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="373" HEIGHT="232" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{q21.eps}}
\end{figure*}">|; 

$key = q/Q;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1065.png"
 ALT="$Q$">|; 

$key = q/{displaymath}z_{cm}=frac{int!int!intz,dx,dy,dz}{int!int!int,dx,dy,dz},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1293.png"
 ALT="\begin{displaymath}
z_{cm} = \frac{\int\!\int\!\int z dx dy dz}{\int\!\int\!\int dx dy dz},
\end{displaymath}">|; 

$key = q/{displaymath}omega=sqrt{frac{9.81}{0.2timescos30^circ}}=7.526,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1098.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{9.81}{0.2\times \cos 30^\circ}} = 7.526 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}p=I.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img891.png"
 ALT="\begin{displaymath}
{\mit\Delta}p = I.
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}'={{bf{0}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1627.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}' = {\bf0}.
\end{displaymath}">|; 

$key = q/{displaymath}phi=tan^{-1}left(frac{v_{g,y}}{v_{g,x}}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="\begin{displaymath}
\phi = \tan^{-1} \left(\frac{v_{g y}}{v_{g x}}\right),
\end{displaymath}">|; 

$key = q/{displaymath}K_f=frac{1}{2},(M+m),v^2=0.5times(0.012+5.2)times1.7^2=7.53,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1009.png"
 ALT="\begin{displaymath}
K_f = \frac{1}{2} (M+m) v^2 = 0.5\times(0.012+5.2)\times 1.7^2 = 7.53 {\rm J}.
\end{displaymath}">|; 

$key = q/z_0=50;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="$z_0=50$">|; 

$key = q/9.834,{rmm,s^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$9.834 {\rm m s^{-2}}$">|; 

$key = q/displaystyle1-lambda;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$\displaystyle 1 - \lambda$">|; 

$key = q/I=(1slash12),M,L^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1475.png"
 ALT="$I=(1/12) M L^2$">|; 

$key = q/v_{g,y}=v_g,sinphi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="$v_{g y}=v_g \sin\phi$">|; 

$key = q/displaystyleX_1=X_2=X_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1693.png"
 ALT="$\displaystyle X_1=X_2=X_3$">|; 

$key = q/I_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1609.png"
 ALT="$I_1$">|; 

$key = q/SP';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1979.png"
 ALT="$SP'$">|; 

$key = q/2.73,{{rm{m,s^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$2.73 
{\rm m s^{-2}}$">|; 

$key = q/10^3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$10^3$">|; 

$key = q/h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$h$">|; 

$key = q/{displaymath}I=frac{5.3times1.3^2}{3}=2.986,{{rm{kg,m^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1495.png"
 ALT="\begin{displaymath}
I = \frac{5.3\times 1.3^2}{3} = 2.986 {\rm kg m^2}.
\end{displaymath}">|; 

$key = q/{displaymath}Usimeq-frac{G,M_oplus,m}{R_oplus}+frac{G,M_oplus,m}{R_oplus^{,2}},z,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1943.png"
 ALT="\begin{displaymath}
U \simeq -\frac{G M_\oplus m}{R_\oplus} + \frac{G M_\oplus m}{R_\oplus^{ 2}} z,
\end{displaymath}">|; 

$key = q/{displaymath}f=S=frac{x}{l,tantheta},M,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1669.png"
 ALT="\begin{displaymath}
f = S = \frac{x}{l \tan\theta}  M g.
\end{displaymath}">|; 

$key = q/displaystylefrac{m,g}{2}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$\displaystyle \frac{m g}{2}.$">|; 

$key = q/W=-W';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img726.png"
 ALT="$W=-W'$">|; 

$key = q/17.11,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1893.png"
 ALT="$17.11 {\rm m/s^2}$">|; 

$key = q/{displaymath}S=M,gslash3+m,gslash2=5000times9.81slash3+1000times9.81slash2=2.13times10^4,{{rm{N},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="553" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1747.png"
 ALT="\begin{displaymath}
S = M g/3 + m g/2 = 5000\times 9.81/3 + 1000\times 9.81 /2 = 2.13\times 10^4 {\rm N},
\end{displaymath}">|; 

$key = q/{displaymath}x_{cm}=frac{m_1,x_1+m_2,x_2}{m_1+m_2}=frac{1}{2}frac{m_2,l_2}{m_1+m_2}=frac{3.4times0.7}{2times8.6}=0.138,{{rm{m},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="474" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1710.png"
 ALT="\begin{displaymath}
x_{cm} = \frac{m_1 x_1+ m_2 x_2}{m_1+m_2} = \frac{1}{2} \f...
...{m_1+m_2}=
\frac{3.4\times 0.7}{2\times 8.6} = 0.138 {\rm m},
\end{displaymath}">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1039.png"
 ALT="$\omega$">|; 

$key = q/{{bf{b}=lambda,{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1274.png"
 ALT="${\bf b} = \lambda {\bf a}$">|; 

$key = q/u=3,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img994.png"
 ALT="$u=3 {\rm m/s}$">|; 

$key = q/T_1,cos60^circ=T_1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="$T_1 \cos 60^\circ = T_1/2$">|; 

$key = q/{displaymath}{{bf{f}=sum_i{{bf{f}_i,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img642.png"
 ALT="\begin{displaymath}
{\bf f} = \sum_i {\bf f}_i,
\end{displaymath}">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\textstyle =$">|; 

$key = q/{displaymath}v'=v-v_t=v-b,omega.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1427.png"
 ALT="\begin{displaymath}
v' = v - v_t = v - b \omega.
\end{displaymath}">|; 

$key = q/A=l^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$A=l^2$">|; 

$key = q/{displaymath}K_B=-{mitDelta}U=5.30times10^4,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="222" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img758.png"
 ALT="\begin{displaymath}
K_B = -{\mit\Delta}U = 5.30\times 10^4 {\rm J}.
\end{displaymath}">|; 

$key = q/b=0.4,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1482.png"
 ALT="$b=0.4 {\rm m}$">|; 

$key = q/displaystylev_0-g,t,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\displaystyle v_0 - g t,$">|; 

$key = q/{displaymath}r=sqrt{x^2+y^2+z^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="153" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="\begin{displaymath}
r = \sqrt{x^2+ y^2 + z^2}.
\end{displaymath}">|; 

$key = q/{displaymath}f=G,frac{m_1,m_2}{r^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1915.png"
 ALT="\begin{displaymath}
f = G \frac{m_1 m_2}{r^2}.
\end{displaymath}">|; 

$key = q/-(1slash2),m,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img591.png"
 ALT="$-(1/2) m v^2$">|; 

$key = q/R,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1191.png"
 ALT="$R \sin\theta$">|; 

$key = q/x=+a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1776.png"
 ALT="$x=+a$">|; 

$key = q/{{bf{d};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="${\bf d}$">|; 

$key = q/I=M,b^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1347.png"
 ALT="$I=M b^2$">|; 

$key = q/v=21,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img984.png"
 ALT="$v=21 {\rm m/s}$">|; 

$key = q/f=50,{{rm{Hz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1060.png"
 ALT="$f=50 {\rm Hz}$">|; 

$key = q/{displaymath}v=frac{(M+m),sqrt{4,g,l}}{m}=frac{1.31timessqrt{4times9.81times0.6}}{0.01}=635.6,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="485" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1229.png"
 ALT="\begin{displaymath}
v = \frac{ (M+m) \sqrt{4 g l} }{ m } =
\frac{1.31\times\sqrt{4\times 9.81\times 0.6}}{0.01}
=635.6 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{displaymath}I_z=I_x+I_y.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1327.png"
 ALT="\begin{displaymath}
I_z = I_x + I_y.
\end{displaymath}">|; 

$key = q/{displaymath}f=mu,M,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1512.png"
 ALT="\begin{displaymath}
f = \mu M g.
\end{displaymath}">|; 

$key = q/{displaymath}K_i=frac{1}{2},m,u^2=0.5times0.012times738.4^2=3.2714,{{rm{kJ}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="392" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1008.png"
 ALT="\begin{displaymath}
K_i = \frac{1}{2} m u^2 = 0.5\times 0.012 \times 738.4^2 = 3.2714 {\rm kJ}.
\end{displaymath}">|; 

$key = q/displaystyleleft(frac{lslash2-x_1}{x_2-x_1}right)M,g.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1644.png"
 ALT="$\displaystyle \left(\frac{l/2-x_1}{x_2-x_1}\right)M g.$">|; 

$key = q/l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$l$">|; 

$key = q/displaystyleb,dot{omega};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1514.png"
 ALT="$\displaystyle b \dot{\omega}$">|; 

$key = q/{displaymath}frac{1}{2},m,v^2-frac{1}{2},m,v_0^{,2}=-m,g,{mitDelta}z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img606.png"
 ALT="\begin{displaymath}
\frac{1}{2} m v^2 - \frac{1}{2} m v_0^{ 2} = -m g {\mit\Delta}z.
\end{displaymath}">|; 

$key = q/i,{{rm{e}^{,i,theta};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1118.png"
 ALT="$i {\rm e}^{ i \theta}$">|; 

$key = q/{displaymath}mu,R>m_B,g,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img564.png"
 ALT="\begin{displaymath}
\mu R &gt; m_B g,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{rocket1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="367" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img866.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{rocket1.eps}}
\end{figure}">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{cysurf.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="541" HEIGHT="227" BORDER="0"
 SRC="|."$dir".q|img1509.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{cysurf.eps}}
\end{figure*}">|; 

$key = q/tau=500,{{rm{N,m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1500.png"
 ALT="$\tau=500 {\rm N m}$">|; 

$key = q/displaystyleI_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1325.png"
 ALT="$\displaystyle I_y$">|; 

$key = q/{displaymath}{{bf{v}(t)=lim_{{mitDelta}trightarrow0}frac{{{bf{r}(t+{mitDelta}t)-{{bf{r}(t)}{{mitDelta}t}=frac{d{{bf{r}}{dt}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="\begin{displaymath}
{\bf v}(t) = \lim_{{\mit\Delta} t\rightarrow 0}\frac{{\bf r}...
...\Delta} t)-{\bf r}(t)}{
{\mit\Delta} t} = \frac{d{\bf r}}{dt}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=5.5in{centerline{epsffile{copernicus.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="619" HEIGHT="629" BORDER="0"
 SRC="|."$dir".q|img1904.png"
 ALT="\begin{figure}
\epsfysize =5.5in
\centerline{\epsffile{copernicus.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{1}{2},m_1,v_{i1}^{,2}=frac{1}{2},m_1,v_{f1}^{,2}+frac{1}{2},m_2,v_{f2}^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img961.png"
 ALT="\begin{displaymath}
\frac{1}{2} m_1 v_{i1}^{ 2} =\frac{1}{2} m_1 v_{f1}^{ 2}+\frac{1}{2} m_2 v_{f2}^{ 2}.
\end{displaymath}">|; 

$key = q/v=500,{{rm{kmslashh};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1216.png"
 ALT="$v=500 {\rm km/h}$">|; 

$key = q/m_2=24,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1717.png"
 ALT="$m_2=24 {\rm kg}$">|; 

$key = q/{displaymath}I=frac{2}{5},M,b^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1361.png"
 ALT="\begin{displaymath}
I = \frac{2}{5} M b^2.
\end{displaymath}">|; 

$key = q/tan60^circ=sqrt{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1702.png"
 ALT="$\tan 60^\circ = \sqrt{3}$">|; 

$key = q/90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="$90^\circ$">|; 

$key = q/displaystylea,sintheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1837.png"
 ALT="$\displaystyle a \sin\theta,$">|; 

$key = q/a_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1106.png"
 ALT="$a_\theta$">|; 

$key = q/{displaymath}A=40times4.047times10^3=1.619times10^5,{{rm{m}^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="\begin{displaymath}
A = 40\times 4.047\times 10^3 = 1.619\times 10^5 {\rm m}^2.
\end{displaymath}">|; 

$key = q/0.5,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="$0.5 {\rm N}$">|; 

$key = q/{displaymath}v=sqrt{r,a_theta}=sqrt{85times0.6}=7.14,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1204.png"
 ALT="\begin{displaymath}
v = \sqrt{r  a_\theta} = \sqrt{85\times 0.6} = 7.14 {\rm m/s}.
\end{displaymath}">|; 

$key = q/I_x=I_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1329.png"
 ALT="$I_x=I_y$">|; 

$key = q/{displaymath}omega^2,r=frac{G,M_ast}{r^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img2044.png"
 ALT="\begin{displaymath}
\omega^2 r = \frac{G M_\ast}{r^2}.
\end{displaymath}">|; 

$key = q/150;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img719.png"
 ALT="$150$">|; 

$key = q/{displaymath}P=p_1+p_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img814.png"
 ALT="\begin{displaymath}
P = p_1 + p_2.
\end{displaymath}">|; 

$key = q/p_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img893.png"
 ALT="$p_i$">|; 

$key = q/{displaymath}Y_3,h,sintheta-X_3,h,costheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1762.png"
 ALT="\begin{displaymath}
Y_3 h \sin\theta - X_3 h \cos\theta = 0,
\end{displaymath}">|; 

$key = q/E_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img694.png"
 ALT="$E_0$">|; 

$key = q/{{bf{f}_{ij};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="${\bf f}_{ij}$">|; 

$key = q/m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="$m_2$">|; 

$key = q/displaystylesqrt{(v_{g,x})^2+(v_{g,y})^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="$\displaystyle \sqrt{(v_{g x})^2 + (v_{g y})^2}$">|; 

$key = q/m,u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1004.png"
 ALT="$m u$">|; 

$key = q/{displaymath}I=2,m,d^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1582.png"
 ALT="\begin{displaymath}
I = 2 m d^2.
\end{displaymath}">|; 

$key = q/-m_1,v_{f1},sintheta_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img958.png"
 ALT="$-m_1 v_{f1} \sin\theta_1$">|; 

$key = q/{displaymath}I=M,frac{int!int,(x^2+y^2),dx,dy}{int!int,dx,dy},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1350.png"
 ALT="\begin{displaymath}
I = M \frac{\int\!\int  (x^2+y^2) dx dy}{\int\!\int  dx dy},
\end{displaymath}">|; 

$key = q/l_1=1.3,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1704.png"
 ALT="$l_1=1.3 {\rm m}$">|; 

$key = q/1,{{rm{hr}=3,600,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$1 {\rm hr}
= 3,600 {\rm s}$">|; 

$key = q/omega_z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1548.png"
 ALT="$\omega_z)$">|; 

$key = q/{displaymath}R,sinphi+T,sintheta-M,g=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1653.png"
 ALT="\begin{displaymath}
R \sin\phi + T \sin\theta - M g = 0.
\end{displaymath}">|; 

$key = q/t=2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$t=2$">|; 

$key = q/k_{{rm{eff};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1873.png"
 ALT="$k_{\rm eff}$">|; 

$key = q/m_B,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img563.png"
 ALT="$m_B  g$">|; 

$key = q/U({{bf{r});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img660.png"
 ALT="$U({\bf r})$">|; 

$key = q/l_2=2.3,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1719.png"
 ALT="$l_2=2.3 {\rm m}$">|; 

$key = q/E_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img696.png"
 ALT="$E_2$">|; 

$key = q/l=0.2,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1097.png"
 ALT="$l=0.2 {\rm m}$">|; 

$key = q/419.3,{{rm{J};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img736.png"
 ALT="$ 419.3 {\rm J}$">|; 

$key = q/{displaymath}{{rm{e}^{,i,theta}=costheta+i,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1116.png"
 ALT="\begin{displaymath}
{\rm e}^{ i \theta} = \cos\theta + i \sin\theta.
\end{displaymath}">|; 

$key = q/{displaymath}Y_1+Y_3=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1752.png"
 ALT="\begin{displaymath}
Y_1+Y_3 = 0.
\end{displaymath}">|; 

$key = q/v_{{rm{esc};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1948.png"
 ALT="$v_{\rm esc}$">|; 

$key = q/460,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1911.png"
 ALT="$460 {\rm m/s}$">|; 

$key = q/{displaymath}v_{cm}=dot{x}_{cm}=frac{m_1,dot{x}_1+m_2,dot{x}_2}{m_1+m_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="229" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img809.png"
 ALT="\begin{displaymath}
v_{cm} = \dot{x}_{cm} = \frac{m_1 \dot{x}_1+m_2 \dot{x}_2}{m_1+m_2}.
\end{displaymath}">|; 

$key = q/10^5,{{rm{N},{{rm{m}^{-2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$10^5 {\rm N} {\rm m}^{-2}$">|; 

$key = q/{displaymath}v'=frac{P}{f+W,sintheta}=frac{160times10^3}{4.57times10^3+3000slash20}=33.90,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="53" BORDER="0"
 SRC="|."$dir".q|img783.png"
 ALT="\begin{displaymath}
v' = \frac{P}{f+W \sin\theta} = \frac{160\times 10^3}{4.57\times 10^3 + 3000/20} = 33.90 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{mitDelta}z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img604.png"
 ALT="${\mit\Delta}z)$">|; 

$key = q/displaystyleM,frac{int,int,y^2,dx,dy}{int,int,dx,dy},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1324.png"
 ALT="$\displaystyle M \frac{\int \int  y^2 dx dy}{\int \int dx dy},$">|; 

$key = q/J;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1407.png"
 ALT="$J$">|; 

$key = q/{displaymath}v^2>4,r,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1160.png"
 ALT="\begin{displaymath}
v^2 &gt; 4 r g.
\end{displaymath}">|; 

$key = q/s=-h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$s=-h$">|; 

$key = q/{displaymath}m_1,v_{i1}+m_2,v_{i2},costheta_i=(m_1+m_2),v_f,costheta_f.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="356" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img965.png"
 ALT="\begin{displaymath}
m_1 v_{i1} + m_2 v_{i2} \cos\theta_i = (m_1+m_2) v_f \cos\theta_f.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{complex1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="288" BORDER="0"
 SRC="|."$dir".q|img1123.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{complex1.eps}}
\end{figure}">|; 

$key = q/{figure*}{epsfysize{=1.5in{centerline{epsffile{skater.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="169" BORDER="0"
 SRC="|."$dir".q|img995.png"
 ALT="\begin{figure*}
\epsfysize =1.5in
\centerline{\epsffile{skater.eps}}
\end{figure*}">|; 

$key = q/{displaymath}t=sqrt{frac{2,h}{g}}=sqrt{frac{2times100}{9.81}}=4.515,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="56" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="\begin{displaymath}
t = \sqrt{\frac{2 h}{g}} = \sqrt{\frac{2\times 100}{9.81}} = 4.515 {\rm s}.
\end{displaymath}">|; 

$key = q/displaystylea;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="$\displaystyle a$">|; 

$key = q/{displaymath}int_0^vfrac{dv}{u}=-int_{m_p}^mfrac{dm}{M+m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img873.png"
 ALT="\begin{displaymath}
\int_0^v\frac{dv}{u}= -\int_{m_p}^m\frac{dm}{M+m}.
\end{displaymath}">|; 

$key = q/bar{f};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img898.png"
 ALT="$\bar{f}$">|; 

$key = q/dslashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img798.png"
 ALT="$d/dt$">|; 

$key = q/{displaymath}frac{[L]}{[T]}=left(frac{[M]}{[T^2][L]}right)^xleft(frac{[M]}{[L^3]}right)^y[L^3]^z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="259" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="\begin{displaymath}
\frac{[L]}{[T]} = \left(\frac{[M]}{[T^2][L]}\right)^x\left(
\frac{[M]}{[L^3]}\right)^y [L^3]^z.
\end{displaymath}">|; 

$key = q/{displaymath}[p]=[M][v]=frac{[M][L]}{[T]}={{rm{kg,m,s^{-1}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="\begin{displaymath}[p]= [M][v] = \frac{[M][L]}{[T]} = {\rm kg m s^{-1}}.
\end{displaymath}">|; 

$key = q/{displaymath}phi=tan^{-1}left(frac{v_{g,y}}{v_{g,x}}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="\begin{displaymath}
\phi = \tan^{-1} \left(\frac{v_{g y}}{v_{g x}}\right).
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{b}times{{bf{a}=-{{bf{a}times{{bf{b}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1278.png"
 ALT="\begin{displaymath}
{\bf b}\times {\bf a} = - {\bf a}\times {\bf b}.
\end{displaymath}">|; 

$key = q/{displaymath}0=m_g,{mitDelta}x_g+m_w,{mitDelta}x_w,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img824.png"
 ALT="\begin{displaymath}
0 = m_g {\mit\Delta}x_g+m_w {\mit\Delta}x_w,
\end{displaymath}">|; 

$key = q/displaystyle0.25,omega,sin(phi).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1859.png"
 ALT="$\displaystyle 0.25 \omega \sin(\phi).$">|; 

$key = q/{displaymath}{{bf{f}=-left(frac{partialU}{partialx},frac{partialU}{partialy},frac{partialU}{partialz}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="189" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img683.png"
 ALT="\begin{displaymath}
{\bf f} = -\left(\frac{\partial U}{\partial x}, \frac{\partial U}{\partial y},
\frac{\partial U}{\partial z}\right).
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{slope.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{slope.eps}}
\end{figure}">|; 

$key = q/30;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img718.png"
 ALT="$30$">|; 

$key = q/{displaymath}f=mu_k,R=0.15times465.9=69.89,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img734.png"
 ALT="\begin{displaymath}
f = \mu_k R = 0.15\times 465.9 = 69.89 {\rm N}.
\end{displaymath}">|; 

$key = q/{displaymath}z_{cm}=frac{a^2left[z^2slash2-2,z^3slash(3,h)+z^4slash(4,h^2)right]h}=frac{a^2,h^2slash12}{a^2,hslash3}=frac{h}{4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="465" HEIGHT="60" BORDER="0"
 SRC="|."$dir".q|img1303.png"
 ALT="\begin{displaymath}
z_{cm} = \frac{ a^2\left[z^2/2 - 2 z^3/(3 h) + z^4/(4 h^2...
... h)\right]_0^h} = \frac{a^2 h^2/12}{a^2 h/3} = \frac{h}{4}.
\end{displaymath}">|; 

$key = q/omega_0=35,{{rm{rad.slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1507.png"
 ALT="$\omega_0=35 {\rm rad./s}$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{constv1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{constv1.eps}}
\end{figure}">|; 

$key = q/|s|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$\vert s\vert$">|; 

$key = q/displaystylev_{i1}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img916.png"
 ALT="$\displaystyle v_{i1}'$">|; 

$key = q/{displaymath}frac{{v''}^2}{l}=g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1233.png"
 ALT="\begin{displaymath}
\frac{{v''}^2}{l} = g.
\end{displaymath}">|; 

$key = q/theta_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img964.png"
 ALT="$\theta_f$">|; 

$key = q/{displaymath}a=frac{R}{m_B},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img560.png"
 ALT="\begin{displaymath}
a = \frac{R}{m_B},
\end{displaymath}">|; 

$key = q/t=0.01,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img985.png"
 ALT="$t=0.01 {\rm s}$">|; 

$key = q/{displaymath}W'=0.5times807.7times0.13^2=6.83,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="278" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img747.png"
 ALT="\begin{displaymath}
W' = 0.5\times 807.7\times 0.13^2 = 6.83 {\rm J}.
\end{displaymath}">|; 

$key = q/displaystylea,costheta,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1836.png"
 ALT="$\displaystyle a \cos\theta,$">|; 

$key = q/10^{18};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$10^{18}$">|; 

$key = q/{displaymath}2,r,g<v^2<5,r,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1175.png"
 ALT="\begin{displaymath}
2 r g &lt; v^2&lt; 5 r g.
\end{displaymath}">|; 

$key = q/T=0.7,{{rm{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1602.png"
 ALT="$T=0.7 {\rm s}$">|; 

$key = q/{{bf{F}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img840.png"
 ALT="${\bf F}_i$">|; 

$key = q/l=0.6,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1222.png"
 ALT="$l=0.6 {\rm m}$">|; 

$key = q/{displaymath}k_{{rm{eff}=frac{k_1,x_1}{x_1+x_2}=frac{k_1}{1+k_1slashk_2}=frac{k_1,k_2}{k_1+k_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="308" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1874.png"
 ALT="\begin{displaymath}
k_{\rm eff} = \frac{k_1 x_1}{x_1+x_2} = \frac{k_1}{1+ k_1/k_2} = \frac{k_1 k_2}{k_1+k_2}.
\end{displaymath}">|; 

$key = q/I=m,l^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1816.png"
 ALT="$I=m l^2$">|; 

$key = q/{displaymath}T=frac{2,pi}{omega}=0.6283,{{rm{s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1877.png"
 ALT="\begin{displaymath}
T = \frac{2 \pi}{\omega} = 0.6283 {\rm s}.
\end{displaymath}">|; 

$key = q/{displaymath}W=tau,{mitDelta}phi.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1395.png"
 ALT="\begin{displaymath}
W = \tau {\mit\Delta}\phi.
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{2},m_1,v_{i1}^2=frac{1}{2},m_1,v_{f1}^{,2}+frac{1}{2},m_2,v_{f2}^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1015.png"
 ALT="\begin{displaymath}
\frac{1}{2} m_1 v_{i1}^2 = \frac{1}{2} m_1 v_{f1}^{ 2} + \frac{1}{2} m_2  v_{f2}^{ 2}.
\end{displaymath}">|; 

$key = q/displaystylefrac{m_1,m_2}{m_1+m_2},g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$\displaystyle \frac{m_1 m_2}{m_1+m_2} g,$">|; 

$key = q/displaystylev_{i1},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img944.png"
 ALT="$\displaystyle v_{i1},$">|; 

$key = q/x(t_A)=x_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img635.png"
 ALT="$x(t_A)=x_A$">|; 

$key = q/{displaymath}f=frac{K_i-K_f}{K_i}=frac{3.2714times10^3-7.53}{3.2714times10^3}=0.9977.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img1010.png"
 ALT="\begin{displaymath}
f = \frac{K_i-K_f}{K_i} = \frac{3.2714\times 10^3-7.53}{3.2714\times 10^3} = 0.9977.
\end{displaymath}">|; 

$key = q/{displaymath}m,g,frac{l}{2},costheta+M,g,x,costheta-S,l,sintheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1737.png"
 ALT="\begin{displaymath}
m g \frac{l}{2} \cos\theta + M g x \cos\theta - S l \sin\theta = 0,
\end{displaymath}">|; 

$key = q/{mitDelta}vslash{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="${\mit\Delta}v/{\mit\Delta} t$">|; 

$key = q/omega,t-phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1782.png"
 ALT="$\omega t-\phi$">|; 

$key = q/{displaymath}f'=mu,m,g=0.25times3times9.81=7.36,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img773.png"
 ALT="\begin{displaymath}
f' = \mu m g = 0.25\times 3\times 9.81 = 7.36 {\rm N}.
\end{displaymath}">|; 

$key = q/m,g,lslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1722.png"
 ALT="$m g l/2$">|; 

$key = q/10^{-18};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$10^{-18}$">|; 

$key = q/{displaymath}W=intmbox{{boldmath{{tau{}!cdot!dmbox{{boldmath{{phi{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="41" BORDER="0"
 SRC="|."$dir".q|img1390.png"
 ALT="\begin{displaymath}
W = \int \mbox{\boldmath$\tau$}\!\cdot\!d \mbox{\boldmath$\phi$}.
\end{displaymath}">|; 

$key = q/{displaymath}omega=frac{2,pi}{T}=frac{2,pi}{0.35}=17.95,{{rm{rad.slashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1881.png"
 ALT="\begin{displaymath}
\omega = \frac{2 \pi}{T} = \frac{2 \pi}{0.35} = 17.95 {\rm rad./s}.
\end{displaymath}">|; 

$key = q/{{rm{ft}^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="${\rm ft}^2$">|; 

$key = q/{bfe}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1121.png"
 ALT="${\bf e}_r$">|; 

$key = q/{displaymath}left.frac{df}{dx}right|_{x_0}>0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img706.png"
 ALT="\begin{displaymath}
\left.\frac{df}{dx}\right\vert _{x_0} &gt;0
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{rocket.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="227" BORDER="0"
 SRC="|."$dir".q|img856.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{rocket.eps}}
\end{figure}">|; 

$key = q/DC;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="$DC$">|; 

$key = q/m_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$m_1$">|; 

$key = q/{figure}{epsfysize{=1in{centerline{epsffile{n3.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="457" HEIGHT="114" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="\begin{figure}
\epsfysize =1in
\centerline{\epsffile{n3.eps}}
\end{figure}">|; 

$key = q/{displaymath}U_A=m,g,h_1=300times9.81times25=7.36times10^4,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img752.png"
 ALT="\begin{displaymath}
U_A = m g h_1 = 300\times 9.81 \times 25 = 7.36\times 10^4 {\rm J}.
\end{displaymath}">|; 

$key = q/{displaymath}v_{f2}=y,v_{i1}=0.5times12=6,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1025.png"
 ALT="\begin{displaymath}
v_{f2} = y v_{i1} = 0.5 \times 12 = 6 {\rm m/s}.
\end{displaymath}">|; 

$key = q/mu=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1456.png"
 ALT="$\mu=0$">|; 

$key = q/{displaymath}v=frac{dx}{dt}=v_0+a,t.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="\begin{displaymath}
v = \frac{dx}{dt}= v_0 + a t.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{pisa.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="438" HEIGHT="228" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{pisa.eps}}
\end{figure}">|; 

$key = q/v_{{rm{max}=a,omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1846.png"
 ALT="$v_{\rm max} = a \omega$">|; 

$key = q/10^{6};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$10^{6}$">|; 

$key = q/{displaymath}R=frac{2,v_0^{,2}}{g},sintheta,costheta=frac{v_0^{,2}}{g},sin2theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="\begin{displaymath}
R = \frac{2 v_0^{ 2}}{g} \sin\theta \cos\theta =
\frac{v_0^{ 2}}{g} \sin 2\theta.
\end{displaymath}">|; 

$key = q/sinthetasimeqtheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1072.png"
 ALT="$\sin\theta \simeq \theta$">|; 

$key = q/{{bf{f}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img619.png"
 ALT="${\bf f}_i$">|; 

$key = q/u=12,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img983.png"
 ALT="$u=12 {\rm m/s}$">|; 

$key = q/v_0=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$v_0=0$">|; 

$key = q/{mitDelta}x=120,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="${\mit\Delta} x = 120 {\rm m}$">|; 

$key = q/displaystylem_1,ddot{x}_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img793.png"
 ALT="$\displaystyle m_1 \ddot{x}_1$">|; 

$key = q/{displaymath}U=m,g,z.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img607.png"
 ALT="\begin{displaymath}
U = m g z.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{cube.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="355" HEIGHT="172" BORDER="0"
 SRC="|."$dir".q|img1289.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{cube.eps}}
\end{figure}">|; 

$key = q/x=1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$x=1/2$">|; 

$key = q/displaystyle{{bf{v}_0,t+frac{1}{2},{{bf{a},t^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$\displaystyle {\bf v}_0 t + \frac{1}{2} {\bf a} t^2,$">|; 

$key = q/rho;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$\rho$">|; 

$key = q/3.8,{{rm{N};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$3.8 {\rm N}$">|; 

$key = q/{{bf{a}!cdot!{{bf{b}={{bf{b}!cdot!{{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1383.png"
 ALT="${\bf a}\!\cdot\!{\bf b}={\bf b}\!\cdot\!{\bf a}$">|; 

$key = q/{displaymath}theta=a,cos(omega,t-phi),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1814.png"
 ALT="\begin{displaymath}
\theta = a \cos(\omega t -\phi),
\end{displaymath}">|; 

$key = q/{displaymath}r=sqrt{x^2+y^2}=sqrt{27^2+12^2}=29.55,{{rm{yd}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="320" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="\begin{displaymath}
r = \sqrt{x^2+y^2} = \sqrt{27^2+12^2} = 29.55 {\rm yd}.
\end{displaymath}">|; 

$key = q/{displaymath}M,g,h=frac{1}{2},M,v^2+frac{1}{2},I,omega^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1432.png"
 ALT="\begin{displaymath}
M g h = \frac{1}{2} M v^2 + \frac{1}{2} I \omega^2.
\end{displaymath}">|; 

$key = q/{displaymath}m_1,dot{x}_1+m_2,dot{x}_2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img811.png"
 ALT="\begin{displaymath}
m_1 \dot{x}_1+m_2 \dot{x}_2
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}=frac{d{{bf{v}}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img526.png"
 ALT="\begin{displaymath}
{\bf a} = \frac{d{\bf v}}{dt},
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{scalar.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="332" HEIGHT="226" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{scalar.eps}}
\end{figure}">|; 

$key = q/d{{bf{r}slashdt={{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$d{\bf r}/dt = {\bf v}$">|; 

$key = q/{displaymath}sum_{i=1,N}m_i,|{{bf{k}times{{bf{r}_i|^2=I_k,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1543.png"
 ALT="\begin{displaymath}
\sum_{i=1,N} m_i 
\vert{\bf k}\times {\bf r}_i\vert^2 = I_k,
\end{displaymath}">|; 

$key = q/L=85,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img970.png"
 ALT="$L=85 {\rm m}$">|; 

$key = q/{displaymath}{mitDelta}K=-{mitDelta}U:{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img584.png"
 ALT="\begin{displaymath}
{\mit\Delta}K = - {\mit\Delta}U:
\end{displaymath}">|; 

$key = q/{displaymath}{mitDelta}p=m,v-(-),m,u=0.35times(21+12)=11.55,{{rm{N,s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="421" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img988.png"
 ALT="\begin{displaymath}
{\mit\Delta} p = m v - (-) m u = 0.35\times(21+12) = 11.55 {\rm N s}.
\end{displaymath}">|; 

$key = q/displaystylefrac{m,g}{1+2,mslashM}=frac{2.6times9.81}{1+2times2.6slash6.4}=14.07,{{rm{N}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="339" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1490.png"
 ALT="$\displaystyle \frac{m g}{1+ 2 m/M}= \frac{2.6\times 9.81}{1+2\times 2.6/6.4}
=14.07 {\rm N}.$">|; 

$key = q/r_0=R_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2021.png"
 ALT="$r_0=R_\oplus$">|; 

$key = q/{displaymath}P=f,v'+W,sintheta,v',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img781.png"
 ALT="\begin{displaymath}
P = f v' + W \sin\theta v',
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}times{{bf{b}cdot{{bf{c}={{bf{a}cdot{{bf{b}times{{bf{c}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1384.png"
 ALT="\begin{displaymath}
{\bf a}\times {\bf b} \cdot {\bf c}= {\bf a}\cdot{\bf b} \times {\bf c}.
\end{displaymath}">|; 

$key = q/d{{bf{v}slashdt={{bf{a};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="$d{\bf v}/dt ={\bf a}$">|; 

$key = q/dt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img711.png"
 ALT="$dt$">|; 

$key = q/{displaymath}v'=frac{m,v}{M+m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="43" BORDER="0"
 SRC="|."$dir".q|img1224.png"
 ALT="\begin{displaymath}
v' = \frac{m v}{M+m}.
\end{displaymath}">|; 

$key = q/{displaymath}f=frac{1}{3},M,g,sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1452.png"
 ALT="\begin{displaymath}
f = \frac{1}{3} M g \sin\theta.
\end{displaymath}">|; 

$key = q/{displaymath}v_f=u,lnleft(1+frac{m_p}{M}right).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img877.png"
 ALT="\begin{displaymath}
v_f = u  \ln\left(1+\frac{m_p}{M}\right).
\end{displaymath}">|; 

$key = q/{displaymath}v^2>5,r,g.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1156.png"
 ALT="\begin{displaymath}
v^2 &gt; 5 r g.
\end{displaymath}">|; 

$key = q/{displaymath}omega=sqrt{frac{k}{m}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1774.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{k}{m}}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}!cdot!{{bf{a}=|{{bf{a}|^2=a^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="\begin{displaymath}
{\bf a} \!\cdot \!{\bf a} = \vert{\bf a}\vert^2 = a^2.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{cyl.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1290.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{cyl.eps}}
\end{figure}">|; 

$key = q/d{{bf{P}slashdt=d{{bf{L}slashdt={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1619.png"
 ALT="$d{\bf P} /dt = d{\bf L}/dt = {\bf0}$">|; 

$key = q/k_2=400,{{rm{Nslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1864.png"
 ALT="$k_2= 400 {\rm N/m}$">|; 

$key = q/g,sintheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$g \sin\theta$">|; 

$key = q/{displaymath}R,cosphi-T,costheta=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1652.png"
 ALT="\begin{displaymath}
R \cos\phi - T \cos\theta = 0.
\end{displaymath}">|; 

$key = q/m_1,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$m_1 g$">|; 

$key = q/{displaymath}m=l,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="\begin{displaymath}
m = l,
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{v}=mbox{{boldmath{{omega{}times{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1280.png"
 ALT="\begin{displaymath}
{\bf v} = \mbox{\boldmath$\omega$}\times {\bf r}.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{p}_3={{bf{p}_1+{{bf{p}_2=(960,,60),{{rm{N,s}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="237" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img998.png"
 ALT="\begin{displaymath}
{\bf p}_3 = {\bf p}_1 + {\bf p}_2 = (960,  60)  {\rm N s}.
\end{displaymath}">|; 

$key = q/theta(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1126.png"
 ALT="$\theta(t)$">|; 

$key = q/{{bf{g};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1629.png"
 ALT="${\bf g}$">|; 

$key = q/{displaymath}frac{d{{bf{L}}{dt}=mbox{{boldmath{{tau{}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1618.png"
 ALT="\begin{displaymath}
\frac{d {\bf L}}{dt} = \mbox{\boldmath$\tau$}.
\end{displaymath}">|; 

$key = q/K=(1slash2),m,v^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img739.png"
 ALT="$K= (1/2) m v^2$">|; 

$key = q/h=g,t^2slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$h= g t^2/2$">|; 

$key = q/15^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img727.png"
 ALT="$15^\circ$">|; 

$key = q/{displaymath}v=lim_{{mitDelta}trightarrow0}frac{{mitDelta}x}{{mitDelta}t}=frac{dx}{dt},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="\begin{displaymath}
v = \lim_{{\mit\Delta}t\rightarrow 0}\frac{{\mit\Delta}x}{{\mit\Delta} t}=
\frac{dx}{dt},
\end{displaymath}">|; 

$key = q/{displaymath}f=m,frac{d^2x}{dt^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img633.png"
 ALT="\begin{displaymath}
f = m \frac{d^2x}{dt^2}.
\end{displaymath}">|; 

$key = q/displaystylev_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="$\displaystyle v_y$">|; 

$key = q/displaystyleY_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1692.png"
 ALT="$\displaystyle Y_2$">|; 

$key = q/{displaymath}z_{cm}=frac{int_0^h,z,dz!int_{-a,(1-zslashh)slash2}^{+a,(1-zslashh)a,(1-zslashh)slash2}^{+a,(1-zslashh)slash2},dx}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="358" HEIGHT="67" BORDER="0"
 SRC="|."$dir".q|img1300.png"
 ALT="\begin{displaymath}
z_{cm} = \frac{\int_0^h z dz\!\int_{-a (1-z/h)/2}^{+a (1...
...a (1-z/h)/2} dy\!
\int_{-a (1-z/h)/2}^{+a (1-z/h)/2} dx}.
\end{displaymath}">|; 

$key = q/displaystyle-2x.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$\displaystyle -2 x.$">|; 

$key = q/{displaymath}R=m,g,(3,costheta-2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1183.png"
 ALT="\begin{displaymath}
R = m g (3 \cos\theta-2).
\end{displaymath}">|; 

$key = q/sigma^2=x^2+y^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1322.png"
 ALT="$\sigma^2 = x^2+y^2$">|; 

$key = q/270^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$270^\circ$">|; 

$key = q/2,l;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1226.png"
 ALT="$2 l$">|; 

$key = q/l=3,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1715.png"
 ALT="$l=3 {\rm m}$">|; 

$key = q/{displaymath}F>frac{(m_A+m_B),g}{mu}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="51" BORDER="0"
 SRC="|."$dir".q|img568.png"
 ALT="\begin{displaymath}
F &gt; \frac{(m_A+m_B) g}{\mu}.
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{27times0.9063-5times9.81times0.4226}{5}=0.7483,{{rm{mslashs^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img545.png"
 ALT="\begin{displaymath}
a = \frac{27\times 0.9063 - 5\times 9.81\times 0.4226}{5} = 0.7483 {\rm m/s^2}.
\end{displaymath}">|; 

$key = q/{displaymath}m,frac{v^2}{r}=m,g,costheta-R.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1182.png"
 ALT="\begin{displaymath}
m \frac{v^2}{r} = m g \cos\theta - R.
\end{displaymath}">|; 

$key = q/displaystylea_theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1138.png"
 ALT="$\displaystyle a_\theta$">|; 

$key = q/x=0.15,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1854.png"
 ALT="$x = 0.15 {\rm m}$">|; 

$key = q/{{bf{f}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="${\bf f}_2$">|; 

$key = q/r=7,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1206.png"
 ALT="$r=7 {\rm m}$">|; 

$key = q/{displaymath}frac{v^2}{r}=frac{G,M_oplus}{r^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2026.png"
 ALT="\begin{displaymath}
\frac{v^2}{r} = \frac{G M_\oplus}{r^2}.
\end{displaymath}">|; 

$key = q/{displaymath}(v_{i2}-v_{i1})=-(v_{f2}-v_{f1}).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img930.png"
 ALT="\begin{displaymath}
(v_{i2}-v_{i1}) = - (v_{f2}-v_{f1}).
\end{displaymath}">|; 

$key = q/f=-dUslashdx=-k,x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img682.png"
 ALT="$f = - dU/dx = - k x$">|; 

$key = q/mu_s>mu_k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img518.png"
 ALT="$\mu_s&gt;\mu_k$">|; 

$key = q/{displaymath}{{bf{a}!cdot!{{bf{b}=a_x,b_x+a_y,b_y+a_z,b_z=|{{bf{a}|,|{{bf{b}|,costheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="330" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1269.png"
 ALT="\begin{displaymath}
{\bf a} \!\cdot\!{\bf b} = a_x b_x+ a_y b_y + a_z b_z = \vert{\bf a}\vert \vert{\bf b}\vert \cos\theta,
\end{displaymath}">|; 

$key = q/{displaymath}R=M,g,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1668.png"
 ALT="\begin{displaymath}
R = M g,
\end{displaymath}">|; 

$key = q/(x_1,y_1,z_1);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$(x_1, y_1, z_1)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pendulum.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="323" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1817.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pendulum.eps}}
\end{figure}">|; 

$key = q/2.532,{{rm{rad.slashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1899.png"
 ALT="$2.532 {\rm rad./s}$">|; 

$key = q/|{{bf{f}|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img612.png"
 ALT="$\vert{\bf f}\vert$">|; 

$key = q/{figure}{epsfysize{=1.5in{centerline{epsffile{semimajor.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="170" BORDER="0"
 SRC="|."$dir".q|img2011.png"
 ALT="\begin{figure}
\epsfysize =1.5in
\centerline{\epsffile{semimajor.eps}}
\end{figure}">|; 

$key = q/{displaymath}W=lim_{Nrightarrowinfty}sum_{i=1}^{N}{{bf{f}_i!cdot!{mitDelta}{{bf{r}_i=int_A^B{{bf{f}({{bf{r})!cdot!d{{bf{r}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img622.png"
 ALT="\begin{displaymath}
W = \lim_{N\rightarrow\infty} \sum_{i=1}^{N} {\bf f}_i\!\cdo...
...\Delta}{\bf r}_i
= \int_A^B {\bf f}({\bf r})\!\cdot\!d{\bf r}.
\end{displaymath}">|; 

$key = q/d^2{{bf{r}slashdt^2={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$d^2{\bf r}/dt^2={\bf0}$">|; 

$key = q/{displaymath}h=frac{v^2}{2,g}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img594.png"
 ALT="\begin{displaymath}
h = \frac{v^2}{2 g}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{knot.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="277" HEIGHT="227" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{knot.eps}}
\end{figure}">|; 

$key = q/{displaymath}{{bf{l}=m,{{bf{r}times{{bf{v}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1972.png"
 ALT="\begin{displaymath}
{\bf l} = m {\bf r}\times{\bf v}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{p_{i1}'^{~2}}{2,m_1}+frac{p_{i2}'^{~2}}{2,m_2}=frac{p_{f1}'^{~2}}{2,m_1}+frac{p_{f2}'^{~2}}{2,m_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img929.png"
 ALT="\begin{displaymath}
\frac{p_{i1}'^{&nbsp;2}}{2 m_1} + \frac{p_{i2}'^{&nbsp;2}}{2 m_2} =
\frac{p_{f1}'^{&nbsp;2}}{2 m_1} + \frac{p_{f2}'^{&nbsp;2}}{2 m_2}.
\end{displaymath}">|; 

$key = q/{displaymath}M,g,lslash3+m,g,lslash2-S,l=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="224" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1746.png"
 ALT="\begin{displaymath}
M g l/3 + m g l/2 - S l = 0.
\end{displaymath}">|; 

$key = q/x_{cm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img801.png"
 ALT="$x_{cm}$">|; 

$key = q/{displaymath}|{{bf{r}|leq|{{bf{r}_1|+|{{bf{r}_2|.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="\begin{displaymath}
\vert{\bf r}\vert \leq \vert{\bf r}_1\vert +
\vert{\bf r}_2\vert.
\end{displaymath}">|; 

$key = q/x(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$x(t)$">|; 

$key = q/{displaymath}{mitDelta}Wneq0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img655.png"
 ALT="\begin{displaymath}
{\mit\Delta}W \neq 0.
\end{displaymath}">|; 

$key = q/h=aslashsqrt{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1291.png"
 ALT="$h=a/\sqrt{2}$">|; 

$key = q/m,omega^2,r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1090.png"
 ALT="$m \omega^2 r$">|; 

$key = q/{{bf{v}_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="${\bf v}_0$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{x.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="275" HEIGHT="271" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{x.eps}}
\end{figure}">|; 

$key = q/{{bf{a}=d{{bf{v}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1397.png"
 ALT="${\bf a} = d{\bf v}/dt$">|; 

$key = q/{displaymath}W=int_{x_A}^{x_B}f(x),dx=-kint_{x_A}^{x_B}x,dx=-left[frac{1}{2},k,x_B^{,2}-frac{1}{2},k,x_A^{,2}right].{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img674.png"
 ALT="\begin{displaymath}
W = \int_{x_A}^{x_B} f(x) dx = -k\int_{x_A}^{x_B} x dx = -...
...
\frac{1}{2} k x_B^{ 2} -\frac{1}{2} k x_A^{ 2} \right].
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{f}_{21}=G,frac{{{bf{r}_1-{{bf{r}_2}{|{{bf{r}_1-{{bf{r}_2|^3}=-{{bf{f}_{21}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1918.png"
 ALT="\begin{displaymath}
{\bf f}_{21} = G \frac{{\bf r}_1-{\bf r}_2}{\vert{\bf r}_1-{\bf r}_2\vert^3} = - {\bf f}_{21}.
\end{displaymath}">|; 

$key = q/displaystylembox{{boldmath{{tau{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1622.png"
 ALT="$\displaystyle \mbox{\boldmath$\tau$}$">|; 

$key = q/-G,M,mslashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1938.png"
 ALT="$-G M m/r$">|; 

$key = q/F<f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img510.png"
 ALT="$F&lt;f$">|; 

$key = q/K=M,v^2slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1405.png"
 ALT="$K = M v^2/2$">|; 

$key = q/m_A=75,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img569.png"
 ALT="$m_A=75 {\rm kg}$">|; 

$key = q/r(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1125.png"
 ALT="$r(t)$">|; 

$key = q/{displaymath}{{bf{v}_i=mbox{{boldmath{{omega{}times{{bf{r}_i.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="31" BORDER="0"
 SRC="|."$dir".q|img1306.png"
 ALT="\begin{displaymath}
{\bf v}_i = \mbox{\boldmath$\omega$}\times {\bf r}_i.
\end{displaymath}">|; 

$key = q/{displaymath}X_3=X=m,gslash2=15times9.81slash2=73.58,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="346" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1760.png"
 ALT="\begin{displaymath}
X_3 = X = m g/2= 15\times 9.81/2 = 73.58 {\rm N}.
\end{displaymath}">|; 

$key = q/{displaymath}x_{{rm{max}=l,mu,tantheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="131" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1672.png"
 ALT="\begin{displaymath}
x_{\rm max} = l \mu \tan\theta.
\end{displaymath}">|; 

$key = q/{displaymath}d=u,t=-0.115times0.738=-0.0849,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="318" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img980.png"
 ALT="\begin{displaymath}
d = u t = -0.115\times 0.738 = -0.0849 {\rm m}.
\end{displaymath}">|; 

$key = q/-x_0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img688.png"
 ALT="$-x_0$">|; 

$key = q/{displaymath}W_i=int_A^B{{bf{f}_i({{bf{r})!cdot!d{{bf{r}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img640.png"
 ALT="\begin{displaymath}
W_i = \int_A^B {\bf f}_i({\bf r})\!\cdot\!d{\bf r}
\end{displaymath}">|; 

$key = q/I={scriptstyleintrho,|hat{mbox{{boldmath{{omega{}}times{{bf{r}|^2,dV};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1398.png"
 ALT="$I =
{\scriptstyle\int \rho  \vert\hat{\mbox{\boldmath$\omega$}}\times {\bf r}\vert^2 dV}$">|; 

$key = q/v=-u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$v=-u$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{traj.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="269" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{traj.eps}}
\end{figure}">|; 

$key = q/m_1=m_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img935.png"
 ALT="$m_1=m_2$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{conical.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img1085.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{conical.eps}}
\end{figure}">|; 

$key = q/displaystyle{{bf{s};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$\displaystyle {\bf s}$">|; 

$key = q/{displaymath}p_{i1}'+p_{i2}'=p_{f1}'+p_{f2}',{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img928.png"
 ALT="\begin{displaymath}
p_{i1}' + p_{i2}' = p_{f1}' + p_{f2}',
\end{displaymath}">|; 

$key = q/E=m,c^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$E=m c^2$">|; 

$key = q/v_f;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img878.png"
 ALT="$v_f$">|; 

$key = q/theta<90^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img614.png"
 ALT="$\theta&lt;90^\circ$">|; 

$key = q/100%;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img950.png"
 ALT="$100\%$">|; 

$key = q/{displaymath}W'=int_0^xf',dx=kint_0^xx,dx=frac{1}{2},k,x^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="293" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img746.png"
 ALT="\begin{displaymath}
W' = \int_0^x f' dx = k\int_0^x x dx = \frac{1}{2} k x^2.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2},I,omega^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1310.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} I \omega^2.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d}{dt},(r^2,dot{theta})=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1970.png"
 ALT="\begin{displaymath}
\frac{d}{dt} (r^2 \dot{\theta}) = 0,
\end{displaymath}">|; 

$key = q/{displaymath}m_1,v_{f1},sintheta=m_2,v_{f2},sintheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1031.png"
 ALT="\begin{displaymath}
m_1 v_{f1} \sin\theta = m_2 v_{f2} \sin\theta.
\end{displaymath}">|; 

$key = q/{displaymath}left|i,{{rm{e}^{,i,theta}right|=sqrt{sin^2theta+cos^2theta}=1.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="40" BORDER="0"
 SRC="|."$dir".q|img1120.png"
 ALT="\begin{displaymath}
\left\vert i {\rm e}^{ i \theta}\right\vert = \sqrt{\sin^2\theta + \cos^2\theta } = 1.
\end{displaymath}">|; 

$key = q/1289.1,{{rm{Nslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1886.png"
 ALT="$ 1289.1 {\rm N/m}$">|; 

$key = q/m_A;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img567.png"
 ALT="$m_A$">|; 

$key = q/{displaymath}I=M,b^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1319.png"
 ALT="\begin{displaymath}
I = M b^2.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{compound.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="542" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1826.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{compound.eps}}
\end{figure}">|; 

$key = q/{displaymath}R=m,g-F,sintheta=50times9.81-95timessin15^circ=465.9,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img733.png"
 ALT="\begin{displaymath}
R = m g - F \sin\theta = 50\times 9.81 - 95\times\sin 15^\circ
= 465.9 {\rm N}.
\end{displaymath}">|; 

$key = q/{{bf{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="${\bf x}$">|; 

$key = q/m=2000,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1198.png"
 ALT="$m=2000 {\rm kg}$">|; 

$key = q/deltasslashdeltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1053.png"
 ALT="$\delta s/\delta t$">|; 

$key = q/displaystyleT+Y_1+Y_3-M,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1683.png"
 ALT="$\displaystyle T + Y_1 + Y_3 - M g$">|; 

$key = q/a_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1105.png"
 ALT="$a_r$">|; 

$key = q/m=m_p;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img871.png"
 ALT="$m=m_p$">|; 

$key = q/{mitDelta}{{bf{r}_i;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img618.png"
 ALT="${\mit\Delta}{\bf r}_i$">|; 

$key = q/{displaymath}[v]=frac{[L]}{[T]}={{rm{m,s^{-1}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="\begin{displaymath}[v]= \frac{[L]}{[T]} = {\rm m s^{-1}}.
\end{displaymath}">|; 

$key = q/{displaymath}I'=I+M,d^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1336.png"
 ALT="\begin{displaymath}
I' = I + M d^2,
\end{displaymath}">|; 

$key = q/E<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img697.png"
 ALT="$E&lt;0$">|; 

$key = q/displaystyle-1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$\displaystyle -1$">|; 

$key = q/displaystylev_0,sintheta,,t-frac{1}{2},g,t^2,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="$\displaystyle v_0 \sin\theta  t - \frac{1}{2} g t^2,$">|; 

$key = q/{displaymath}omega=sqrt{frac{g}{l}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1825.png"
 ALT="\begin{displaymath}
\omega = \sqrt{\frac{g}{l}}.
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=omega_x,hat{{bf{x}+omega_y,hat{{bf{y}+omega_z,hat{{bf{z}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="186" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img1263.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}= \omega_x \hat{\bf x} + \omega_y \hat{\bf y} + \omega_z \hat{\bf z}
\end{displaymath}">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$v$">|; 

$key = q/z=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$z=0$">|; 

$key = q/1,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="$1 {\rm m/s^2}$">|; 

$key = q/m=1.2,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img968.png"
 ALT="$m=1.2 {\rm kg}$">|; 

$key = q/a(t);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$a(t)$">|; 

$key = q/T_1,sin60^circ=sqrt{3},T_1slash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="$T_1 \sin 60^\circ = \sqrt{3} T_1/2$">|; 

$key = q/{displaymath}m_1,v_{f1},sintheta_1=m_2,v_{f2},sintheta_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img960.png"
 ALT="\begin{displaymath}
m_1 v_{f1} \sin\theta_1= m_2 v_{f2} \sin\theta_2.
\end{displaymath}">|; 

$key = q/costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1778.png"
 ALT="$\cos\theta$">|; 

$key = q/m_1=5,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1026.png"
 ALT="$m_1=5 {\rm kg}$">|; 

$key = q/z=h;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1295.png"
 ALT="$z=h$">|; 

$key = q/{displaymath}{mitDelta}K=f,{mitDeltah}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img598.png"
 ALT="\begin{displaymath}
{\mit\Delta}K = f {\mit\Delta h}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{work2.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="511" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img625.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{work2.eps}}
\end{figure}">|; 

$key = q/m,g,costheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="$m g \cos\theta$">|; 

$key = q/{displaymath}omega(t)=omega_0+alpha,t,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1249.png"
 ALT="\begin{displaymath}
\omega(t) = \omega_0 + \alpha t,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{newton.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1916.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{newton.eps}}
\end{figure}">|; 

$key = q/PP';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1980.png"
 ALT="$PP'$">|; 

$key = q/{displaymath}1=x+3,y,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1019.png"
 ALT="\begin{displaymath}
1 = x + 3 y,
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{pivot.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="341" BORDER="0"
 SRC="|."$dir".q|img1379.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{pivot.eps}}
\end{figure}">|; 

$key = q/v^2=v_0^{,2}-2,g,s;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img572.png"
 ALT="$v^2 = v_0^{ 2} - 2 g s$">|; 

$key = q/{displaymath}v_{{rm{esc}=sqrt{frac{2,G,M}{R}}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img1952.png"
 ALT="\begin{displaymath}
v_{\rm esc} = \sqrt{\frac{2 G M}{R}}.
\end{displaymath}">|; 

$key = q/r,(1-costheta);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1180.png"
 ALT="$r (1-\cos\theta)$">|; 

$key = q/{{bf{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1308.png"
 ALT="${\bf k}$">|; 

$key = q/h_2=7,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img750.png"
 ALT="$h_2=7 {\rm m}$">|; 

$key = q/{displaymath}a=lim_{{mitDelta}trightarrow0}frac{{mitDelta}v}{{mitDelta}t}=frac{dv}{dt}=frac{d^2x}{dt^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="\begin{displaymath}
a = \lim_{{\mit\Delta}t\rightarrow 0}\frac{{\mit\Delta}v}{{\mit\Delta} t}=
\frac{dv}{dt}=\frac{d^2 x}{dt^2}.
\end{displaymath}">|; 

$key = q/{displaymath}h=-frac{1}{2},g,t^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="\begin{displaymath}
h = -\frac{1}{2} g t^2,
\end{displaymath}">|; 

$key = q/I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img888.png"
 ALT="$I$">|; 

$key = q/displaystyle0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$\displaystyle 0,$">|; 

$key = q/v_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img829.png"
 ALT="$v_r$">|; 

$key = q/{displaymath}bar{f}=frac{I}{t}=frac{11.55}{0.01}=1155.0,{{rm{N}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img990.png"
 ALT="\begin{displaymath}
\bar{f} = \frac{I}{t}= \frac{11.55}{0.01} = 1155.0 {\rm N}.
\end{displaymath}">|; 

$key = q/U(R);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img662.png"
 ALT="$U(R)$">|; 

$key = q/{displaymath}u_1=frac{u_0,r_0}{r_1}=frac{54times0.57}{35}=0.879,{{rm{kmslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img2039.png"
 ALT="\begin{displaymath}
u_1 = \frac{u_0 r_0}{r_1} = \frac{54\times 0.57}{35} = 0.879 {\rm km/s}.
\end{displaymath}">|; 

$key = q/delta{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1070.png"
 ALT="$\delta{\bf v}$">|; 

$key = q/displaystyle(85,cos225^circ,,85,sin225^circ)=(-6.010times10^1,,-6.010times10^1).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="477" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="$\displaystyle (85 \cos 225^\circ, 85 \sin 225^\circ) =(-6.010\times 10^1, -6.010\times 10^1).$">|; 

$key = q/y=v_{f2}slashv_{i1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1017.png"
 ALT="$y=v_{f2}/v_{i1}$">|; 

$key = q/x_Brightarrowx;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img677.png"
 ALT="$x_B\rightarrow x$">|; 

$key = q/omega=vslashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1076.png"
 ALT="$\omega=v/r$">|; 

$key = q/{displaymath}mbox{{boldmath{{tau{}={{bf{r}times{{bf{f},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1386.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\tau$}= {\bf r}\times{\bf f},
\end{displaymath}">|; 

$key = q/{displaymath}a=frac{2,T-W}{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="112" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img551.png"
 ALT="\begin{displaymath}
a = \frac{2 T-W}{m}.
\end{displaymath}">|; 

$key = q/m=300,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img748.png"
 ALT="$m= 300 {\rm kg}$">|; 

$key = q/v_{g,x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="$v_{g x}$">|; 

$key = q/{displaymath}{{bf{s}={{bf{v}_0,t+frac{1}{2},{{bf{a},t^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="\begin{displaymath}
{\bf s} = {\bf v}_0 t + \frac{1}{2} {\bf a} t^2.
\end{displaymath}">|; 

$key = q/E=K+U=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1950.png"
 ALT="$E = K+U = 0$">|; 

$key = q/{displaymath}x_{cm}=frac{m_g,x_g+m_w,x_w}{m_g+m_w},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img822.png"
 ALT="\begin{displaymath}
x_{cm} = \frac{m_g x_g + m_w x_w}{m_g+m_w},
\end{displaymath}">|; 

$key = q/displaystylemu,g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1513.png"
 ALT="$\displaystyle \mu g,$">|; 

$key = q/m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="$m g$">|; 

$key = q/{v'}^2slashr;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1149.png"
 ALT="${v'}^2/r$">|; 

$key = q/r,omega^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1211.png"
 ALT="$r \omega^2$">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{friction.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img511.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{friction.eps}}
\end{figure}">|; 

$key = q/I_x=I_y=I_z=I;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="139" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1561.png"
 ALT="$I_x=I_y=I_z=I$">|; 

$key = q/E>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img698.png"
 ALT="$E&gt;0$">|; 

$key = q/v'=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1174.png"
 ALT="$v'=0$">|; 

$key = q/costheta>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1167.png"
 ALT="$\cos\theta&gt;0$">|; 

$key = q/{displaymath}M,g,(lslash2),costheta+X,l,sintheta=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img1699.png"
 ALT="\begin{displaymath}
M g (l/2) \cos\theta + X l \sin\theta = 0,
\end{displaymath}">|; 

$key = q/E=m,c;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$E=m c$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{circle1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="343" BORDER="0"
 SRC="|."$dir".q|img1064.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{circle1.eps}}
\end{figure}">|; 

$key = q/M;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$M$">|; 

$key = q/{figure*}{epsfysize{=2.5in{centerline{epsffile{e53.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img547.png"
 ALT="\begin{figure*}
\epsfysize =2.5in
\centerline{\epsffile{e53.eps}}
\end{figure*}">|; 

$key = q/pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1045.png"
 ALT="$\pi$">|; 

$key = q/zllR_oplus;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1942.png"
 ALT="$z\ll R_\oplus$">|; 

$key = q/v_{i1}=4.5,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1027.png"
 ALT="$v_{i1} = 4.5 {\rm m/s}$">|; 

$key = q/{displaymath}R=u,t=5times4.515=22.58,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="250" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="\begin{displaymath}
R = u t = 5\times 4.515 = 22.58 {\rm m}.
\end{displaymath}">|; 

$key = q/R_oplus=6.378times10^6,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1922.png"
 ALT="$R_\oplus = 6.378\times 10^6 {\rm m}$">|; 

$key = q/{figure*}{epsfysize{=2in{centerline{epsffile{q52.eps}}{{figure*};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="507" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img729.png"
 ALT="\begin{figure*}
\epsfysize =2in
\centerline{\epsffile{q52.eps}}
\end{figure*}">|; 

$key = q/M=120,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img991.png"
 ALT="$M=120 {\rm kg}$">|; 

$key = q/{displaymath}{v'}^2=frac{r,R}{m}-r,g,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1164.png"
 ALT="\begin{displaymath}
{v'}^2 = \frac{r R}{m} - r g \cos\theta.
\end{displaymath}">|; 

$key = q/{displaymath}J'=-left(1-frac{3,h}{2,l}right)J=-left(1-frac{h}{h_0}right)J,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="54" BORDER="0"
 SRC="|."$dir".q|img1423.png"
 ALT="\begin{displaymath}
J' = -\left(1 - \frac{3 h}{2 l}\right) J=-\left(1 - \frac{h}{h_0}\right) J,
\end{displaymath}">|; 

$key = q/{displaymath}frac{1}{2},(M+m),{v'}^2=(M+m),2,g,l,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1227.png"
 ALT="\begin{displaymath}
\frac{1}{2} (M+m) {v'}^2 = (M+m) 2 g l,
\end{displaymath}">|; 

$key = q/{displaymath}mbox{{boldmath{{omega{}=omega,{{bf{k},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1307.png"
 ALT="\begin{displaymath}
\mbox{\boldmath$\omega$}= \omega {\bf k},
\end{displaymath}">|; 

$key = q/dot{{bf{r}={{bf{v}={{bf{p}slashm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1530.png"
 ALT="$\dot{\bf r} = {\bf v} = {\bf p}/m$">|; 

$key = q/theta=8.65^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1713.png"
 ALT="$\theta=8.65^\circ$">|; 

$key = q/75,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img556.png"
 ALT="$75 {\rm kg}$">|; 

$key = q/{displaymath}frac{dp}{dt}=f,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img890.png"
 ALT="\begin{displaymath}
\frac{dp}{dt} = f,
\end{displaymath}">|; 

$key = q/{{bf{r}=(x,y,z);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="${\bf r}=(x,y,z)$">|; 

$key = q/{figure}{epsfysize{=3in{centerline{epsffile{shm.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="369" HEIGHT="325" BORDER="0"
 SRC="|."$dir".q|img1791.png"
 ALT="\begin{figure}
\epsfysize =3in
\centerline{\epsffile{shm.eps}}
\end{figure}">|; 

$key = q/{displaymath}v=sqrt{frac{2,K}{m}}=sqrt{frac{2times131.3}{50}}=2.29,{{rm{mslashs}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="303" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img740.png"
 ALT="\begin{displaymath}
v = \sqrt{\frac{2 K}{m}} = \sqrt{\frac{2\times 131.3}{50}} = 2.29 {\rm m/s}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{kepler.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="470" HEIGHT="229" BORDER="0"
 SRC="|."$dir".q|img1907.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{kepler.eps}}
\end{figure}">|; 

$key = q/{displaymath}frac{m,{v'}^2}{r}=R-m,g,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1163.png"
 ALT="\begin{displaymath}
\frac{m {v'}^2}{r} = R - m g \cos\theta.
\end{displaymath}">|; 

$key = q/phi=5.3times2,pi=33.30,{{rm{rad.};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1462.png"
 ALT="$\phi=5.3\times 2 \pi = 33.30 {\rm rad.}$">|; 

$key = q/u_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2035.png"
 ALT="$u_1$">|; 

$key = q/{{bf{f}_g+{{bf{f}_R={{bf{0};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="${\bf f}_g + {\bf f}_R = {\bf0}$">|; 

$key = q/f_{21};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img787.png"
 ALT="$f_{21}$">|; 

$key = q/U=(1slash2),k,x^2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img681.png"
 ALT="$U=(1/2) k x^2$">|; 

$key = q/U;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img663.png"
 ALT="$U$">|; 

$key = q/m_1,g,l_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1723.png"
 ALT="$m_1 g l_1$">|; 

$key = q/displaystylev_{f2}';MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img922.png"
 ALT="$\displaystyle v_{f2}'$">|; 

$key = q/t+{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$t+{\mit\Delta} t$">|; 

$key = q/{{bf{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="${\bf v}$">|; 

$key = q/X_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1678.png"
 ALT="$X_3$">|; 

$key = q/mu,R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img562.png"
 ALT="$\mu R$">|; 

$key = q/{displaymath}1~{{rm{acre}=43560times(0.3048)^2=4.047times10^3,{{rm{m}^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="365" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="\begin{displaymath}
1&nbsp;{\rm acre} = 43560 \times (0.3048)^2 = 4.047\times 10^3 {\rm m}^2.
\end{displaymath}">|; 

$key = q/{displaymath}y_{cm}=frac{m_1,y_1+m_2,y_2}{m_1+m_2}=frac{m_1,l_1slash2+m_2,l_1}{m2times1.3slash2+3.4times1.3}{8.6}=0.907,{{rm{m}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="622" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1711.png"
 ALT="\begin{displaymath}
y_{cm} = \frac{m_1 y_1+ m_2 y_2}{m_1+m_2} = \frac{m_1 l_1...
...
= \frac{5.2\times 1.3/2+3.4\times 1.3}{8.6} = 0.907 {\rm m}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d{{bf{l}}{dt}=dot{{bf{r}times{{bf{p}+{{bf{r}timesdot{{bf{p}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="46" BORDER="0"
 SRC="|."$dir".q|img1529.png"
 ALT="\begin{displaymath}
\frac{d{\bf l}}{dt} = \dot{\bf r} \times {\bf p} + {\bf r} \times \dot{\bf p}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2in{centerline{epsffile{cylinder.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="230" BORDER="0"
 SRC="|."$dir".q|img1428.png"
 ALT="\begin{figure}
\epsfysize =2in
\centerline{\epsffile{cylinder.eps}}
\end{figure}">|; 

$key = q/45^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="$45^\circ$">|; 

$key = q/{displaymath}T,sintheta=m,omega^2,r.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1091.png"
 ALT="\begin{displaymath}
T \sin\theta = m \omega^2 r.
\end{displaymath}">|; 

$key = q/{displaymath}W'=f',h=-m,g,h.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img723.png"
 ALT="\begin{displaymath}
W' = f' h= - m g h.
\end{displaymath}">|; 

$key = q/{displaymath}-m,g+frac{sqrt{3},T_1}{2}+frac{T_2}{2}=0.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="201" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="\begin{displaymath}
-m g+\frac{\sqrt{3} T_1}{2} + \frac{T_2}{2} = 0.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{impulse1.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img884.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{impulse1.eps}}
\end{figure}">|; 

$key = q/f'=-m,g;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img721.png"
 ALT="$f'=-m g$">|; 

$key = q/displaystyleleft(frac{G,M_oplus}{omega^2}right)^{1slash3}=left(frac{(6.673times1imes(5.97times10^{24})}{(7.27times10^{-5})^2}right)^{1slash3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="427" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1958.png"
 ALT="$\displaystyle \left(\frac{G M_\oplus}{\omega^2}\right)^{1/3} =
\left(\frac{(6....
...times 10^{-11})\times(5.97\times 10^{24})}{(7.27\times 10^{-5})^2}\right)^{1/3}$">|; 

$key = q/displaystyle{{bf{v}_a;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$\displaystyle {\bf v}_a$">|; 

$key = q/[T];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$[T]$">|; 

$key = q/{displaymath}x=1+t+frac{t^2}{2}-frac{t^4}{4}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="\begin{displaymath}
x = 1 + t + \frac{t^2}{2} - \frac{t^4}{4}.
\end{displaymath}">|; 

$key = q/0.0059,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1914.png"
 ALT="$0.0059 {\rm m/s^2}$">|; 

$key = q/r,deltatheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1981.png"
 ALT="$r \delta\theta$">|; 

$key = q/{displaymath}{{bf{a}+{{bf{b}={{bf{b}+{{bf{a}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img1254.png"
 ALT="\begin{displaymath}
{\bf a} + {\bf b} = {\bf b} + {\bf a}.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{line.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="286" BORDER="0"
 SRC="|."$dir".q|img651.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{line.eps}}
\end{figure}">|; 

$key = q/{{bf{r}=r,{{bf{e}_r;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1973.png"
 ALT="${\bf r} = r {\bf e}_r$">|; 

$key = q/{displaymath}m,{{bf{a}={{bf{f}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1966.png"
 ALT="\begin{displaymath}
m {\bf a} = {\bf f}.
\end{displaymath}">|; 

$key = q/{displaymath}K=frac{1}{2},m,v^2,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img581.png"
 ALT="\begin{displaymath}
K = \frac{1}{2} m v^2,
\end{displaymath}">|; 

$key = q/I_z>I_x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1333.png"
 ALT="$I_z&gt;I_x$">|; 

$key = q/{displaymath}0=(M+m),dv+u,dm.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="197" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img869.png"
 ALT="\begin{displaymath}
0 = (M+m) dv + u dm.
\end{displaymath}">|; 

$key = q/displaystyle-p_{i1}'=p_{i2}'=mu,(v_{i2}-v_{i1}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img924.png"
 ALT="$\displaystyle -p_{i1}' = p_{i2}' = \mu (v_{i2}-v_{i1}),$">|; 

$key = q/{displaymath}phi=tan^{-1}!left(frac{v_0}{omega,x_0}right),{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img1801.png"
 ALT="\begin{displaymath}
\phi =\tan^{-1}\!\left(\frac{v_0}{\omega x_0}\right),
\end{displaymath}">|; 

$key = q/P=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img831.png"
 ALT="$P=0$">|; 

$key = q/[L]slash[T^2];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$[L]/[T^2]$">|; 

$key = q/omega=sqrt{kslashm};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="50" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1882.png"
 ALT="$\omega=\sqrt{k/m}$">|; 

$key = q/0.06228,{{rm{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1887.png"
 ALT="$0.06228 {\rm m}$">|; 

$key = q/{displaymath}U=frac{1}{2},k,x^2=frac{k,a^2,cos^2(omega,t-phi)}{2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1805.png"
 ALT="\begin{displaymath}
U = \frac{1}{2} k x^2= \frac{k a^2 \cos^2(\omega t-\phi)}{2}.
\end{displaymath}">|; 

$key = q/{displaymath}frac{d^2u}{dtheta^2}+u=frac{G,M_odot}{h^2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img1990.png"
 ALT="\begin{displaymath}
\frac{d^2 u}{d\theta^2} + u = \frac{G M_\odot}{h^2}.
\end{displaymath}">|; 

$key = q/fproptox;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1768.png"
 ALT="$f\propto x$">|; 

$key = q/{displaymath}M,dot{v}=f,sintheta,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1364.png"
 ALT="\begin{displaymath}
M \dot{v} = f \sin\theta,
\end{displaymath}">|; 

$key = q/W_1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img645.png"
 ALT="$W_1$">|; 

$key = q/displaystyley+z,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$\displaystyle y + z,$">|; 

$key = q/mneql;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$m\neq l$">|; 

$key = q/displaystylea,cos(omega,t-pislash2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="141" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1839.png"
 ALT="$\displaystyle a \cos(\omega t - \pi/2).$">|; 

$key = q/displaystylefrac{2,m_1,m_2}{m_1+m_2},g,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img498.png"
 ALT="$\displaystyle \frac{2 m_1 m_2}{m_1+m_2} g,$">|; 

$key = q/{displaymath}W=-frac{6.03times0.43}{sin35^circ}=-4.52,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="243" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img771.png"
 ALT="\begin{displaymath}
W = -\frac{6.03 \times 0.43 }{\sin 35^\circ} =- 4.52 {\rm J}.
\end{displaymath}">|; 

$key = q/W=intmbox{{boldmath{{tau{}!cdot!dmbox{{boldmath{{phi{};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1402.png"
 ALT="$W = \int \mbox{\boldmath$\tau$}\!\cdot\!d\mbox{\boldmath$\phi$}$">|; 

$key = q/mbox{{boldmath{{omega{}=dmbox{{boldmath{{phi{}slashdt;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1396.png"
 ALT="$\mbox{\boldmath$\omega$}= d\mbox{\boldmath$\phi$}/dt$">|; 

$key = q/{displaymath}int_A^B{{bf{f}!cdot!d{{bf{r}=-int_B^A{{bf{f}!cdot!d{{bf{r},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img652.png"
 ALT="\begin{displaymath}
\int_A^B {\bf f}\!\cdot\!d{\bf r} = - \int_B^A {\bf f}\!\cdot\!d{\bf r},
\end{displaymath}">|; 

$key = q/{displaymath}0=M,u+m,v,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img972.png"
 ALT="\begin{displaymath}
0 = M u + m v,
\end{displaymath}">|; 

$key = q/x=0.13;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img744.png"
 ALT="$x=0.13$">|; 

$key = q/{displaymath}I=frac{1}{2},M,b^2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1359.png"
 ALT="\begin{displaymath}
I = \frac{1}{2} M b^2.
\end{displaymath}">|; 

$key = q/9.780,{rmm,s^{-2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$9.780 {\rm m s^{-2}}$">|; 

$key = q/{figure}{epsfysize{=2.5in{centerline{epsffile{fvec.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="296" HEIGHT="285" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="\begin{figure}
\epsfysize =2.5in
\centerline{\epsffile{fvec.eps}}
\end{figure}">|; 

$key = q/g=9.81,{{rm{mslashs^2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$g=9.81 {\rm m/s^2}$">|; 

$key = q/|theta|;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1821.png"
 ALT="$\vert\theta\vert$">|; 

$key = q/{displaymath}U=-frac{G,M_oplus,m}{R_oplus+z},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="128" HEIGHT="50" BORDER="0"
 SRC="|."$dir".q|img1939.png"
 ALT="\begin{displaymath}
U = -\frac{G M_\oplus m}{R_\oplus +z},
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{a}!cdot!{{bf{s}=|{{bf{a}|,|{{bf{s}|,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="33" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="\begin{displaymath}
{\bf a} \! \cdot \! {\bf s} = \vert{\bf a}\vert  \vert{\bf s}\vert  \cos\theta.
\end{displaymath}">|; 

$key = q/{displaymath}I=frac{1}{3},M,l^2{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1422.png"
 ALT="\begin{displaymath}
I = \frac{1}{3} M l^2
\end{displaymath}">|; 

$key = q/L_1=L_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1613.png"
 ALT="$L_1=L_2$">|; 

$key = q/displaystyle(-2.101times10^2,,1.997times10^2).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="$\displaystyle (-2.101\times 10^2,   1.997\times 10^2).$">|; 

$key = q/{displaymath}v_{cm}=frac{m_1,v_{i1}+m_2,v_{i2}}{m_1+m_2}=frac{m_1,v_{f1}+m_2,v_{f2}}{m_1+m_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img914.png"
 ALT="\begin{displaymath}
v_{cm} = \frac{m_1 v_{i1} + m_2 v_{i2}}{m_1+m_2}= \frac{m_1 v_{f1} + m_2 v_{f2}}{m_1+m_2}.
\end{displaymath}">|; 

$key = q/k>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1767.png"
 ALT="$k&gt;0$">|; 

$key = q/{displaymath}s=r,theta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="28" BORDER="0"
 SRC="|."$dir".q|img1048.png"
 ALT="\begin{displaymath}
s = r \theta.
\end{displaymath}">|; 

$key = q/{displaymath}omega_2=frac{I_1}{I_2},omega_2.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img1614.png"
 ALT="\begin{displaymath}
\omega_2 = \frac{I_1}{I_2} \omega_2.
\end{displaymath}">|; 

$key = q/e<1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1997.png"
 ALT="$e&lt;1$">|; 

$key = q/2,b;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1585.png"
 ALT="$2 b$">|; 

$key = q/v_{i1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img906.png"
 ALT="$v_{i1}$">|; 

$key = q/{displaymath}I_z=M,frac{int,int,(x^2+y^2),dx,dy}{int,int,dx,dy},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="221" HEIGHT="52" BORDER="0"
 SRC="|."$dir".q|img1321.png"
 ALT="\begin{displaymath}
I_z = M \frac{\int \int  (x^2+y^2) dx dy}{\int \int dx dy},
\end{displaymath}">|; 

$key = q/0.75,{{rm{mslashs};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1851.png"
 ALT="$0.75 {\rm m/s}$">|; 

$key = q/displaystylea_y;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="$\displaystyle a_y$">|; 

$key = q/197.46^circ;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="$197.46^\circ$">|; 

$key = q/v_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img576.png"
 ALT="$v_2$">|; 

$key = q/{displaymath}I=frac{1}{12},M,(a^2+b^2).{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="45" BORDER="0"
 SRC="|."$dir".q|img1358.png"
 ALT="\begin{displaymath}
I = \frac{1}{12} M (a^2+b^2).
\end{displaymath}">|; 

$key = q/{displaymath}a=g-frac{T}{m_2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="\begin{displaymath}
a =g - \frac{T}{m_2}.
\end{displaymath}">|; 

$key = q/{displaymath}K=-{mitDelta}U+W=12.65-4.52=8.13,{{rm{J}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="30" BORDER="0"
 SRC="|."$dir".q|img772.png"
 ALT="\begin{displaymath}
K = - {\mit\Delta}U + W = 12.65 - 4.52 = 8.13 {\rm J}.
\end{displaymath}">|; 

$key = q/{displaymath}U(x_B)-U(x_A)=-int_{x_A}^{x_B}f(x),dx=frac{1}{2},k,x_B^{,2}-frac{1}{2},k,x_A^{,2}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img675.png"
 ALT="\begin{displaymath}
U(x_B) - U(x_A) = - \int_{x_A}^{x_B} f(x) dx =
\frac{1}{2} k x_B^{ 2} -\frac{1}{2} k x_A^{ 2}.
\end{displaymath}">|; 

$key = q/deltatheta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1050.png"
 ALT="$\delta\theta$">|; 

$key = q/mbox{{boldmath{{omega{}_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1260.png"
 ALT="$\mbox{\boldmath$\omega$}_2$">|; 

$key = q/m=,15,{{rm{kg};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1714.png"
 ALT="$m= 15 {\rm kg}$">|; 

$key = q/{displaymath}frac{dU}{dx}=frac{d^2U}{dx^2}=0{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="48" BORDER="0"
 SRC="|."$dir".q|img708.png"
 ALT="\begin{displaymath}
\frac{dU}{dx}=\frac{d^2 U}{dx^2}=0
\end{displaymath}">|; 

$key = q/{displaymath}tantheta>mu.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img514.png"
 ALT="\begin{displaymath}
\tan\theta &gt; \mu.
\end{displaymath}">|; 

$key = q/{mitDelta}t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="${\mit\Delta} t$">|; 

$key = q/{displaymath}m_2,v_{i2},sintheta_i=(m_1+m_2),v_f,sintheta_f.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="34" BORDER="0"
 SRC="|."$dir".q|img966.png"
 ALT="\begin{displaymath}
m_2 v_{i2} \sin\theta_i = (m_1+m_2) v_f \sin\theta_f.
\end{displaymath}">|; 

$key = q/{displaymath}{{bf{F}_i=m_i,{{bf{g},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1628.png"
 ALT="\begin{displaymath}
{\bf F}_i = m_i {\bf g},
\end{displaymath}">|; 

$key = q/{displaymath}U=-M,g,h=-M,g,r,costheta.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="32" BORDER="0"
 SRC="|."$dir".q|img1637.png"
 ALT="\begin{displaymath}
U = - M g h = -M g r \cos\theta.
\end{displaymath}">|; 

$key = q/{figure}{epsfysize{=1.75in{centerline{epsffile{ski.eps}}{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="607" HEIGHT="199" BORDER="0"
 SRC="|."$dir".q|img1179.png"
 ALT="\begin{figure}
\epsfysize =1.75in
\centerline{\epsffile{ski.eps}}
\end{figure}">|; 

$key = q/{{bf{L};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1537.png"
 ALT="${\bf L}$">|; 

$key = q/displaystyle{{bf{0},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1621.png"
 ALT="$\displaystyle {\bf0},$">|; 

1;

