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.