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Conservation of Angular Momentum

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  • 0:03 Conservation of…
  • 1:52 Angular Momentum of an…
  • 3:23 Example Calculation
  • 5:24 Lesson Summary
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Lesson Transcript
Instructor: David Wood

David has taught Honors Physics, AP Physics, IB Physics and general science courses. He has a Masters in Education, and a Bachelors in Physics.

After watching this video you will be able to explain what is meant by conservation of angular momentum. You will also be able to use this concept along with the equation to solve problems. A short quiz will follow.

Conservation of Angular Momentum

If you've ever seen a model of a satellite orbiting around a planet, you might have noticed that when they get near to the planet, they're moving super fast. The closer they are to the planet, the faster they move, and as they move back out again they slow down. In general, this phenomenon that you see so often in gravitational orbits is because of conservation of angular momentum.

Conservation of momentum says that momentum in the universe is never created or destroyed, it only moves from one place to another. And angular momentum is conserved in the same way. But what is momentum?

Momentum is a quantity of motion of an object. You might say that a football player has a lot of momentum, because he's both heavy and moving fast. So, we know that momentum is related to velocity and mass. If you have more of either of these, you have more momentum -- linear momentum is equal to m * v.

Angular momentum is similar, except that it relates to rotational motion: instead of mass and velocity, we have the rotational quantities of the moment of inertia and rotational velocity. We could define it by saying that angular momentum is the momentum of an object that is either rotating or in circular motion, and is equal to the product of the moment of inertia and the angular velocity. Though it's worth noting that this equation that defines angular momentum needs to be modified when you look at non-circular orbits.

These two terms, both 'moment of inertia' and 'angular velocity' are covered in more detail in other lessons. But, in summary, moment of inertia is a number that reflects both the mass (linear inertia) and the way that mass is distributed. And the angular velocity is a measure of the angle the object rotates through each second.

Angular Momentum of an Object in Orbit

In the case of a satellite orbiting around a planet, the angular momentum is also dependent on the radius of the orbit. For example, at a given speed, the angular momentum of a satellite will increase as the radius of the orbit increases. So, something moving in a larger orbit will have more angular momentum.

Here is the equation that describes this motion.

L = m * v * r * sin theta

This tells us that the angular momentum of an object, L, measured in kilogram meters squared per second, is equal to the mass of the object, m, in kilograms, multiplied by the velocity, v, in meters per second, multiplied by the radius of the orbit, r, in meters, multiplied by sine of the angle between the radius and the velocity, theta. In a perfectly circular orbit, this angle is 90 degrees, and sine 90 is 1, so then the sine theta part just disappears. This angle can be in either radians or degrees, as long as you have your calculator in the right mode.

Because of this equation, when a comet moves around the Sun in a wildly elliptical orbit, it has to move faster when it gets closer to the Sun. When the radius of the orbit is decreased, this would cause the angular momentum to decrease, so the velocity has to increase to compensate. This keeps the angular momentum of the object constant -- angular momentum must always be conserved.

Example Calculation

Let's go through an example, let's say we have one of those wildly elliptical comets. And at a particular point in the orbit it has a mass of 3 x 10^3 kilograms, a velocity of 2 x 10^4 meters per second, a radius of 4 x 10^13 meters, and is orbiting at an angle of 80 degrees to the star. When the comet reaches its closest point of approach, at a radius of 2 x 10^11 meters, and an angle of 90 degrees, how fast will it be moving?

First of all, let's write down what we know.

  • We know that the mass, m, is 3 x 10^3 kilograms
  • That the initial velocity, v, is 2 x 10^4 meters per second
  • That the initial radius, r, is 4 x 10^13 meters
  • And that the angle, theta, is 80 degrees

We also know a couple of things at the closest point of approach: we know that the final radius, rf, is 2 x 10^11 meters and the final angle, theta-f is 90 degrees. And we're asked to figure out the final velocity, vf.

To solve this problem, we have to apply conservation of momentum. We know everything about the comet in the initial position, so we can calculate its angular momentum pretty easily. And we know most everything about it afterwards, except of course its velocity, which we're trying to find. Remember that the mass is the same before and after, so you can use that for both positions.

Rather than solving two equations, we can combine this into one, by saying that the angular momentum before equals the angular momentum after. Which means:

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