Question

An ice skater is spinning about a vertical axis with arms fully extended. If the arms are pulled in closer to the body and assuming no change in the skater's elevation, which of the following statements are true? There are two answers.

a) Kinetic energy remains the same.
b) Angular momentum decreases.
c) Kinetic energy increases.
d) Angular momentum is conserved.

Answer 1

I know that C is correct and A is not, but I can't help you with the other two.

Answer 2

Just like linear momentum, angular momentum is always conserved unless acted on by an external torque. In this case, there is no external torque (the "shape" of the ice skater is just changing), so D automatically must be right.

If you want to consider the math behind this, remember that L=Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. When the skater pulls their arms in, I decreases (note that I=Σmr, and r decreases), but this is compensated for by an increase in ω (the skater spins faster), meaning there is no change in L.

As @hhopke mentioned, C is correct as well because the skater necessarily has more kinetic energy because they are spinning faster. Mathematically, KE = (½)Iω^2. As I explained above, I decreases and ω increases. These changes cancel each other when calculating angular momentum. Here, however, we have the SQUARE of ω, meaning the increase in ω outweighs the decrease in I, and KE has a net increase!

Answer 3

Thanks. I picked c and d as the answers and was told by the test makers that the answers are actually a and d.

Answer 4

Kinetic energy definitely increases. The skater does work to move their arms in. By conservation of energy, this work has to go somewhere, and that somewhere is additional kinetic energy.

Answer 5

Great! This confirms my thoughts exactly!