Conservation of Angular Momentum

Once a skater has generated (or an object has obtained) angular momentum, external forces may act to reduce the angular momentum of the skater. However, in situations, where there are no external forces producing torque about the axis of rotation, the skater's angular momentum will remain constant. This is called the conservation of angular momentum and holds true for any object.

While we illustrated this concept with a figure skating spin, angular momentum is not actually conserved in a spin due to the friction of the ice. In other words, even though a skater can increase and decrease his or her angular velocity by changing body position, as the spin progresses the skater's angular momentum gets smaller. Eventually, no matter the skater's body position, he or she will come to a stop.

However, during jumps, when the skater is rotating in the air, his or her angular momentum is conserved. This means that however much angular momentum the skater generated during take-off (by applying forces to the ice), he or she can not change it in the air. This is a very critical concept for skaters to understand, because they need to generate enough angular momentum so that they will be able to complete their jumps.

In each of the following single, double, and triple axel jumps, the skater has 0.5 seconds of flight time. The skater generates the maximum amount of angular momentum of which he is capable for each jump. Knowing the amount of angular momentum, and the average rotation speed he must attain to complete the 1-1/2, 2-1/2 and 3-1/1 revolutions in 0.5 seconds, calculate the average moment of inertia of each jump.

When the skater completes a triple jump, he must attain a much tighter body position than in a single jump. The tighter body position lowers his moment of inertia so that he can rotate fast enough to complete three revolutions before landing.

However, a skater does not maintain a constant body position during a jump. In flight, you will see a skater bring his or her arms into his or her body. Recall that a skater will be able to produce more angular momentum in an open body position (where he has a greater moment of inertia) than in a closed position, so at the instant of take-off the skater wants to be in an open position. If a skater stayed in this open position, he would not be able to increase his rotation speed during the jump, and thus would not be able to complete very many rotations in the air.

Look at the three jumps again (use your BACK button), and notice how the skater completes a single jump in a fairly open body position, but is very tight in the triple. This is because, he has similar amounts of angular momentum in all three jumps, and is adjusting his rotation speed for the different jumps by using different body positions.

Up until know we have only talked about increasing rotation speed by lowering the moment of inertia. Since angular momentum is conserved in the air, the skaters can also decrease rotation speed by increasing their moment of inertia. For example, when a skater wants to stop rotating to prepare for landing he or she open his or her arms. Like wise, in some simpler jumps (single and doubles), skaters will adjust their body position in flight to obtain dramatic increases and decreases in rotation speed for artistic flare. In the more complicated jumps, such as triples and quadruples, skaters do not have this luxury. Most rotates fast as possible during the entire jump to complete the 3 to 4 revolutions.

To calculate how changing the body position changes the rotation speed during the jumps, we just need to know the angular momentum of the skater and the moments of inertia for the different body positions.

One of the limiting factors of skaters being able to perform a jump is arm strength. Skaters can produce such high angular momenta at take-off, that they may not be physically strong enough to counteract the g-forces experienced during the rotation to bring their arms into a tight rotating position. The g-force felt by the arms during a jump or spin may exceed 4 G's.

The conservation of angular momentum holds true for any object in the air or experiencing no external torque. While it is particularly important for sports such as gymnastics, diving, arial skiing, and figure skating, it is also important in designing helicopters and other types of equipment which have rotational components. To gain an understanding of the importance of angular momentum in activities which do not involve changing your body position to control angular velocity, you can do a simple experiment with a turn table (rotating stool) and a bicycle wheel.

While sitting or standing on a stool or turn table, hold a bicycle wheel by the axle in your hands. Have someonw start the bicycle wheel spinning. Next, turn the bicycle wheel 90 degrees to the right or left, and then turn the wheel back the other direction.

Questions

What happens to the person on the stool? How can you explain this?

Solution

The person and the bicycle have zero angular momentum about he vertical axis (spinning axis) of the stool. The wheel does have angular momentum about the axle of the wheel (a horizontal axis), but angular momentum about this axis is not conserved. In other words, there are many external forces which keep the person from rotating about the horizontal axis. However, the person is free to rotate about the spinning axis of the stool since we assume the stool has no friction. Thus, when the person turns the wheel 90 degrees, the wheel has angular momentum about it's vertical axis. Since no external forces have acted, angular momentum about the system's vertical axis must be conserved or remain zero. Thus, the person rotates in a way opposite to that of the spinning wheel to counteract the angular momentum of the wheel. Notice that the person does not spin as fast as the wheel is rotating since the person has a larger moment of inertia. It is the angular momentum of the wheel and person which must be exactly opposite each other, not their rotating velocities, as is illustrated in the following equation:

Hsystem = Hwheel + Hperson

0 = Hwheel + Hperson

Hwheel = - Hsystem

Iwheelwwheel = - Ipersonwperson

The rotation you observed of the person in the opposite direction of the wheel is very similar to the way a person can rotate his or her arms and legs in the air to stop or start the body from rotating. For example, long jumpers move their arms and legs in the air to keep from falling forward. Similarly, if you fall from a building, by rotating your arms forward you could get your body to rotate backwards. The important fact is that the total system angular momentum must stay the same. Thus, any positive angular momentum of one segment must be counteracted by a negative angular momentum of another segment.

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