Classical Mechanics (Kibble & Berkshire) · Problem 10.23

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

Physics — Mechanics · integrable-systems

Problem

A particle

Q

has mass

2m

and two other particles

P

R

, each of mass

m

, are connected to

Q

by light inextensible strings of length

a

. The system is free to move on a smooth horizontal table. Initially

P

Q

R

are at the points

(0,a)

(0,0)

(0,-a)

respectively so that they lie in a straight line with the strings taut.

Q

is then projected in the positive

x

-direction with speed

u

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

x

(the displacement of

Q

) and

\theta

(the angle turned by each of the strings). Show that

\theta

satisfies the equation

\dot{\theta}^2 = \frac{u^2}{a^2}\left(\frac{1}{2-\cos^2\theta}\right)

and deduce that

P

and

R

will collide after a time

\frac{a}{u}\int_0^{\pi/2}\left[2-\cos^2\theta\right]^{\frac{1}{2}}\,d\theta.