Rotation physics

Question

A student sits on a freely rotating stool holding two weights, each of mass 3.09 kg. When his arms are extended horizontally, the weights are 1.08 m from the axis of rotation and he rotates with an angular speed of 0.756 rad/s. The moment of inertia of the student plus stool is 3.09 kg·m² and is assumed to be constant. The student pulls the weights inward horizontally to a position 0.302 m from the rotation axis.

(a) Find the new angular speed of the student in rad/s.

(b) Find the change in kinetic energy of the rotating system before and after he pulls the weights inward.

Answer 1

(a) The conservation of angular momentum states that the initial angular momentum of the system is equal to the final angular momentum of the system. Therefore:

I1ω1 = I2ω2

and since the moment of inertia is assumed to be constant, that is I1 = I2, we can also deduce that ω1 = ω2.

(b) Since I1 = I2 and ω1 = ω2, the kinetic energy of the rotating system is the same before and after he pulls the weights inward. Practically, this is because the change in potential energy in the form of pulling weights inward is equivalent to the resultant change in kinetic energy.

Answer 2

When weights are brought in, torque is 0 as the force is radial and towards the axis. This means that the angular momentum is conserved:

I₁ω₁ = I₂ω₂ or I and ω are inverse.

For the kinetic energy, use E_K = (1/2)Iω² and find the difference before and after.