Physics at the Olympics

Congratulations to Yu-Na Kim, who won the Olympic gold medal in ladies' figure skating last week. I could gush on and on about Yu-Na Kim and her exquisite skating. Normal people have strengths and weaknesses, and the same goes for skaters. Some skaters have great artistry but lackluster jumps, or athletic jumping but unrefined artistry. Yu-Na Kim is very rare in that all the elements of her skating are excellent. It really isn't fair. But I love watching her.

If you were watching figure skating at the Olympics last week, you were watching physics in action! There are some very simple laws of physics at work that can make or break great skating.

Take a “scratch spin”. That's one of the first real spins that a skater learns, and the technique is very simple. Despite its simplicity, it looks very cool.



See what I mean?

So how can a skater keep spinning for so long? And if you watched the video of the scratch spin, you might have noticed that the skater actually spins faster as the spin progresses. So how is she increasing her angular velocity as time passes?

Know the answer?




It's conservation of angular momentum.

Let's call angular momentum “L”. In high school physics you learn that angular momentum is the moment of inertia (I) times the angular velocity (ω).

L = Iω

You can think of the moment of inertia as the resistance of an object to rotating about its axis. If you have a cylinder or a disk or a sphere, there is a certain resistance for that object to spin.

The moment of inertia is proportional to the mass times the square of the radius:

I = MR²

The larger the radius of an object, the larger the resistance to make it spin. Try spinning two coins of equal mass on the ground. If the second coin has a larger radius than the first coin, it will be harder to make it spin as fast as the first coin. In fact, since the moment of inertia is proportional to the square of the radius, doubling the radius would quadruple the moment of inertia.

Angular velocity (ω) is the number of revolutions per second. When a skater spins, she wants to maximize her angular velocity – she wants to spin fast!

When a skater starts a scratch spin, she spreads her arms out wide and extends them to both sides. This creates a large radius from the center of her body. She also sticks her free leg out for extra radius points.

r_a is the radius resulting from the arm, and r_l is the radius resulting from the leg. Here r_a and r_l are maximum values.

Then she slowly pulls her arms towards her chest. She also brings her free leg in towards her spinning leg. This decreases her spinning radius.

r_a and r_l are decreasing, so ω increases.

As we said, the laws of physics dictate that angular momentum is conserved. This means that “L” is a fixed value. By decreasing her radius, the skater decreases her moment of inertia (I). In order for L to remain fixed, the angular velocity (ω) must increase.

Here r_a and r_l are minimum values, so ω is large.

And that's exactly what you see when you watch a skater decrease her spinning radius (and therefore her moment of inertia). The speed of the spin increases. There is no extra pushing or external forces involved – just the skater's muscles working to pull her arms and leg inward.

So the next time you watch a skater end their performance with one of those long spin combinations – okay, he's doing a camel spin, into a sit spin, into some contorted position, into a reverse sit spin, and now oh my god he's suddenly spinning so much faster it's crazy and the crowd is on its feet...!!! You can think, “oh, well that's just conservation of angular momentum”. Sweet and simple.

Thanks to Jen Leslie for the idea to write about the science behind the Olympics! Check out her blog at Scientific (mis)Communications.

More info:
Scientific American on the physics of figure skating
http://www.bsharp.org/physics/spins