Exercises

1. A mass stands on a platform which 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 $10^{-2}\,{\rm m}$.

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

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

4. 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 mass of the displaced liquid), show that the period of small amplitude oscillations about the equilibrium position is
   
   $$T = 2\pi\,\sqrt{\frac{V}{g\,A}}.$$
   
   

5. 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.

6. 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.

7. Suppose that a hole is drilled through a laminar (i.e., flat) object of mass $M$, which is then suspended in a frictionless manner from a horizontal axis passing through the hole, such that it is free to rotate in a vertical plane. Suppose that the moment of inertia of the object about the axis is $I$, and that the distance of the hole from the object's center of mass is $d$. Find the frequency of small angle oscillations of the object about its equilibrium position. Hence, find the frequency of small angle oscillations of a compound pendulum consisting of a uniform rod of mass $M$ and length $l$ suspended vertically from a horizontal axis passing through one of its ends.

8. A pendulum consists of a uniform circular disk of radius $r$ which 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.

9. A particle of mass $m$ executes one-dimensional simple harmonic oscillation under the action of a conservative force such that its instantaneous $x$ coordinate 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. Find the average values of the kinetic and potential energies of the particle over a single cycle of the oscillation.

10. 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) $\phi=\pi/2$, and (c) $\phi=-\pi/2$. In each case, plot the trajectory of the particle in the $x$-$y$ plane.

11. 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.

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

13. Demonstrate that
    
    $$E = \frac{1}{2}\,m\,l^2\,\dot{\theta}^{\,2} + m\,g\,l\,(\cos\theta-1)$$
    
    
    is a constant of the motion of a simple pendulum whose time evolution equation is given by (50). (Do not make the small angle approximation.) Hence, show that the amplitude of the motion, $\theta_0$, can be written
    
    $$\theta_0 = 2\,\sin^{-1}\left(\frac{E}{2\,m\,g\,l}\right)^{1/2}.$$
    
    
    Finally, demonstrate that the period of the motion is determined by
    
    $$\frac{T}{T_0} = \frac{1}{\pi}\int_0^{\theta_0}\frac{d\theta}{\sqrt{\sin^2(\theta_0/2)-\sin^2(\theta/2)}},$$
    
    
    where $T_0$ is the period of small angle oscillations. Verify that $T/T_0\rightarrow 1$ as $\theta_0\rightarrow 0$. Does the period increase, or decrease, as the amplitude of the motion increases?