# Spinning ice skater and Coriolis force

Among the explanations for the Coriolis force I've seen, there is a mention of the ice skater's rotational acceleration by pulling arms in radially.

I've always understood the spinning ice skater as conservation of angular momentum. However, I do see that the Coriolis force cross product formulation of $$-2m\ \vec{\omega} \times \vec{v}$$ (cross product of rotational velocity and the velocity of pulling the arm in) yields a vector that is tangential to the arm's spiral sweep and should cause rotational acceleration.

Can someone confirm that the rotating ice skater system is a satisfactory example of this? I've already read through an analysis of forces in the inertial frame that is posted here, which seems pretty thorough but doesn't mention Coriolis so I'm not sure that my perspective is correct.

• My view of this is that the author of the excellent answer you linked concentrated on a very full account of the conservation of angular momentum, that he simply didn't think of the viewpoint from a reference frame that incorporated the Coriolis Force. I would think, obviously enough I suppose, if you could "map" every aspect of the ice skater to a standard CF example, you could confirm/ refute it. My point is that just because the author didn't mention it probably has no significance.
– user108787
Nov 3, 2016 at 0:09
• Right. Angular momentum is conserved. The term "coriolis force" is just the name for the force she feels making her speed up.. Nov 3, 2016 at 0:14
• Physics is full of examples that can be described by different approaches, yet they describe the same reality. Quite often one can choose between explicit integration of a force, and between applying conservation laws to some quantity. Or using variational principle. All should yield the same result. Aug 12, 2021 at 9:36

When the rotating skater pulls his arm in close to the body, his moment of inertia $I$ decreases. As the angular momentum $L=I\omega$, where $\omega$ is the angular velocity, is conserved, this means that the angular velocity $\omega$ has to increase so that $L$ stays the same. This has nothing to do with the Coriolis force.

Note: The Coriolis force is an apparent force that like the centrifugal force only appears in a rotational frame of reference. The Coriolis force is not seen in an inertial frame of an observer.

The rotational acceleration gained by a spinning ice skater can indeed be explained using the Coriolis force. Yes, there is no Coriolis pseudoforce in the inertial frame. However, the fact of the matter is that by contracting her arms, a sideways (tangential) force must somehow act on her body, for otherwise she cannot angularly accelerate. The question is, can this sideways force be explained from a fundamental viewpoint?

If we want to work directly in the inertial frame, we observe that when the ice skater contracts her arms (along what she thinks are straight lines), they are actually traveling along curved paths. This requires a lateral force to be exerted on her arms, and if she exerts this lateral force herself (with her own body), then in the inertial frame, this must, by Newton's Third Law, induce a tangential counter-force on her body that rotationally accelerates her.

But now let's see what this implies in the rotational frame: Since a sideways force must be exerted to simply move the skater's arms in a straight line in this frame, this means there must be an opposing sideways pseudoforce that acts on any radially moving object in this frame. This pseudoforce cannot be the centrifugal force, since that only acts along the radial direction (and thus cannot push the skater's arms sideways). Thus, we conclude that there must be another pseudoforce apart from the centrifugal force in the rotating frame, and this must give any radially moving object a sideways push. This pseudoforce is precisely what we call the Coriolis force.

Thus, the rotational acceleration caused by the ice skater's contracting her arms can be legitimately attributed to the existence of a Coriolis pseudoforce in her rotating frame that she needs to counter to move her arms radially. Of course, since the same rotational acceleration can also be explained using conservation of angular momentum in the non-rotating frame, this shows that the Coriolis force and the conservation of angular momentum are intimately connected.

The Coriolis force is always directed perpendicular to the movement of a body. So, it doesn't exert any influence on the speed and kinetic energy of the ice skater (in fact, it could only change his direction). In other words, it does no work in this situation.

On the other hand, the force one has to apply a force inward to counter the centrifugal force does do a mechanical work: this is the work the ice skater has to do by his muscles in order to keep spinning (that is, the increase in rotation is a result of him pulling in his arms).

Any of this is in contradiction with the answer you referred to. The author of that answer didn't considered a non-inertial frame in his explanation, that's all.