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Angular Momentum vs. Linear Momentum

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  • 0:05 Angular Momentum vs…
  • 0:45 Equation
  • 2:51 Example Calculation
  • 4:07 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 lesson, you will be able to explain the difference between angular and linear momentum and solve problems using angular momentum equations. A short quiz will follow.

Angular Momentum vs. Linear Momentum

Momentum is a concept we understand intuitively. If a linebacker slams into you, you most certainly understand momentum. Ouch!

A large football player has more momentum because of how much mass he has. And a fast-moving football player also has more momentum. And so stopping that momentum is much harder... and more painful.

But what about objects moving in a circle? It turns out that we can't quite imagine angular momentum in the same way. 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. Angular momentum is measured in kilogram meters squared per second.

It's pretty hard to picture in a real-life context, so let's look at it mathematically.

Equation

There is more than one equation for angular momentum. Momentum, p, is mass multiplied by velocity, so to figure out the angular version of momentum, L, we just replace those translational (or linear) quantities with angular ones. The angular equivalent of mass is moment of inertia. And the angular equivalent of velocity is angular velocity. So angular momentum is moment of inertia, measured in kilogram meters squared, multiplied by angular velocity, measured in radians per second.

Radians, as we discussed in another lesson, are just an alternative to degrees. Whereas there are 360 degrees in a circle, there are 2-pi radians in a circle. And moment of inertia is a number that represents how much mass an object has AND how that mass is distributed -- differently shaped objects have different equations to calculate their moments of inertia, but they always relate to the mass of the object and the radius of the various parts of the object.

But the definition equation doesn't make it much easier to imagine angular momentum either. So let's try this alternative definition that works for a particle in orbit. It also works pretty well for a spherical object in orbit, like a planet. This tells us that the angular momentum of a particle 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 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.

So for example, an object moving slowly in a large orbit will have the same angular momentum as an object moving faster in a small orbit. Because it will have a bigger value of r and a smaller value of v -- the two quantities balance each other out. This is why comets orbiting around the Sun move faster when they're near to the Sun and slower when they're farther out -- to keep their angular momentum the same at all times.

Example Calculation

Let's say a planet is orbiting the star in a wildly elliptical orbit (like Pluto for example). The planet has a velocity of 3 x 10^4 m/s, a mass of 6 x 10^24 kg. If at a particular moment it's orbiting at an angle of 85 degrees and a radius of 1.5 x 10^11 meters, how much angular momentum does it have?

Well first of all, we should write down what we know. We know that the velocity, v, is equal to 3 x 10^4 m/s, and the mass, m, is equal to 6 x 10^24 kg. We also know that the angle, theta, is 85 degrees, and the radius, r, is 1.5 x 10^11 m. We're asked to figure out the angular momentum, L. All of the values given are in the L = mvr sin (theta) equation, so that's the one we'll use.

Since L is already the subject of the equation, all we have to do here is plug the numbers in and solve. 6 x 10^24 multiplied by 3 x 10^4 multiplied by 1.5 x 10^11 multiplied by sine 85 gives us an angular momentum of 2.69 x 10^40 kilogram meters squared per second. And that's it! We're done.

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