Spark image

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:

Kinetic energy of a mass m swung round on a string = ½mv2 = ½mw2r2

where w is the angular velocity of the mass. (Think of the action of a hammer thrower: he may swing a 7.5 kg hammer round his head once a second and use the kinetic energy so gained to project it some 80 m when it is released.

Student investigation
Make a detailed study of the effects of moment of inertia in athletics. If possible, obtain numerical values of the angular velocity in events such as the hammer, the discus and the long jump.

What rotational problems might there be in the different methods of high jumping, that is, the flop technique and the straddle?


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

These equations for angular motion can be compared with the similar ones for linear motion.


Linear motion   Angular equivalent
Displacement (s)   Angular displacement (q)
Velocity (v)   Angular velocity (w)
Acceleration (a)   Angular acceleration (a)
Momentum = mu   Angular momentum = Iw
Kinetic energy = ½mv2   Rotational kinetic energy = ½Iw2
Force = ma   Torque = Ia
s = vt   q = wt
s = ut + ½at2   q = qo + ½at2
v2 = u2 + 2as   w12 = wo2 + 2aq
v = u + at   w1 = wo + at
 
 
 
© Keith Gibbs 2013