We can now understand why Earth keeps on spinning. As we saw in the previous example,

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

implying that

If the change in angular momentum

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 Figure 3, 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

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

### Example 4: **Calculating the Angular Momentum of a Spinning Skater**

Suppose an ice skater, such as the one in Figure 3, is spinning at 0.800 rev/ s with her arms extended. She has a moment of inertia of

*Strategy*

In the first part of the problem, we are looking for the skater’s angular velocity

*Solution for (a)*

Because torque is negligible (as discussed above), the conservation of angular momentum given in

or

Solving for

*Solution for (b) *

Rotational kinetic energy is given by

The initial value is found by substituting known values into the equation and converting the angular velocity to rad/s:

The final rotational kinetic energy is

Substituting known values into this equation gives

*Discussion*

In both parts, there is an impressive increase. First, the final angular velocity is large, although most world-class skaters can achieve spin rates about this great. Second, the final kinetic energy is much greater than the initial kinetic energy. The increase in rotational kinetic energy comes from work done by the skater in pulling in her arms. This work is internal work that depletes some of the skater’s food energy.

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 Figure 4.)

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.

**Check Your Undestanding**

Is angular momentum completely analogous to linear momentum? What, if any, are their differences?

#### Solution

Yes, angular and linear momentums are completely analogous. While they are exact analogs they have different units and are not directly inter-convertible like forms of energy are.

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