You should recall that linear momentum is defined by: p = mu
The corresponding rotational quantity is angular momentum, defined by: L = Iw
where I is the rotational inertia of the rotating body and w is the angular velocity.
As the textbook indicates, we can start with Newton's second law for rotation and combine this with the definition of angular acceleration:
t = Ia = I(Dw/Dt) = D(Iw)/Dt = DL/Dt
This tells us that torque is the time rate of change of angular momentum, just as force is the time rate of change of linear momentum. If the net torque is zero, the angular momentum is constant. Like rotational inertia, angular momentum depends not only on the mass of the body or system, but also on the distribution of the mass. Linear momentum is conserved if all the forces in an interaction come from within the system. Likewise, angular momentum is a conserved quantity in rotational interactions when the torques originate within the system. An increase in the angular momentum of one component will result in a corresponding angular momentum decrease in other components of the system.
For problems involving conservation of angular momentum, it is important to identify all the rotating members both before and after the interaction. Equations [8.15] and [8.16] in the text can be misleading in this regard. A clearer form of this important conservation law will indicate explicitly that the behavior of all components of the system before and after the interaction must be considered:
If St = 0 then SIiwi = SIfwf
Look at Figure 8.30 in the text. The angular velocity changes, because the mass distribution changes. There is a demonstration done in class that calls for an application of this principle. A disk is dropped onto another rotating disk and the two rotate together. The moment of inertia of the system before the interaction is I1, and after the interaction it is (I1 + I2). There is no change in mass distribution for either cylinder, but before the interaction only one of them is part of the rotating system, whereas after the interaction both cylinders are rotating with a common angular velocity.
Look at the movie clip at the beginning of this section. Can you see that the diver's angular momentum has more than one component? How might you describe his rotation about different axes? Is his total angular momentum conserved during the dive? How does gravity enter the picture?