Question

a skater with arms initally extend stats spining on the ice at 3rad/s she then pulls her arms in close to her body which is the result

Answer 1

When the skater was spinning initially with extended arms the inertal will be more for the body as arms are at greater distance from the axis of rotation which is the body. Let this inertia be I₁. The angular speed now is w₁ = 3 rad/sec. So the initial angular momentum is L₁ = I₁*w₁ = 3I₁.

Then she pulls her arms close to her body which decreases the inertia as arms come close to the axis. So let this lesser inertia be I₂ and now the angular speed of rotation be w₂. Now the angular momentum is L₂ = I₂*w₂.

Since there was no external torque acting on the system that is the skater and her arms during the pulling in of arms, the angular momentum of the system will be conserved. That implies L₁ = L₂, implies 3I₁ = I₂*w₂,

implies w₂ = 3*(I₁/I₂), from this eqaution we can say that (I₁/I₂) is greater than 1 as I₁ > I₂. So w₂ > 3 rad/sec. Which implies that as the skater pulls her arms in, she begins to rotate faster.