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Angular Momentum and Its Conservation for AP Physics C

By — McGraw-Hill Professional
Updated on Feb 11, 2011

Practice problems for these concepts can be found at:

Rotational Motion Practice Problems for AP Physics B & C

It probably won't surprise you by this point that momentum, too, has a rotational form. It's called angular momentum (abbreviated, oddly, as L), and it is found by this formula:

This formula makes intuitive sense. If you think of angular momentum as, roughly, the amount of effort it would take to make a rotating object stop spinning, then it should seem logical that an object with a large moment of inertia or with a high angular velocity (or both) would be pretty tough to bring to rest.

For a point particle, this formula can be rewritten as

where v is linear velocity, and r is either (1) the radius of rotation, if the particle is moving in a circle, or (2) distance of closest approach if the particle is moving in a straight line (see Figure 16.1).

Angular Momentum and Its Conservation

Wait a minute! How can an object moving in a straight line have angular momentum?!? Well, for the purposes of the AP exam, it suffices just to know that if a particle moves in a straight line, then relative to some point P not on that line, the particle has an angular momentum. But if you want a slightly more satisfying—if less precise—explanation, consider this image. You're standing outside in an open field, and an airplane passes high overhead. You first see it come over the horizon far in front of you, then it flies toward you until it's directly over where you're standing, and then it keeps flying until it finally disappears beneath the opposite horizon. Did the plane fly in an arc over your head or in a straight path? It would be hard for you to tell, right? In other words, when a particle moves in a straight line, an observer who's not on that line would think that the particle sort of looked like it were traveling in a circle.

As with linear momentum, angular momentum is conserved in a closed system; that is, when no external torques act on the objects in the system. Among the most famous examples of conservation of angular momentum is a satellite's orbit around a planet. As shown in Figure 16.2, a satellite will travel in an elliptical orbit around a planet. This means that the satellite is closer to the planet at some times than at others.

Angular Momentum and Its Conservation

Obviously, at point A, the satellite is farther from the center of rotation than at point B. Conservation of angular momentum tells us that, correspondingly, the angular speed at point A must be less than at point B.1

1Note the consistency with Kepler's law of equal areas in equal times, as discussed in Chapter 15.

The other really famous example of conservation of angular momentum involves a spinning figure skater. When a skater spinning with his or her arms outstretched suddenly brings the arms in close to the body, the speed of rotation dramatically increases. Moment of inertia decreased, so angular speed increased.

You can demonstrate this phenomenon yourself ! Sit in a desk chair that spins, and with your legs outstretched, push off forcefully and start spinning. Then tuck in your feet. Dizzying, isn't it?

Practice problems for these concepts can be found at:

Rotational Motion Practice Problems for AP Physics B & C

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