Angular momentum of a rigid object

A motorcycle wheel has almost all its mass concentrated at the outside. If the wheel has mass m and radius r , and the time required for one revolution is T, what is the spin part of its angular momentum?

This is an example of the commonly encountered special case of rigid motion, as opposed to the rotation of a system like a hurricane in which the different parts take different amounts of time to go around. We don't really have to go through a laborious process of adding up contributions from all the many parts of a wheel, because they are all at about the same distance from the axis, and are all moving around the axis at about the same speed. The velocity is all perpendicular to the spokes,

v = v

= (circumference)/T

= 2πr/T ,

and the angular momentum of the wheel about its center is

L = mvr

= m(2πr/T)r

= 2πmr²/T .