Spark image

Moment of inertia

We can think of a solid rigid body as made up of many particles of masses m1, m2, m3... at distances r1, r2, r3... from the centre of rotation; the total rotational kinetic energy of the body will be the sum of the energies of all the particles. Then

Total kinetic energy = ½ m1ω2r12 + ½ m2ω2r22 + ½ m3ω2r32 + …..

Notice that since the body is rigid the angular velocity (ω) is the same for all particles although the linear velocity will be greater for particles further from the axis of rotation. We can write this as:



kinetic energy = ½ ω2 Σ mr2


where Σ mr2 represents the sum of all terms like m1r12.

If we now compare the expression with that for linear kinetic energy we see that they are very similar.

The term Smr2 takes the place of mass in the linear equation and it is known as the moment of inertia of the body (I).
The units for moment of inertia are kg m2.

Rotational kinetic energy = ½ Iω2

Since power is the rate at which work is done, or at which energy is transformed from one type to another, we can write:

Energy = power x time = ½ Iω2


Unlike mass, the moment of inertia of a body may be variable; it depends not only on the mass of the rotating object but also on how the mass is distributed about the axis of rotation. Therefore a wheel with a heavy rim will have a bigger moment of inertia than a uniform disc of the same mass and radius.


Example problem
Consider two wheels, both of mass 4 kg and both of radius 0.3 m. One wheel has all its mass concentrated in a heavy rim and the other is a uniform thin flat disc.
Calculate the rotational kinetic energy of both if they are rotated at 10 rev s-1.

(a) Disc with heavy rim:
Kinetic energy = ½ mω2r2
Since all the mass is concentrated in the rim.
Therefore: Kinetic energy = 710.6J

(b) Uniform disc:
Kinetic energy = ½ Iω2 = 355.3 J

(The formula for the moment of inertia of a flat disc is given in the file of moment of inertia formulae)
 
 
 
© Keith Gibbs 2013