Exercises

1. Derive Equation (2.12).

2. Consider a system consisting of $N$ point particles. Let ${\bf r}_i$ be the position vector of the $i$th particle, and let ${\bf F}_i$ be the external force acting on this particle. Any internal forces are assumed to be central in nature. The resultant force and torque (about the origin) acting on the system are

   $${\bf F} =\sum_{i=1,N} {\bf F}_i,$$

   $$\mbox{\boldmath$\tau$} =\sum_{i=1,N} {\bf r}_i\times{\bf F}_i,$$

   respectively. A point of action of the resultant force is defined as a point whose position vector ${\bf r}$ satisfies
   $${\bf r}\times{\bf F} =$$   $\tau$$$.$$
   Demonstrate that there are an infinite number of possible points of action lying on the straight line
   $${\bf r} = \frac{{\bf F}\times\mbox{\boldmath $\tau$}}{F^{\,2}} + \lambda\,\frac{{\bf F}}{F},$$
   where $\lambda$ is arbitrary. This straight line is known as the line of action of the resultant force.

3. Consider an isolated system consisting of two extended bodies (which can, of course, be modeled as collections of point particles), $A$ and $B$. Let ${\bf F}_{A}$ be the resultant force acting on $A$ due to $B$, and let ${\bf F}_{B}$ be the resultant force acting on $B$ due to $A$. Demonstrate that ${\bf F}_{B}= -{\bf F}_{A}$, and that both forces have the same line of action.

4. An extended body is acted upon by two resultant forces, ${\bf F}_1$ and ${\bf F}_2$. Show that these forces can be only replaced by a single equivalent force, ${\bf F} = {\bf F}_1+{\bf F}_2$, provided:
   1. ${\bf F}_1$ and ${\bf F}_2$ are parallel (or antiparallel). In this case, the line of action of ${\bf F}$ is parallel to those of ${\bf F}_1$ and ${\bf F}_2$.
   2. ${\bf F}_1$ and ${\bf F}_2$ are not parallel (or antiparallel), but their lines of action cross at a point (with one proviso). In this case, the line of action of ${\bf F}$ passes through the crossing point.

5. Deduce that if an isolated system consists of three extended bodies, $A$, $B$, and $C$, where ${\bf F}_{A}$ is the resultant force acting on $A$ (due to $B$ and $C$), ${\bf F}_{B}$ is the resultant force acting on $B$, and ${\bf F}_C$ is the resultant force acting on $C$, then ${\bf F}_A+{\bf F}_B+{\bf F}_C = {\bf0}$ and the forces either all have parallel lines of action or have lines of action that cross at a common point.

6. A particle of mass $m$ moves in one dimension and has an instantaneous displacement $x$. The particle is released at rest from $x=a$, subject to the force $f(x) = - c\,x^{\,-2}$, where $a$, $c>0$. Demonstrate that the time needed for the particle to reach $x=0$ is
   $$\pi\left(\frac{m\,a^{\,3}}{8\,c}\right)^{1/2}.$$
   (Modified from Fowles and Cassiday 2005.)

7. A particle of mass $m$ moves in one dimension and has an instantaneous displacement $x$. The particle is released at rest from $x=a$, subject to the force $f(x)=-m\,\mu\,(a^{\,5}/x^{\,2})^{1/3}$, where $a, \mu>0$. Show that the particle will reach the origin with a speed $a\sqrt{6\,\mu}$ after a time $(8/15)\,(6/\mu)^{1/2}$ has elapsed. (Modified from Smart 1951.)

8. A particle moves in one dimension and has an instantaneous displacement $x$. The particle is released at rest from $x=a$ and accelerates such that $\skew{3}\ddot{x}=\mu\,(x + a^{\,4}/x^{\,3})$, where $a>0$, and $\ddot{\phantom{x}}\equiv d^{\,2}/dt^{\,2}$. Show that the particle will reach the origin after a time $\pi/(4\sqrt{\mu})$ has elapsed, and that its speed is then infinite. (Modified from Smart 1951.)

9. A particle of mass $m$, moving in one dimension with an initial (i.e., at $t=-\infty$) velocity $\varv_0$, is subject to a force
   $$f(t) = \frac{p_0\,\delta t}{\pi} \,\frac{1}{(t-t_0)^{\,2} + (\delta t)^{\,2}}.$$
   Find the velocity as a function of time. Show that as $\delta t\rightarrow 0$ the motion approaches motion at constant velocity with an abrupt change in velocity, by an amount $p_0/m$, at $t=t_0$ .

10. A particle of mass $m$ moving in one dimension is subject to a force
    $$f(x) = -k\,x+\frac{a}{x^{\,3}},$$
    where $k, a>0$. Find the potential energy, $U(x)$. Find the equilibrium points. Are they stable or unstable? Determine the angular frequency of small-amplitude oscillations about any stable equilibrium points.

11. A particle moving in one dimension with simple harmonic motion has speeds $u$ and $\varv$ at displacements $a$ and $b$, respectively, from its mean position. Show that the period of the motion is
    $$T = 2\pi\left(\frac{b^{\,2}-a^{\,2}}{u^{\,2}-\varv^{\,2}}\right)^{1/2}.$$
    Find the amplitude. (From Smart 1951.)

12. The potential energy for the force between two atoms in a diatomic molecule has the approximate form
    $$U(x) = -\frac{a}{x^{\,6}}+ \frac{b}{x^{\,12}},$$
    where $x$ is the distance between the atoms, and $a, b$ are positive constants. Find the force.
    1. Assuming that one of the atoms is relatively heavy and remains at rest, while the other, whose mass is $m$, moves in a straight line, find the equilibrium distance and the period of small oscillations about the equilibrium position.
    2. Assuming that both atoms have the same mass $m$, and move in a straight line, find the equilibrium distance and the period of small oscillations about the equilibrium position.

13. Two light springs have spring constants $k_1$ and $k_2$, respectively, and are used in a vertical orientation to support an object of mass $m$. Show that the angular frequency of oscillation is $[(k_1+k_2)/m]^{1/2}$ if the springs are connected in parallel, and $[k_1\,k_2/(k_1+k_2)\,m]^{1/2}$ if the springs are connected in series.

14. A body of uniform cross-sectional area $A$ and mass density $\rho$ floats in a liquid of density $\rho_0$ (where $\rho<\rho_0$), and at equilibrium displaces a volume $V$. Show that the period of small oscillations about the equilibrium position is
    $$T = 2\pi\,\sqrt{\frac{V}{g\,A}}$$
    (neglecting the phenomenon of added mass).

15. A particle of mass $m$ executes one-dimensional simple harmonic oscillation under the action of a conservative force such that its instantaneous displacement is
    $$x(t) = a\,\cos(\omega\,t-\phi).$$
    Find the average values of $x$, $x^{\,2}$, $\dot{x}$, and $\dot{x}^{\,2}$ over a single cycle of the oscillation. Here, $\dot{\phantom{x}}\equiv d/dt$. Find the average values of the kinetic and potential energies of the particle over a single cycle of the oscillation.

16. Using the notation of Section 2.9, show that the total momentum and angular momentum of a two-body system take the form

    $${\bf P} = M\,\dot{\bf r}_{cm},$$

    $${\bf L} = M\,{\bf r}_{cm}\times \dot{\bf r}_{cm} + \mu\,{\bf r}\times \dot{\bf r},$$

    respectively, where $M=m_1+m_2$, and $\dot{\phantom{r}}\equiv d/dt$.
    1. If the force acting between the bodies is conservative, such that ${\bf f}=-\nabla U$, demonstrate that the total energy of the system is written
       $$E=\frac{1}{2}\,M\,\dot{r}_{cm}^{\,2} + \frac{1}{2}\,\mu\,\dot{r}^{\,2} + U.$$
       Show, from the equation of motion, $\mu\,\ddot{\bf r} = -\nabla U$, that $E$ is constant in time.
    2. If the force acting between the particles is central, so that ${\bf f}\propto {\bf r}$, demonstrate, from the equation of motion, $\mu\,\ddot{\bf r} = {\bf f}$, that ${\bf L}$ is constant in time.