Exercises

1. A horizontal rod $AB$ rotates with constant angular velocity $\omega$ about its mid-point $O$. A particle $P$ is attached to it by equal-length strings $AP$, $BP$. If $\theta$ is the inclination of the plane $APB$ to the vertical, prove that
   $$\frac{d^{\,2}\theta}{dt^{\,2}} -\omega^{\,2}\,\sin\theta\,\cos\theta = -\frac{g}{l}\,\sin\theta,$$
   where $l=OP$. Deduce the condition that the vertical position of $OP$ should be stable.

2. A double pendulum consists of two simple pendula, with one pendulum suspended from the bob of the other. Suppose that the two pendula have equal lengths, $l$, and bobs of equal mass, $m$, and are confined to move in the same vertical plane. Let $\theta$ and $\phi$—the angles that the upper and lower pendula make with the downward vertical (respectively)—be the generalized coordinates. Demonstrate that Lagrange's equations of motion for the system are

   $$2\,\skew{5}\ddot{\theta} + \cos(\theta-\phi)\,\skew{5}\ddot{\phi} + \sin(\theta-\phi)\,\skew{5}\dot{\phi}^{\,2} + \frac{2\,g}{l}\,\sin\theta = 0,$$

   $$\skew{5}\ddot{\phi} + \cos(\theta-\phi)\,\skew{5}\ddot{\theta} -\sin(\theta-\phi)\,\skew{5}\dot{\theta}^{\,2} + \frac{g}{l}\,\sin\phi = 0.$$

3. Consider an elastic pendulum consisting of a bob of mass $m$ attached to a light elastic string of stiffness $k$ and unstreatched length $l$. Let $x$ be the extension of the string, and $\theta$ the angle that the string makes with the downward vertical. Assume that any motion is confined to a vertical plane. Demonstrate that Lagrange's equations of motion for the system are

   $$\skew{3}\ddot{x} - (l+x)\,\skew{5}\dot{\theta}^{\,2} - g\,\cos\theta + \frac{k}{m}\,x = 0,$$

   $$\skew{5}\ddot{\theta} + \frac{2\,\skew{3}\dot{x}\,\skew{5}\dot{\theta}}{l+x}+ \frac{g}{l+x}\,\sin\theta = 0.$$

4. A disk of mass $M$ and radius $R$ rolls without slipping down a plane inclined at an angle $\alpha$ to the horizontal. The disk has a short weightless axle of negligible radius. From this axle is suspended a simple pendulum of length $l< R$ whose bob is of mass $m$. Assume that the motion of the pendulum takes place in the plane of the disk. Let $s$ be the displacement of the center of mass of the disk down the slope, and let $\theta$ be the angle subtended between the pendulum and the downward vertical. Demonstrate that Lagrange's equations of motion for the system are

   $$\left(\frac{3}{2}\,M+ m\right)\skew{3}\ddot{s} + m\,l\,\cos(\alph... ... - m\,l\,\sin(\alpha+\theta)\,\skew{5}\dot{\theta}^{\,2} - (M+m)\,g\,\sin\alpha = 0,$$

   $$\skew{5}\ddot{\theta} + \cos(\alpha+\theta)\,\frac{\skew{3}\ddot{s}}{l} + \frac{g}{l}\,\sin\theta = 0.$$

   

5. A vertical circular hoop of radius $a$ is rotated in a vertical plane about a point $P$ on its circumference at the constant angular velocity $\omega$. A bead of mass $m$ slides without friction on the hoop. Let the generalized coordinate be the angle $\theta$ shown in the diagram. Here, $x$ is a horizontal Cartesian coordinate, $z$ a vertical Cartesian coordinate, and $C$ the center of the hoop. Demonstrate that the equation of motion of the system is
   $$\skew{5}\ddot{\theta} + \omega^2\,\sin\theta + \frac{g}{a}\,\cos(\omega\,t+\theta) = 0.$$
   (Modified from Fowles and Cassiday 2005.)

6. The kinetic energy of a rotating rigid object with an axis of symmetry can be written
   $$K = \frac{1}{2}\left[{\cal I}_\perp\,\skew{5}\dot{\theta}^{\,2} +... ...hi}\,\skew{5}\dot{\psi} + {\cal I}_\parallel\,\skew{5}\dot{\psi}^{\,2}\right],$$
   where ${\cal I}_\parallel$ is the moment of inertia about the symmetry axis, ${\cal I}_\perp$ is the moment of inertia about an axis perpendicular to the symmetry axis, and $\theta$, $\phi$, $\psi$ are the three Euler angles. (See Chapter 8.) Suppose that the object is rotating freely. Find the momenta conjugate to the Euler angles. Which of these momenta are conserved? Find Lagrange's equations of motion for the system. Demonstrate that if the system is precessing steadily (which implies that $\theta$, $\skew{5}\dot{\phi}$, and $\skew{5}\dot{\psi}$ are constants) then
   $$\skew{5}\dot{\psi} = \left(\frac{{\cal I}_\perp-{\cal I}_\parallel}{{\cal I}_\parallel}\right)\,\cos\theta\,\,\skew{5}\dot{\phi}.$$

7. Demonstrate that the components of acceleration in the spherical coordinate system are

   $$a_r = \skew{3}\ddot{r} - r\,\skew{5}\dot{\theta}^{\,2} -r\,\sin^2\theta\,\,\skew{5}\dot{\phi}^{\,2},$$

   $$a_\theta = \frac{1}{r}\,\frac{d}{dt}(r^{\,2}\,\skew{5}\dot{\theta})-r\,\sin\theta\,\cos\theta\,\,\skew{5}\dot{\phi}^{\,2},$$

   $$a_\phi = \frac{1}{r\,\sin\theta}\,\frac{d}{dt}(r^{\,2}\,\sin^2\theta\,\,\skew{5}\dot{\phi}).$$

   (From Lamb 1923.)

8. A particle is constrained to move on a smooth spherical surface of radius $a$. Suppose that the particle is projected with velocity $\varv$ along the horizontal great circle. Demonstrate that the particle subsequently falls a vertical height $a\,{\rm e}^{-u}$, where
   $$\sinh u = \frac{\varv^{\,2}}{4\,g\,a}.$$
   Show that if $\varv^{\,2}$ is large compared to $4\,g\,a$ then this height becomes approximately $2\,g\,a^{\,2}/\varv^{\,2}$. (From Lamb 1923.)

9. Consider a nonconservative system in which the dissipative forces take the form $f_i = -k_i\,\skew{3}\dot{x}_i$, where the $x_i$ are Cartesian coordinates, and the $k_i$ are all positive. Demonstrate that the dissipative forces can be incorporated into the Lagrangian formalism provided that Lagrange's equations of motion are modified to read
   $$\frac{d}{dt}\!\left(\frac{\partial {\cal L}}{\partial \skew{3}\do... ...al L}}{\partial q_i} + \frac{\partial {\cal R}}{\partial \skew{3}\dot{q}_i}=0,$$
   where
   $${\cal R} = \frac{1}{2} \sum_i k_i\,\skew{3}\dot{x}_i^{\,2}$$
   is termed the Rayleigh dissipation function.