Rotation of rigid bodies

In this section we are going to look at the case of rigid bodies whose mass is spread over a definite area. The behaviour of rotating objects is of considerable importance in our lives: the results of our considerations can be applied to rotating car wheels, flywheels, the rotation of high divers and many other things.

A rotating object has kinetic energy associated with it. Flywheels can be used to store rotational kinetic energy; for example, current to energise the electromagnets of the proton accelerator at the Rutherford laboratory in Oxfordshire is provided by a generator this driven by a 5 m diameter flywheel; this allows pulses of electricity to be obtained without putting a sudden large drain on the mains supply.

Rotational energy is also important in stability, which is due to the angular momentum of the body (discussed below); many washing machines have a disc of concrete fixed to the base of the drum to prevent vibrations due to uneven loading with clothes.

If we think of a mass m on a string being swung round in a circle of radius r (Figure 1) then its kinetic energy at any instant is given by:

Student investigation
The effect of moment of inertia may be studied using the apparatus shown in the diagram. It consists of a bar, pivoted at the centre, along which two masses may be moved and fixed at varying distances.

The bar is fixed to an axle and may be set in rotation by wrapping a string round the axle and fixing a weight to the free end of the string and allowing the weight to fall to the ground. Investigate the following:

(a) the angular velocity of the system after varying times with the masses in a fixed position, and
(b) the angular velocity after the string has left the axle for different positions of the masses.

Linear and rotational equations