Classical Mechanics (Kibble & Berkshire) · Problem 10.24

The angular momentum principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · integrable-systems

Problem

A uniform rod of length

2a

has its lower end in contact with a smooth horizontal table. Initially the rod is released from rest in a position making an angle of

60^\circ

with the upward vertical. Express the conservation of linear momentum and energy for this system in terms of the coordinates

x

(the horizontal displacement of the centre of mass of the rod) and

\theta

(the angle between the rod and the upward vertical). Deduce that the centre of mass of the rod moves in a vertical straight line, and that

\theta

satisfies the equation

\dot{\theta}^2 = \frac{3g}{a}\left(\frac{1-2\cos\theta}{4-3\cos^2\theta}\right).

Find how long it takes for the rod to hit the table.