Exercises

1. If a train of mass $M$ is subject to a retarding force $M\,(a+b\,v^2)$, show that if the engines are shut off when the speed is $v_0$ then the train will come to rest in a time
   
   $$\frac{1}{\sqrt{a\,b}}\,\tan^{-1}\left(\sqrt{\frac{b}{a}}\,v_0\right),$$
   
   
   after traveling a distance
   
   $$\frac{1}{2\,b}\,\ln\left(1+\frac{b\,v_0^{\,2}}{a}\right).$$
   
   

2. A particle is projected vertically upward from the Earth's surface with a velocity which would, if gravity were uniform, carry it to a height $h$. Show that if the variation of gravity with height is allowed for, but the resistance of air is neglected, then the height reached will be greater by $h^2/(R-h)$, where $R$ is the Earth's radius.

3. A particle is projected vertically upward from the Earth's surface with a velocity just sufficient for it to reach infinity (neglecting air resistance). Prove that the time needed to reach a height $h$ is
   
   $$\frac{1}{3}\left(\frac{2\,R}{g}\right)^{1/2}\,\left[\left(1+\frac{h}{R}\right)^{3/2}-1\right].$$
   
   
   where $R$ is the Earth's radius, and $g$ its surface gravitational acceleration.

4. A particle of mass $m$ is constrained to move in one dimension such that its instantaneous displacement is $x$. The particle is released at rest from $x=b$, and is subject to a force of the form $f(x) = - k\,x^{-2}$. Show that the time required for the particle to reach the origin is
   
   $$\pi\left(\frac{m\,b^3}{8\,k}\right)^{1/2}.$$
   
   

5. A block of mass $m$ slides along a horizontal surface which is lubricated with heavy oil such that the block suffers a viscous retarding force of the form
   
   $$F = - c\,v^n,$$
   
   
   where $c>0$ is a constant, and $v$ is the block's instantaneous velocity. If the initial speed is $v_0$ at time $t=0$, find $v$ and the displacement $x$ as functions of time $t$. Also find $v$ as a function of $x$. Show that for $n=1/2$ the block does not travel further than $2\,m\,v_0^{3/2}/(3\,c)$.

6. A particle is projected vertically upward in a constant gravitational field with an initial speed $v_0$. Show that if there is a retarding force proportional to the square of the speed then the speed of the particle when it returns to the initial position is
   
   $$\frac{v_0\,v_t}{\sqrt{v_0^{\,2} + v_t^{\,2}}},$$
   
   
   where $v_t$ is the terminal speed.

7. A particle of mass $m$ moves (in one dimension) in a medium under the influence of a retarding force of the form $m\,k\,(v^3+a^2\,v)$, where $v$ is the particle speed, and $k$ and $a$ are positive constants. Show that for any value of the initial speed the particle will never move a distance greater than $\pi/(2\,k\,a)$, and will only come to rest as $t\rightarrow \infty$.

8. 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 in parallel, and $[k_1\,k_2/(k_1+k_2)\,m]^{1/2}$ if the springs are in series.

9. 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
   
   $$T = 2\pi\,\sqrt{\frac{V}{g\,A}}.$$
   
   

10. Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant.

11. If the amplitude of a damped harmonic oscillator decreases to $1/e$ of its initial value after $n$ periods show that the ratio of the period of oscillation to the period of the same oscillator with no damping is
    
    $$\left(1+\frac{1}{4\pi^2\,n^2}\right)^{1/2}\simeq 1 + \frac{1}{8\pi^2\,n^2}.$$
    
    

12. Consider a damped driven oscillator whose equation of motion is
    
    $$\frac{d^2 x}{dt^2} + 2\,\nu\,\frac{dx}{dt} + \omega_0^{\,2} \,x = F(t).$$
    
    
    Let $x=0$ and $dx/dt = v_0$ at $t=0$.
    1. Find the solution for $t>0$ when $F = \sin(\omega\,t)$.
    2. Find the solution for $t>0$ when $F= \sin^2(\omega\,t)$.

13. Obtain the time asymptotic response of a damped linear oscillator of natural frequency $\omega_0$ and damping coefficient $\nu$ to a square-wave periodic forcing function of amplitude $F_0=m\,\omega_0^{\,2}\,X_0$ and frequency $\omega$. Thus, $F(t) = F_0$ for $-\pi/2< \omega\,t< \pi/2$, $3\pi/2<\omega\,t<5\pi/2$, etc., and $F(t)=-F_0$ for $\pi/2<\omega\,t<3\pi/2$, $5\pi/2<\omega\,t<7\pi/2$, etc.