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# Exercises

1. A horizontal rod rotates with constant angular velocity about its mid-point . A particle is attached to it by equal-length strings , . If is the inclination of the plane to the vertical, prove that

where . Deduce the condition that the vertical position of 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, , and bobs of equal mass, , and are confined to move in the same vertical plane. Let and --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

 and

3. Consider an elastic pendulum consisting of a bob of mass attached to a light elastic string of stiffness and unstreatched length . Let be the extension of the string, and 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

 and

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

 and

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Chapter06/fig6.01.eps

5. A vertical circular hoop of radius is rotated in a vertical plane about a point on its circumference at the constant angular velocity . A bead of mass slides without friction on the hoop. Let the generalized coordinate be the angle shown in the diagram. Here, is a horizontal Cartesian coordinate, a vertical Cartesian coordinate, and the center of the hoop. Demonstrate that the equation of motion of the system is

(Modified from Fowles and Cassiday 2005.)

6. The kinetic energy of a rotating rigid object with an axis of symmetry can be written

where is the moment of inertia about the symmetry axis, is the moment of inertia about an axis perpendicular to the symmetry axis, and , , 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 , , and are constants) then

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

 and

(From Lamb 1923.)

8. A particle is constrained to move on a smooth spherical surface of radius . Suppose that the particle is projected with velocity along the horizontal great circle. Demonstrate that the particle subsequently falls a vertical height , where

Show that if is large compared to then this height becomes approximately . (From Lamb 1923.)

9. Consider a nonconservative system in which the dissipative forces take the form , where the are Cartesian coordinates, and the 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

where

is termed the Rayleigh dissipation function.

Next: Rigid body rotation Up: Lagrangian mechanics Previous: Generalized momenta
Richard Fitzpatrick 2016-03-31