# Conservation of Angular Momentum

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Ever notice how flying acrobats, gymnasts, ice-skaters and half-pipe snowboarders, tuck in their arms and scrunch up their bodies while spinning in the air? By keeping their arms and legs tucked close to their centers of mass, they are able to rotate faster. This is because, just like linear momentum, the momentum of rotation, called angular momentum, is also conserved. Angular momentum depends on the speed of rotation and the distribution of weight from the center of mass. In this unit you'll learn the basics of angular momentum and its conservation.

## Watch the Video

## Circus Physics: Conservation of Angular Momentum

Watch Alex and Robinson flip high above the ring as they fly from trapeze to trapeze. Conservation of angular momentum means that the tighter they tuck, the faster they spin.

### Questions to Consider While Watching the Video

- Why do the fliers scrunch up in the air while flipping and twisting?
- What happens to the rate at which they spin when they change shape in the air?

## Digging Deeper

Here's a diagram showing Alex as he flips in the air:

Recall that linear momentum is the product of mass and velocity. Angular momentum, *L*, is very similar. It is the product of rotational velocity, *ω*, and moment of inertia, *I*. In the Linear Momentum unit we learned that moment of inertia is like mass, except that it also takes into account the mass's distance from the center of rotation.

When Alex is tucked, the mass in his arms and legs is closer to his center of rotation, so his moment of inertia, *i*, is small, and his rotational velocity, *ω*, is large. To slow his rotation in the air, he extends his legs, pushing more of his mass away from his center of rotation. This makes his moment of inertia, *I*, larger. Because angular momentum must be conserved, his rotational velocity, *ω*, slows. Notice that in both the tuck, and the layout, I and *ω* change, but their product, *L*, always stays the same.

This same concept is what allows ice-skaters to spin so rapidly:

This ice-skater starts her spin with arms and leg outstretched, making her moment of inertia large and rotational velocity small. When she pulls in her leg and arms, the moment of inertia gets small, and rotational velocity increases to keep the overall angular momentum, L, equal in both positions.

## Your Turn

Use the concepts and formulas from this unit to figure out the following:

If Alex flips once every two seconds with arms and legs outstretched, how fast can he spin when he tucks them in? Assume his moment of inertia when tucked is 2/3 of what it is when outstretched.

##### Answer: Once every 1.33 seconds

If his moment of inertia is L when outstretched, then it is 2/3 x *L *when tucked. His angular momentum stays the same, however, so we can set the equation for angular momentum before the tuck, equal to the one for after:

*I* x (½ rotation/second) = (2/3) x *I* x ( ? rotation/second)

Since I appears on both sides of the equation, it cancels out, leaving:

(½) = (2/3) x (?)

Solving for ? gives:

(¾) rotation/second

To find how long it takes for a full rotation, divide this by 1 rotation:

1 rotation / (¾) rotation/second ≈ 1.33 seconds)

## Further Explanation

If you'd like to learn more about flying acts and conservation of angular momentum, check out these links: