A professor sits on a rotating stool that is spinning at 10.0 rpm while she holds a heavy weight in each of her hands. Her outstretched hands are 0.785 m from the axis of rotation, which passes through her head into the center of the stool. When she symmetrically pulls the weights in closer to her body, her angular speed increases to 32.5 rpm. Neglecting the mass of the professor, how far are the weights from the rotational axis after she pulls her arms in

In this case the System is formed by the teacher with the two masses, so the forces during movement are internal and the angular momentum is conserved

Initial.

L₀ = I₀ w₀

Final

Lf = I w

L₀ = Lf

I₀ w₀ = I w

The moment of inertia is

I₀ = m r₀²

I = m r²

Let's replace

m r₀² w₀ = m r² w

r² = r₀² w₀ / w

Angular velocity

w₀ = 10 rpm (2pi rad / 1 rev) (1 min / 60 s) = 1.047 rad / s

w = 32.5 rpm = 3.403 rad / s

Let's calculate

r = √(0.785 1.047 / 3.403)

r = 0.491 m