Mass Orbiting on Table

A particle of mass M is constrained to move on a horizontal plane. A second particle, of mass m, is constrained to a vertical line. The two particles are connected by a massless string which passes through a hole in the plane (see Figure 1.1). The motion is frictionless.

Figure 1.1

a) Find the Lagrangian of the system and derive the equations of motion.

b) Show that the orbit is stable with respect to small changes in the radius, and find the frequency of small oscillations.

SOLUTION

a) We can write the Lagrangian in terms of the length r of the string on the table and the angle θ (see Figure 1.2):

The equations of motion are

(1)

(2)

Figure 1.2

we have angular momentum conservation:  

b) The equilibrium position is defined by taking the derivative of U_eff where

∂²U_eff/∂r² > 0, so the orbit is stable with respect to a small perturbation in the radius. The frequency of small oscillations is given by