# Four mechanics problems I can't solve.

1. May 14, 2007

### Walczyk

I cannot solve these for the life of me. They are from my book, chapters regarding mechanics of rigid bodies, and lagrangian mechanics.. I need help badly, I'm having much trouble.

A small thin disk of radius r and mass m is attached rigidly to the face of a second thin disk of radius R and mass M as shown below. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle and released. Find the period of the motion.

A billard ball of radius a is initially spinning about a horizontal axis with angular speed w_0 and with zero forward speed. If the coefficient of slidign friction between the ball and the billiard table is µ_k, find the dsitance the ball travels before slipping ceases to occur.

The point of support of a simple pendulum is being elevated at a constant acceperation a, so that the height of the support is 1/2at^2, and its vertical velocity is at. Find the differential equation of motion for small oscillations of the pendulum by Lagrange's method. Show that the period of the pendulum is 2*pi*sqrt(l/(g+a)), where l is the length of the pendulum.

A heavy elastic spring of uniform stiffness and density supports a block of mass m. If m' is the mass of the spring and k its stiffness, show that the period of oscillations is 2*pi*sqrt((m+(m'/3))/k). This problem shows the effect of the mass of the spring on the period of oscillation. (Hilt: To set up the Lagrangian function for the system, assume that the velocity of any part of the spring is proportional to its distance from the point of suspension.)

2. May 14, 2007

### Reshma

You will have to show us some work that you have done, only then we can help.