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
mr
2 represents the sum of all terms like
m
1r
12.
If we now compare the expression with that for
linear kinetic energy we see that they are very similar.
The term
Smr
2 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
m
2.
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)
