Exercises

1. A mass stands on a platform that executes simple harmonic oscillation in a vertical direction at a frequency of $5\,{\rm Hz}$. Show that the mass loses contact with the platform when the displacement exceeds $10^{-2}\,{\rm m}$. [From Pain 1999.]

2. A small body rests on a horizontal diaphragm of a loudspeaker that is supplied with an alternating current of constant amplitude but variable frequency. If the diaphragm executes simple harmonic oscillation in the vertical direction of amplitude $10\,\mu{\rm m}$, at all frequencies, find the greatest frequency (in hertz) for which the small body stays in contact with the diaphragm.

3. A mass $m$ is attached to the mid-point of a stretched string of negligible mass, length $2\,l$, and uniform tension $T$. Let $x\ll l$ be the transverse displacement of the mass from its equilibrium position. Show that the displacement executes simple harmonic oscillation at the angular frequency $\omega=\sqrt{2\,T/l\,m}$.

4. 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 small amplitude oscillations about the equilibrium state 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.

5. A mass $M$ is suspended at the end of a uniform spring of unstretched length $l$ and spring constant $k$. If the mass of the spring is $m$ and the velocity of an element $dy$ of its length is proportional to its distance $y$ from the fixed end of the spring, show that the kinetic energy of this element is
   $$\frac{1}{2}\left(\frac{m}{l}\,dy\right)\left(\frac{y}{l}\,v\right)^2,$$
   where $v$ is the velocity of the suspended mass. Hence, by integrating over the length of the spring, show that its total kinetic energy is $(1/6)\,m\,v^{\,2}$. Finally, deduce, from energy conservation arguments, that the angular oscillation frequency of the system is given by
   $$\omega=\sqrt{\frac{k}{M+m/3}}.$$
   [From Pain 1999.]

6. 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 $V$. Making use of Archimedes' principle (that the buoyancy force acting on a partially submerged body is equal to the weight of the displaced liquid), show that the period of small amplitude oscillations about the equilibrium position is
   $$T = 2\pi\,\sqrt{\frac{V}{g\,A}}.$$

7. A U-tube of constant cross-sectional area $A$ consists of a horizontal section connected at either end to two vertical sections. Suppose that the tube is filled with an incompressible liquid of mass density $\rho$. Let the total length of the liquid column be $l$. (Where $l$ exceeds the length of the horizontal section.) Suppose that the surface of the liquid in one of the vertical sections is initially displaced (vertically) a small distance $h$ from its equilibrium position. Show that the surface displacement subsequently executes simple harmonic oscillation at the angular frequency $\omega=\sqrt{2\,g/l}$, where $g$ is the acceleration due to gravity.

8. A particle of mass $m$ slides in a frictionless semi-circular depression in the ground of radius $R$. Find the angular frequency of small amplitude oscillations about the particle's equilibrium position, assuming that the oscillations are essentially one-dimensional, so that the particle passes through the lowest point of the depression during each oscillation cycle.

9. 1. Imagine a straight tunnel passing through the center of the Earth, which is regarded as a sphere of radius $R$ and uniform mass density. A particle is dropped into the tunnel from the surface. Show that the particle undergoes simple harmonic oscillation at the angular frequency $\omega=\sqrt{g/R}$, where $g$ is the gravitational acceleration at Earth's surface. (Hint: The gravitational acceleration at a point inside a spherically symmetric mass distribution is the same as if all of the mass interior to the point were concentrated at the center, and all of the mass exterior to the point were neglected.) Estimate how long it takes the particle to reach the other end of the tunnel.

   2. Assuming that the tunnel is smooth (i.e., ignoring friction), show that motion is simple harmonic even if the tunnel does not pass through the center of the Earth, and that the travel time from one end of the tunnel to the other is the same as before.
   [From Ingard 1988.]

10. A particle executing simple harmonic oscillation in one dimension has speeds $u$ and $v$ at displacements $a$ and $b$, respectively, from its equilibrium position.
    1. Show that the period of the motion can be written
       $$T = 2\pi\left(\frac{b^{\,2}-a^{\,2}}{u^{\,2}-v^{\,2}}\right)^{1/2}.$$

    2. Show that the amplitude of the motion can be written
       $$A= \left(\frac{u^{\,2}\,b^{\,2}-v^{\,2}\,a^{\,2}}{u^{\,2}-v^{\,2}}\right)^{1/2}.$$

11. If a thin wire is twisted through an angle $\theta$ then a restoring torque $\tau = - k\,\theta$ develops, where $k>0$ is known as the torsional force constant. Consider a so-called torsional pendulum, which consists of a horizontal disk of mass $M$, and moment of inertia $I$, suspended at its center from a thin vertical wire of negligible mass and length $l$, whose other end is attached to a fixed support. The disk is free to rotate about a vertical axis passing through the suspension point, but such rotation twists the wire. Find the frequency of torsional oscillations of the disk about its equilibrium position.

12. A circular hoop of diameter $d$ hangs on a nail. What is the period of its small amplitude oscillations? [From French 1971.]

13. A compound pendulum consists of a uniform bar of length $l$ that pivots about one of its ends. Show that the pendulum has the same period of oscillation as a simple pendulum of length $(2/3)\,l$.

14. A compound pendulum consists of a uniform circular disk of radius $r$ that is free to turn about a horizontal axis perpendicular to its plane. Find the position of the axis for which the periodic time is a minimum.

15. A laminar object of mass $M$ has a moment of inertia $I_0$ about a perpendicular axis passing through its center of mass. Suppose that the object is converted into a compound pendulum by suspending it about a horizontal axis perpendicular to its plane. Show that the minimum effective length of the pendulum occurs when the distance of the suspension point from the center of gravity is equal to the radius of gyration, $k=(I_0/M)^{1/2}$.

16. A uniform disk of mass $M$ and radius $a$ rolls without slipping over a rough horizontal surface. Suppose that a small mass $m$ is attached to the edge of the disk. Show that the angular frequency of small amplitude oscillations of the disk about its equilibrium position is
    $$\omega =\sqrt{\frac{2}{3}\,\frac{m}{M}\,\frac{g}{a}}.$$

17. A body hung at the end of a light vertical spring stretches the spring statically to twice its original length. The system can be set into motion either as a simple pendulum or as a mass-spring oscillator. Determine the ratio between the periods of these motions. (In the pendulum mode of motion, assume the length of the spring to be constant.) [From Ingard 1988.]

18. Show that the average speed of a particle executing simple harmonic oscillation is $2/\pi$ times the maximum speed. [From Ingard 1988.]

19. A particle of mass $m$ executes one-dimensional simple harmonic oscillation such that its instantaneous $x$ coordinate is
    $$x(t) = a\,\cos(\omega\,t-\phi).$$
    1. Find the average values of $x$, $x^{\,2}$, $\dot{x}$, and $\dot{x}^{\,2}$ over a single cycle of the oscillation.
    2. Find the average values of the kinetic and potential energies of the particle over a single cycle of the oscillation.

20. A particle executes two-dimensional simple harmonic oscillation such that its instantaneous coordinates in the $x$-$y$ plane are

    $$x(t) =a\,\cos(\omega\,t),$$

    $$y(t) =a\,\cos(\omega\,t-\phi).$$

    Describe the motion when (a) $\phi = 0$, (b) $\phi=\pi/4$, and (c) $\phi=\pi/2$. In each case, plot the trajectory of the particle in the $x$-$y$ plane.

21. An LC circuit is such that at $t=0$ the capacitor is uncharged and a current $I_0$ flows through the inductor. Find an expression for the charge $Q$ stored on the positive plate of the capacitor as a function of time.

22. 1. A simple pendulum of mass $m$ and length $l$ is such that $\theta(0)=0$ and $\skew{3}\dot{\theta}(0) = \omega_0$. Find the subsequent motion, $\theta(t)$, assuming that its amplitude remains small.
    2. Suppose, instead, that $\theta(0)=\theta_0$ and $\skew{3}\dot{\theta}(0) = 0$. Find the subsequent motion.
    3. Suppose, finally, that $\theta(0)=\theta_0$ and $\skew{3}\dot{\theta}(0) = \omega_0$. Find the subsequent motion.

23. 1. Demonstrate that
       $$E = \frac{1}{2}\,m\,l^{\,2}\,\skew{3}\dot{\theta}^{\,2} + m\,g\,l\,(1-\cos\theta)$$
       is a constant of the motion of a simple pendulum whose time evolution equation is given by Equation (1.50). (Do not make the small-angle approximation.)

    2. Show that the amplitude of the motion, $\skew{3}\hat{\theta}$, can be written
       $$\skew{3}\hat{\theta} = 2\,\sin^{-1}\left(\frac{E}{2\,m\,g\,l}\right)^{1/2}.$$

    3. Demonstrate that the period of the motion is determined by
       $$\frac{T}{T_0} = \frac{1}{\pi}\int_0^{\skew{3}\hat{\theta}}\frac{d\theta}{\sqrt{\sin^2(\skew{3}\hat{\theta}/2)-\sin^2(\theta/2)}},$$
       where $T_0$ is the period of small-angle oscillations.

    4. Making use of the substitution $\sin u =\sin(\theta/2)/\sin(\skew{3}\hat{\theta}/2)$, show that the previous expression transforms to
       $$\frac{T}{T_0} = \frac{2}{\pi}\int_0^{\pi/2}\frac{du}{\sqrt{1-\sin^2(\skew{3}\hat{\theta}/2)\,\sin^2 u}}.$$
       Hence, deduce that
       $$\frac{T}{T_0} = 1+ \frac{\skew{3}\hat{\theta}^{\,2}}{16}+{\cal O}\left(\skew{3}\hat{\theta}^{\,4}\right).$$