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In figure skating, if you are doing a spin on ice and you leave your arms spread out it creates resistance. If you pull the arms in toward the body, you create less resistance so you spin more easily. How do I express this with mathematical equations in physics?

 Figure Skater Yu-Na Kim
Figure Skater Yu-Na Kim. Image Credit: YN 08-A09 via flickr

The physical phenomenon involved is not really resistance; it's inertia. Sometimes, however, the term 'inertia' is explained as 'resistance to changes in motion'.

For ordinary motion, inertia has close ties to mass. The main equations that help us understand spinning motion (like the figure skater) involve the conservation of angular momentum. In spinning motion, unlike regular "push-pull" motion, the way the mass distributes with respect to the axis of the rotation becomes vitally important.

The "axis" is that imaginary line about which an object is spinning. You can imagine a figure skater having an "axle" line that runs from the skater's feet through her head while remaining perpendicular to the ice surface.

To describe the angular momentum of the spinning skater, we need just a couple ingredients: the speed that the skater is spinning — the angular speed denoted with the Greek Letter (small case) omega — and the moment of inertia, usually given by the letter "I". The moment of inertia measures how much an object fights against this spinning motion, and it varies for different shapes.

The angular momentum of the spinning skater is the product of the rotational inertia, "I", and the rotational speed, "omega".

Conservation of angular momentum implies that this product remains constant. So if you change the "I", the "omega" must change in the other direction. To understand how I can change, we just need to think about how the mass is distributed relative to the axis.

The further the mass is from the rotational axis, the larger the "I". In the example of a figure skater with outstretched arms, "I" is larger than it is when the skater pulls the arms close to the axis. So, when the skater goes from larger "I" (outstretched arms) to smaller "I" (arms held close), the "omega" has to get bigger. Consequently, a skater with arms held close will spin much faster.

The equations look like this:

Equations for Angular Momentum

Answered by:

Toni Sauncy, PhD
Interim Director of Society of Physics Students and Sigma Pi Sigma
American Institute of Physics

Submitted by:

Zohara from Vermont