7Rigid bodies
IA Dynamics and Relativity
7.1 Angular velocity
We’ll first consider the cases where there is just one particle, moving in a circle
of radius s about the z axis. Its position and velocity vectors are
r = (s cos θ, s sin θ, z)
˙
r = (−s
˙
θ sin θ, s
˙
θ cos θ, 0).
We can write
˙
r = ω ×r,
where
ω =
˙
θ
ˆ
z
is the angular velocity vector.
In general, we write
ω =
˙
θ
ˆ
n = ω
ˆ
n,
where
ˆ
n is a unit vector parallel to the rotation axis.
The kinetic energy of this particle is thus
T =
1
2
m|
˙
r|
2
=
1
2
ms
2
˙
θ
2
=
1
2
Iω
2
where
I
=
ms
2
is the moment of inertia. This is the counterpart of “mass” in
rotational motion.
Definition (Moment of inertia). The moment of inertia of a particle is
I = ms
2
= m|
ˆ
n × r|
2
,
where s is the distance of the particle from the axis of rotation.