# Law of Conservation of Angular Momentum

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Instructor: Damien Howard

Damien has a master's degree in physics and has taught physics lab to college students.

Learn what we mean by conservation of angular momentum. Then, explore examples of how it affects the real world by looking at swivel chairs, asteroids, and figure skaters!

## What Are Conservation Laws?

One important concept that you'll see pop up a few times in a physics course is the idea of conservation. You may already have a general sense of what conservation means. You've probably heard of people concerned with conserving our natural resources. They want to save non-renewable resources like oil and coal from being used up. So you know in the general sense conservation means the act of saving something instead of using it up.

It turns out that physics conservation works in quite a similar way. When we talk about conservation of energy, linear momentum, or angular momentum we are talking about the total amount of energy, or momentum, in a system being preserved. In this lesson, we'll focus on learning about conservation of angular momentum.

## Conservation of Angular Momentum

Any object that rotates has angular momentum while it's spinning. This can be anything from obvious things like a top spinning on a table to things we might not think about like a doorknob turning. Angular momentum gives us a measurement of an object's ability to keep spinning. The more angular momentum something has, the more it will want to keep rotating. We write angular momentum (L) mathematically as moment of inertia (I) multiplied by angular velocity (w).

The law of conservation of angular momentum states that angular momentum is conserved when there is zero net torque applied to a system, where the system is the object or objects that are rotating. Torque and angular momentum are related through the angular impulse equation. Angular impulse equals net torque (tau) times a change in time (t) which in turn equals a change in angular momentum.

When the angular momentum of a system is conserved, it means that there is no change in total angular momentum. In our equation we get this when net torque equals zero:

If you think about it, this makes sense. Whenever you apply a torque to an object, you change its angular momentum. For example, imagine applying torque to a swivel chair by spinning it. When you spin the chair you give it an angular velocity, and therefore, an angular momentum as well. Since the chair went from standing still with zero angular momentum to having some after you spin it, the angular momentum is changing and it can't be conserved.

Now instead, imagine an asteroid spinning freely as it flies through space. There is currently nothing adding any torque to the asteroid, so its angular momentum is conserved. This means if we were to look at its angular momentum in March and then again later in May, we would see it is unchanged.

We can see how this works out by looking at our earlier equation for conservation of angular momentum. A change in angular momentum can be written as final angular momentum minus initial angular momentum. Rearranging the equation, we get initial angular momentum equal to final angular momentum. In our asteroid example, the initial angular momentum would be from when it was looked at in March, and the final angular momentum when it was looked at again in May.

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