3.11: Angular Momentum and Its Conservation

Learning Objectives

• Understand the analogy between angular momentum and linear momentum.
• Observe the relationship between torque and angular momentum.
• Apply the law of conservation of angular momentum.

Why does Earth keep on spinning? What started its spinning to begin with? 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 \(L\) as

\[L=I \omega. \nonumber \]

This equation is an analog to the definition of linear momentum as \(p=m v\). Units for linear momentum are \(\mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) while units for angular momentum are \(\mathrm{kg} \cdot \mathrm{m}^{2} / \mathrm{s}\). As we would expect, an object that has a large moment of inertia \(I\), such as Earth, has a very large angular momentum. An object that has a large angular velocity \(\omega\), 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 \(L\). The relationship between torque and angular momentum is

\[\text { net } \tau=\frac{\Delta L}{\Delta t}. \nonumber \]

This expression is exactly analogous to the relationship between force and linear momentum, \(F=\Delta p / \Delta t\). The equation \(\text { net } \tau=\frac{\Delta L}{\Delta t}\) is 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. We can rearrange the previous equation with some algebra to get \(\Delta L=(\operatorname{net} \tau) \Delta t\). 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 \(\text { net } \tau=\frac{\Delta L}{\Delta t}\) for the situation in which the net torque is zero. In that case,

\[\operatorname{net} \tau=0 \nonumber \]

implying that

\[\frac{\Delta L}{\Delta t}=0. \nonumber \]

If the change in angular momentum \(\Delta L\) is zero, then the angular momentum is constant; thus,

\[L=\text { constant }(\text { net } \tau=0) \nonumber \]

or

\[L=L^{\prime}(\operatorname{net} \tau=0) . \nonumber \]

These expressions are the law of conservation of angular momentum. Conservation laws are as scarce as they are important.

MAKING CONNECTIONS: CONSERVATION LAWS

Angular momentum, like 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.

Before looking at some examples of the conservation of angular momentum, let us consider what it means that angular momentum is conserved. For any rotating object experiencing no net torque, we can express the conservation of angular momentum with the following equation:

\[L=L^{\prime}. \nonumber \]

Here, the primed quantity might represent the angular momentum after some change occurs to the system. (Primed quantity here means the same symbol but with a ' symbol after it. This is a common convention in physics to show how measures of the same variable relate after some change has occurred.)

We must ask ourselves what change could possibly be occurring in this system if there is not net torque. We also saw changes in systems where linear momentum is conserved. For example, a part might fall off of a rocket, decreasing the mass of the remaining rocket. The linear momentum of the system composed of both the rocket and the piece that fell from it will remain the same even as those two objects are separated. Just as linear momentum is composed of mass and velocity, angular momentum is composed of angular velocity and moment of inertia. We can express the equation as such,

\[I \omega=I^{\prime} \omega^{\prime}, \nonumber \]

Looking at this equation, we can see how changes might be possible to angular velocity if we make changes to the moment of inertia. Unlike mass, moment of inertia is not conserved, so it is possible to effect great changes in the angular velocity simply by moving the parts of the system relative to the axis of rotation. It turns out that moment of inertia becomes smaller as the mass of a spinning object moves closer to the axis of rotation. By changing the location of the masses that make up a spinning object, we can effect great changes in the angular velocity without expending much force to do so. We will discuss some examples of this in phenomena that you may be aware of.

An example of conservation of angular momentum is seen in Figure \(\PageIndex{2}\), 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 \(F\) and \(R\) are small, and so \(\tau\) 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 changing the moment of inertia can result in a corresponding change to angular momentum, as shown in the previous equation. When the skater pulls her arms in \(I^{\prime}\), becomes smaller as more of her mass gets closer to the axis of rotation. Because \(I^{\prime}\) is smaller, the angular velocity \(\omega^{\prime}\) must increase to keep the angular momentum constant. The change can be dramatic, as the following example shows.

Figure \(\PageIndex{2}\): (a) An ice skater is spinning on the tip of her skate with her arms extended. Her angular momentum is conserved because the net torque on her is negligibly small. In the next image, her rate of spin increases greatly when she pulls in her arms, decreasing her moment of inertia. The work she does to pull in her arms results in an increase in rotational kinetic 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 \(\PageIndex{3}\).)

Figure \(\PageIndex{3}\): The Solar System coalesced from a cloud of gas and dust that was originally rotating. The orbital motions and spins of the planets are in the same direction as the original spin and conserve the angular momentum of the parent cloud.

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.

Exercise \(\PageIndex{1}\)

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

Answer

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.

Section Summary

• Every rotational phenomenon has a direct translational analog , likewise angular momentum \(L\) can be defined as \(L=I \omega\).
• This equation is an analog to the definition of linear momentum as \(p=m v\). The relationship between torque and angular momentum is \(\text { net } \tau=\frac{\Delta L}{\Delta t}\).
• 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.

Glossary

Contributors

Curated from resources found in Introduction to Physics published by OpenStax.

rotational kinetic energy

the kinetic energy due to the rotation of an object. This is part of its total kinetic energy