Assignment 4

Due: Monday, 4 November 2002

17. A mass m hangs on a massless rod of length L that is suspended from the edge of a horizontal disk of radius a. The rod is attached to the disk by a pivot so that the angle the rod makes relative to the downward can change. The mass is constrained to move in the plane containing the axis of the disk and the pivot. The disk rotates about its axis at a fixed angular velocity. Gravity is in the direction opposite to the rotation. Write down the Lagrangian. Derive the equations of motion. Find an expression for the equilibrium angle the mass makes with the vertical.

18. Write down the Lagrangian for the motion of a particle of mass m in the spherical potential, V(r). Derive the equations of motion. Identify the different terms in the equations of motion. (Are the terms known by other names?) Are any coordinates ignorable (are there any conserved quantities)? If so, which?

19. Find the Euler-Lagrange equation describing the brachistochrone curve for a particle moving inside a spherical Earth of uniform mass density. Obtain the first integral for this problem. Assume that the motion takes place in the equatorial plane of the Earth.

20. A uniform hoop of mass m and radius r rolls without slipping on a fixed cylinder of radius R. The only external force is gravity. If the hoop starts rolling from the top of the cylinder, at which point does the hoop fall off the cylinder?

21. A sphere of moment-of-inertia I and radius r rolls on the lower half of the inner surface of a hollow cylinder of inner radius R > 2r. Gravity is the only external force. What is the Lagrangian? What are the equations of motion? Find the frequency of small oscillations of the sphere about equilibrium.