Exercises

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

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 [画像:$h^2/(R-h)$], 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
   
   [画像:\begin{displaymath} \frac{1}{3}\left(\frac{2,円R}{g}\right)^{1/2},円\left[\left(1+\frac{h}{R}\right)^{3/2}-1\right]. \end{displaymath}]
   
   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 $x=b,ドル and is subject to a force of the form [画像:$f(x) = - k,円x^{-2}$]. Show that the time required for the particle to reach the origin is
   
   [画像:\begin{displaymath} \pi\left(\frac{m,円b^3}{8,円k}\right)^{1/2}. \end{displaymath}]
   
   
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
   
   \begin{displaymath} F = - c,円v^n, \end{displaymath}
   
   where [画像:$c>0$] is a constant, and $v$ is the block's instantaneous velocity. If the initial speed is $v_0 $ at time $t=0,ドル find $v$ and the displacement $x$ as functions of time $t$. Also find $v$ as a function of $x$. Show that for [画像:$n=1/2$] the block does not travel further than [画像:2ドル,円m,円v_0^{3/2}/(3,円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
   
   [画像:\begin{displaymath} \frac{v_0,円v_t}{\sqrt{v_0^{,2円} + v_t^{,2円}}}, \end{displaymath}]
   
   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 [画像:$m,円k,円(v^3+a^2,円v)$], 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 [画像:$\pi/(2,円k,円a)$], and will only come to rest as $t\rightarrow \infty$.

8. Two light springs have spring constants $k_1$ and $k_2,ドル respectively, and are used in a vertical orientation to support an object of mass $m$. Show that the angular frequency of oscillation is [画像:$[(k_1+k_2)/m]^{1/2}$] if the springs are in parallel, and [画像:$[k_1,円k_2/(k_1+k_2),円m]^{1/2}$] 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 [画像:$\rho<\rho_0$]), and at equilibrium displaces a volume $V$. Show that the period of small oscillations about the equilibrium position is
   
   [画像:\begin{displaymath} T = 2\pi,円\sqrt{\frac{V}{g,円A}}. \end{displaymath}]
   
   

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ドル/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
    
    [画像:\begin{displaymath} \left(1+\frac{1}{4\pi^2,円n^2}\right)^{1/2}\simeq 1 + \frac{1}{8\pi^2,円n^2}. \end{displaymath}]
    
    

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

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 [画像:$F_0=m,円\omega_0^{,2円},円X_0$] and frequency $\omega$. Thus, [画像:$F(t) = F_0$] for [画像:$-\pi/2< \omega,円t< \pi/2$], [画像:3ドル\pi/2<\omega,円t<5\pi/2$], etc., and [画像:$F(t)=-F_0$] for [画像:$\pi/2<\omega,円t<3\pi/2$], [画像:5ドル\pi/2<\omega,円t<7\pi/2$], etc.