Classical Mechanics (Kibble & Berkshire) · Problem 11.10

Lagrange's equations and conservation principles — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · general-rigid-pendulum

Problem

General rigid pendulum A rigid body of general shape has mass

M

and can rotate freely about a fixed horizontal axis. The centre of mass of the body is distance

h

from the rotation axis, and the moment of inertia of the body about the rotation axis is

I

. Show that the period of small oscillations of the body about the downward equilibrium position is

2\pi\left(\frac{I}{Mgh}\right)^{1/2}.

Deduce the period of small oscillations of a uniform rod of length

2a

, pivoted about a horizontal axis perpendicular to the rod and distance

b

from its centre.