Angular Momentum
The rotational analogue of linear momentum is angular
momentum, L. After
torque and equilibrium, angular momentum is the aspect of rotational
motion most likely to be tested on SAT II Physics. For the test,
you will probably have to deal only with the angular momentum of
a particle or body moving in a circular trajectory. In such a case,
we can define angular momentum in terms of moment of inertia and
angular velocity, just as we can define linear momentum in terms
of mass and velocity:
The angular momentum vector always points in the same
direction as the angular velocity vector.
Angular Momentum of a Single Particle
Let’s take the example of a tetherball of mass m swinging
about on a rope of length r:
The tetherball has a moment of inertia of
I
= mr2 and an angular
velocity of
=
v/r. Substituting
these values into the formula for linear momentum we get:
This is the value we would expect from the cross product
definition we saw earlier of angular momentum. The momentum, p = mv of
a particle moving in a circle is always tangent to the circle and
perpendicular to the radius. Therefore, when a particle is moving
in a circle,
Newton’s Second Law and Conservation of Angular
Momentum
In the previous chapter, we saw that the net force acting
on an object is equal to the rate of change of the object’s momentum
with time. Similarly, the net torque acting on an object is equal
to the rate of change of the object’s angular momentum with time:
If the net torque action on a rigid body is zero, then
the angular momentum of the body is constant or conserved. The law
of conservation of angular momentum is another one of nature’s
beautiful properties, as well as a very useful means of solving
problems. It is likely that angular momentum will be tested in a
conceptual manner on SAT II Physics.
Example
|
|
|
One
of Brian Boitano’s crowd-pleasing skating moves involves initiating
a spin with his arms extended and then moving his arms closer to
his body. As he does so, he spins at a faster and faster rate. Which
of the following laws best explains this phenomenon? |
| (A) |
Conservation of Mechanical Energy |
| (B) |
Conservation of Angular Momentum |
| (C) |
Conservation of Linear Momentum |
| (D) |
Newton’s First Law |
| (E) |
Newton’s Second Law |
|
Given the context, the answer to this question is no secret:
it’s B, the conservation of angular momentum. Explaining
why is the interesting part.
As Brian spins on the ice, the net torque acting on him
is zero, so angular momentum is conserved. That means that
I
is a conserved quantity.
I is
proportional to
R2,
the distance of the parts of Brian’s body from his axis of rotation.
As he draws his arms in toward his body, his mass is more closely
concentrated about his axis of rotation, so
I decreases. Because
I
must remain constant,
must increase as
I decreases.
As a result, Brian’s angular velocity increases as he draws his
arms in toward his body.
