Physics: What is her momentum of Inertia?

Question

A figure skater can increase her angular velocity from a speed of 2.00 revolutions in 3.00 seconds to 5.00 revolutions in 2.00 seconds by bringing her arms in. If her moment of inertia was 4.60 kg m2 with her arms out, what is it with her arms in?

Answer 1

Let θ is angular change in position, w is angular velocity, t is time, I is inertia.

 

What We know

Arms out                                                                      Arms in

θ_out = 2.00 revolutions                                                  θ_in = 5.00 revolution

t = 3.00 sec                                                                  t = 2.00 sec

I_out = 4.60 kgm²                                                           I_in = ??????

 

Let w represent angular velocity. w = θ/t

w_out = 2.00/3.00 rev/sec = 0.667 rev/sec                       w_in = 5.00/2.00 rev/sec = 2.50 rev/sec

 

Let L represent angular momentum. L = Iw. Conservation of angular momentum tells us that L_out = L_in so,

(I_out)(w_out) = (I_in)(w_in)

(I_out)(w_out)/(w_in) = (I_in)

( 4.60 kgm²)(0.667 rev/sec )/( 2.50 rev/sec) = (I_in)

I_in = 1.227