Exercises

1. A horizontal rod $AB$ rotates with constant angular velocity $\omega$ about its midpoint $O$. A particle $P$ is attached to it by equal 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. If the two pendula have equal lengths, $l$, and have bobs of equal mass, $m$, and if both pendula are confined to move in the same vertical plane, find Lagrange's equations of motion for the system. Use $\theta$ and $\phi$--the angles the upper and lower pendulums make with the downward vertical (respectively)--as the generalized coordinates. Do not assume small angles.

3. The surface of the Diskworld is a disk of radius $R$ which rotates uniformly about a perpendicular axis passing through its center with angular velocity ${\mit\Omega}$. Diskworld gravitational acceleration is of magnitude $g$, and is everywhere directed normal to the disk. Find the Lagrangian of a projectile of mass $m$ using co-rotating cylindrical polar coordinates as the generalized coordinates. What are the momenta conjugate to each coordinate? Are any of these momenta conserved? Find Lagrange's equations of motion for the projectile.

4. Find Lagrange's equations of motion for an elastic pendulum consisting of a particle of mass $m$ attached to an elastic string of stiffness $k$ and unstretched length $l_0$. Assume that the motion takes place in a vertical plane.

5. 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. Find Lagrange's equations of motion of the system.

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   Chapter09/fig9.04.eps
6. 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. Find the kinetic energy, the potential energy, the Lagrangian, and Largrange's equation of motion of the bead, respectively, in terms of the angular coordinate $\theta$ shown in the above diagram. Here, $x$ is a horizontal Cartesian coordinate, $z$ a vertical Cartesian coordinate, and $C$ the center of the hoop. Show that the beam oscillates like a pendulum about the point on the rim diagrammatically opposite the point about which the hoop rotates. What is the effective length of the pendulum?

7. Consider a spherical pendulum of length $l$. Suppose that the string is initially horizontal, and the bob is rotating horizontally with tangental velocity $v$. Demonstrate that, at its lowest subsequent point, the bob will have fallen a vertical height $l\,{\rm e}^{-u}$, where
   
   $$\sinh u = \frac{v^2}{4\,g\,l}.$$
   
   
   Show that if $v^2$ is large compared to $4\,g\,l$ then this result becomes approximately $2\,g\,l^2/v^2$.

8. The kinetic energy of a rotating rigid object with an axis of symmetry can be written
   
   $$K = \frac{1}{2}\left[I_\perp\,\dot{\theta}^{\,2} + (I_\perp\... ...\dot{\phi}\,\dot{\psi} + I_\parallel\,\dot{\psi}^{\,2}\right],$$
   
   
   where $I_\parallel$ is the moment of inertia about the symmetry axis, $I_\perp$ is the moment of inertia about an axis perpendicular to the symmetry axis, and $\theta$, $\phi$, $\psi$ are the three Euler angles. 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 $\theta$, $\dot{\phi}$, and $\dot{\psi}$ are constants) then
   
   $$\dot{\psi} = \left(\frac{I_\perp-I_\parallel}{I_\parallel}\right)\cos\theta\,\dot{\phi}.$$
   
   

9. Consider a nonconservative system in which the dissipative forces take the form $f_i = -k_i\,\dot{x}_i$, 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
   
   $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\righ... ...al L}{\partial q_i} + \frac{\partial R}{\partial \dot{q}_i}=0,$$
   
   
   where
   
   $$R = \frac{1}{2} \sum_i k_i\,\dot{x}_i^{\,2}$$
   
   
   is termed the Rayleigh Dissipation Function.