Classical Mechanics (Kibble & Berkshire) · Problem 5.18

Energy conservation — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · normal-modes

Problem

A particle

P

of mass

3m

is suspended from a fixed point

O

by a light inextensible string of length

a

. A second particle

Q

of mass

m

is in turn suspended from

P

by a second string of length

a

. The system moves in a vertical plane through

O

. Show that the linearised equations of motion for small oscillations near the downward vertical are

4\ddot{\theta} + \ddot{\phi} + 4n^2\theta = 0,

\ddot{\theta} + \ddot{\phi} + n^2\phi = 0,

where

\theta

and

\phi

are the angles that the two strings make with the downward vertical, and

n^2 = g/a

. Find the normal frequencies and the forms of the normal modes for this system.