11.3 Conservation of Angular Momentum – University Physics Volume 1.pdf

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11 Angular Momentum
11.3 Conservation of Angular
Momentum
Learning Objectives
By the end of this section, you will be able to:
Apply conservation of angular momentum to determine the angular velocity of a ro‐
tating system in which the moment of inertia is changing
Explain how the rotational kinetic energy changes when a system undergoes changes
in both moment of inertia and angular velocity

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So far, we have looked at the angular momentum of systems consisting of point particles and rigid
bodies. We have also analyzed the torques involved, using the expression that relates the external
net torque to the change in angular momentum, (Figure) . Examples of systems that obey this
equation include a freely spinning bicycle tire that slows over time due to torque arising from
friction, or the slowing of Earth’s rotation over millions of years due to frictional forces exerted
on tidal deformations.
However, suppose there is no net external torque on the system,
In this case, (Figure) becomes the law of conservation of angular momentum .
Law of Conservation of Angular Momentum
The angular momentum of a system of particles around a point in a fixed inertial reference frame
is conserved if there is no net external torque around that point:
or
Note that the total angular momentum
is conserved. Any of the individual angular momenta can change as long as their sum remains
constant. This law is analogous to linear momentum being conserved when the external force on a
system is zero.
As an example of conservation of angular momentum, (Figure) shows an ice skater executing a
spin. The net torque on her is very close to zero because there is relatively little friction between
her skates and the ice. Also, the friction is exerted very close to the pivot point. Both
are small, so

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is negligible. Consequently, she can spin for quite some time. She can also increase her rate of
spin by pulling her arms and legs in. Why does pulling her arms and legs in increase her rate of
spin? The answer is that her angular momentum is constant, so that
or
where the primed quantities refer to conditions after she has pulled in her arms and reduced her
moment of inertia. Because
is smaller, the angular velocity
must increase to keep the angular momentum constant.