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Imagine an ice skater spinning around its axis. It is well-known that if she extends her arms, she is spinning slower and by moving her arms towards the body, she is spinning faster.

But what happens if she only extends one arm and the other arms is left close to the body?

In that case, the center of mass (COM) moves. But, as no external force is acting, the COM cannot start to rotate. Hence, the rotation must still be around the COM and the ice skater should start to move in a cirle around the new COM.

Is this true? Or is she still spinning around the same axis?

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Suppose you mark "x" on the ground at the point about which she is rotating. Now she extends her arm.

In that case, the center of mass (COM) moves. But, as no external force is acting, the COM cannot start to rotate. Hence, the rotation must still be around the COM and the ice skater should start to move in a cirle around the new COM.

And now the axis passing through COM also passes through the point "x" which you earlier marked on the ground. (Conservation of Momentum).enter image description here

EDIT: To make things a bit more clear, I will say as the hand arm moves away(supposedly left), the centre of mass of the body will be stationary at the axis of rotation as no external foce is applied. And to keep the center of mass stationary the rest body will move to right of the axis.

All in all COM will be stationary at the initial axis and rotation will be about it(or axis you may say as they coincide for that matter).

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