Classical Mechanics (Kibble & Berkshire) · Problem 11.2

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

Physics — Mechanics · angular-momentum-fairground-target

Problem

A fairground target consists of a uniform circular disk of mass

M

and radius

a

that can turn freely about a diameter which is fixed in a vertical position. Initially the target is at rest. A bullet of mass

m

is moving with speed

u

along a horizontal straight line at right angles to the target. The bullet embeds itself in the target at a point distance

b

from the rotation axis. Find the final angular speed of the target. [The moment of inertia of the disk about its rotation axis is

Ma^2/4

.] Show also that the energy lost in the impact is

\frac{1}{2}mu^2\left(\frac{Ma^2}{Ma^2+4mb^2}\right).