Aphysics professor that doesn’t get easily embarrassed stands at the center of a frictionless turntable with arms outstretched and a 5.0-kg dumbbell in each hand. he is set rotating about the vertical axis, making one revolution in 2.0s. find his angular velocity if he pulls the dumbbells to his stomach. his moment of inertia (without the dumbbells) is 3.0 kg*m^2 with arms outstretched and 2.2 kg*m^2 with his hands at his stomach. the dumbbells are 1.0 m from the axis initially and 0.20 m at the end. how about kinetic energy before and after? explain from where this energy (if any) came from?

Apply conservation of angular momentum:
L = Iw = const.
L = angular momentum, I = moment of inertia, w = angular velocity, L must stay constant.

L must stay the same before and after the professor brings the dumbbells closer to himself.

His initial angular velocity is 2π radians divided by 2.0 seconds, or π rad/s. His initial moment of inertia is 3.0kg•m^2

His final moment of inertia is 2.2kg•m^2.

Calculate the initial angular velocity:
L = 3.0π

Final angular velocity:
L = 2.2w

Set the initial and final angular momentum equal to each other and solve for the final angular velocity w:

3.0π = 2.2w
w = 1.4π rad/s

The rotational energy is given by:
KE = 0.5Iw^2

Initial rotational energy:
KE = 0.5(3.0)(π)^2 = 14.8J

Final rotational energy:
KE = 0.5(2.2)(1.4)^2 = 21.3J

There is an increase in rotational energy. Where did this energy come from? It came from changing the moment of inertia. The professor had to exert a radially inward force to pull in the dumbbells, doing work that increases his rotational energy.

5 seconds

Explanation:

Acceleration = final velocity - initial velocity all over time taken

so time = change in velocity divided by acceleration