# Activity Guide: Conservation of Angular Momentum

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Students are likely to have noticed that 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, these athletes 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 students will learn the basics of angular momentum and its conservation.

## How to Incorporate the Video into Instruction

Use the video to introduce angular momentum and its conservation or to reinforce a previous lesson on these topics. If you watch the video in class, have students pause video at points where the various fliers and skaters change their angular velocity.

### Questions to Ask Students Before Watching the Video

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

## Watch the Video

### Questions to Kick-Start Class Discussion After the Video

1. How are flying acrobats and ice-skaters able to speed up or slow down their rotations? Does this require some outside force?
No outside force is required, just the ability to change mass distribution while spinning. You can't change your mass very quickly, but you can change how it is distributed.
2. What is moment of inertia?
Moment of inertia is roughly the product of mass and its distance from the center of mass, or center of rotation.
3. What is angular momentum?
This is the rotational analog of linear momentum—the "momentum of rotation".
4. What is changing (physically), as the flier/skater changes shape?
Mass is located at a larger or smaller distance from the pivot point/axis of rotation.

### Connections to Everyday Life

Students who are fans of extreme sports will be very familiar with tucking the body to spin faster...half-pipe snowboarders/skateboarders, aerial skiers, and big-air competitors all do this. Gyroscopes are another common example. Astronauts in space must be anchored down to use any sort of wrench or rotational tool. Cars with flywheels must have two of them, spinning opposite to each other, or else have one positioned horizontally, in order to be able to turn.

## Suggested Classroom Activities

### Activity 1: Gyroscopes

Given toy gyroscopes, students can get an intuitive sense for how conservation of angular momentum, "feels." Changing the plane of rotation is difficult because angular momentum is conserved. Ask them to balance one on top of the other, replicating a very stable version of the Nanjing acrobats (Center of Mass). Hanging the gyroscope "sideways" on the end of a string is another good demonstration.

### Activity 2: Spinning Students

If you have access to a few office-type swivel chairs, students can replicate the spinning ice-skater or acrobat. Have a student sit in the chair holding a textbook in each hand, arms outstretched. Ask another student to spin the first one around and let go. Once spinning freely, ask the seated student to pull in the textbooks toward his or her torso. This will cause to the student to spin significantly faster. Be sure to use caution!

If you have a free-spinning bicycle wheel and a heavy-duty turntable, you can have students feel a different kind of angular momentum conservation. While standing on the turntable, have the student hold the wheel's axle with both arms outstretched. Spin the wheel up to a good clip and ask the student to try and turn the wheel. Changing the plane of the wheel's rotation must be offset by a counter-rotation in a different plane in order for angular momentum to be conserved. This will cause the student to begin spinning on the turntable. Done correctly, it's quite a memorable experience. Using a solid rubber tube instead of an inflatable one will make the effect more pronounced. Another fun demo is to suspend the spinning wheel from a rope attached to one side of the axle; the wheel cannot flip over because angular momentum must be conserved.

### Activity 3: Conservation of Angular Momentum Video Analysis

The filming of Anna in the Russian Barre routine allows students to see in slow motion how she speeds up and slows down in a tight spin while in the air. You can use VideoPoint qualitatively to see this by interpreting the graphs of position vs time.

To do this activity you will need to download the video "Video Analysis: Conservation of Angular Momentum (QT)". Use VideoPoint or similar software for graphing and analysis.

While it would be great to show conservation of angular momentum directly as she goes through her routine, it is difficult to make these tedious calculations given the unknown moment of inertia. However students can appreciate the slow motion video and they can mark the position of a point on her body and watch its progression. In the figure below the position of her knee was marked during a spin.

It is helpful to note that a person's belly button generally is quite close to their center of mass. So in this figure we are looking at the speed of her knee around her belly button. Students might note from the graph that the knee is moving fast while in the tight spin and then slows as she comes out of the spin. To do a more detailed calculation requires students to find the slope of each curve in the x and y directions, and then use Pythagoras's theorem to find the total speed at any given time.