# Rotational Motion and Angular Momentum

## Angular Momentum and Its Conservation


### Learning Objectives

By the end of this section, you will be able to:
1. Understand the analogy between angular momentum and linear momentum.
2. Observe the relationship between torque and angular momentum.
3. Apply the law of conservation of angular momentum.
Why does Earth keep on spinning? What started it spinning to begin with? And how does an ice skater manage to spin faster and faster simply by pulling her arms in? Why does she not have to exert a torque to spin faster? Questions like these have answers based in angular momentum, the rotational analog to linear momentum.
By now the pattern is clear—every rotational phenomenon has a direct translational analog. It seems quite reasonable, then, to define angular momentum  as
This equation is an analog to the definition of linear momentum as . Units for linear momentum are  while units for angular momentum are . As we would expect, an object that has a large moment of inertia , such as Earth, has a very large angular momentum. An object that has a large angular velocity , such as a centrifuge, also has a rather large angular momentum.
When you push a merry-go-round, spin a bike wheel, or open a door, you exert a torque. If the torque you exert is greater than opposing torques, then the rotation accelerates, and angular momentum increases. The greater the net torque, the more rapid the increase in . The relationship between torque and angular momentum is
This expression is exactly analogous to the relationship between force and linear momentum, . The equation  is very fundamental and broadly applicable. It is, in fact, the rotational form of Newton’s second law.

### Conservation of Angular Momentum

We can now understand why Earth keeps on spinning. As we saw in the previous example, . This equation means that, to change angular momentum, a torque must act over some period of time. Because Earth has a large angular momentum, a large torque acting over a long time is needed to change its rate of spin. So what external torques are there? Tidal friction exerts torque that is slowing Earth’s rotation, but tens of millions of years must pass before the change is very significant. Recent research indicates the length of the day was 18 h some 900 million years ago. Only the tides exert significant retarding torques on Earth, and so it will continue to spin, although ever more slowly, for many billions of years.
What we have here is, in fact, another conservation law. If the net torque is zero, then angular momentum is constant or conserved. We can see this rigorously by considering  for the situation in which the net torque is zero. In that case,
implying that
If the change in angular momentum  is zero, then the angular momentum is constant; thus,
or
These expressions are the law of conservation of angular momentum. Conservation laws are as scarce as they are important.
An example of conservation of angular momentum is seen in , in which an ice skater is 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 and because the friction is exerted very close to the pivot point. (Both  and  are small, and so  is negligibly small.) Consequently, she can spin for quite some time. She can do something else, too. She can 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
Expressing this equation in terms of the moment of inertia,
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. The change can be dramatic, as the following example shows.

There are several other examples of objects that increase their rate of spin because something reduced their moment of inertia. Tornadoes are one example. Storm systems that create tornadoes are slowly rotating. When the radius of rotation narrows, even in a local region, angular velocity increases, sometimes to the furious level of a tornado. Earth is another example. Our planet was born from a huge cloud of gas and dust, the rotation of which came from turbulence in an even larger cloud. Gravitational forces caused the cloud to contract, and the rotation rate increased as a result. (See .)
In case of human motion, one would not expect angular momentum to be conserved when a body interacts with the environment as its foot pushes off the ground. Astronauts floating in space aboard the International Space Station have no angular momentum relative to the inside of the ship if they are motionless. Their bodies will continue to have this zero value no matter how they twist about as long as they do not give themselves a push off the side of the vessel.

### Test Prep for AP Courses


### Section Summary

1. Every rotational phenomenon has a direct translational analog , likewise angular momentum  can be defined as 
2. This equation is an analog to the definition of linear momentum as . The relationship between torque and angular momentum is 
    
3. Angular momentum, like energy and linear momentum, is conserved. This universally applicable law is another sign of underlying unity in physical laws. Angular momentum is conserved when net external torque is zero, just as linear momentum is conserved when the net external force is zero. 

### Conceptual Questions


### Problems & Exercises
