A student sitting on a frictionless rotating stool has rotational inertia 0.96 kg*m^2 about a vertical axis through her center of mass when her arms are tight to her chest. The stool rotates at 7.20 rad/s and has negligible mass. The student extends her arms until her hands, each holding a 4.4 kg mass, are 0.72 m from the rotation axis. If each arm is modeled as a 0.72m long uniform rod of mass of 4.8kg and her total body mass is 65kg, how do I find the final rotational velocity?

Question

anonymous · Answer

We need to use the conservation of momentum. Rotational momentum is defined as$$P = I \omega$$

We need to find the new moment of inertia when she extends her arms. $$I_e = I_b + I_a + I_w$$where $I_e$ is the moment of inertia with her arms extended, $I_b$ is the moment of inertia of her body, $I_a$ is the moment of inertia of her arms, and $I_w$ is the moment of inertia of the weights she is holding. With the information given, I don't see any way to solve for $I_b$ (we don't know her body's mass distribution about her axis of rotation). For the sake of this example, we will assume it is the same as when her arms are tucked. $$I_e = I_b + {m_a L^2 \over 3} + m_w L^2$$$$I_e = 0.96 + {4.8 (0.72)^2 \over 3} + 4.4 (0.72)^2$$

Now, back to conservation of momentum$$I \omega = I_e \omega'$$where $I$ is the moment of inertia of the girl with her arms tucked, $\omega$ is the angular velocity with her arms tucked, $I_e$ is the moment of inertia of the girl when her arms are extended, and $\omega'$ is her new angular velocity after extending her arms.

anonymous · Answer

For this I get a result of 1.70, which the program says is incorrect. I said $$I _{i}*W _{i}/I _{f}$$ where Ii is the initial inertia of .96 and Wi being the initial angular velocity of 7.2 and If being what you call Ie, the inertia after she stretches her arms, which I calculated to 4.07. Do you have any idea what I am doing wrong?

anonymous · Answer

Yes. The assumption that $I_b = 0.96$ could be wrong. 

And my expression for $I_e$ is wrong. Sorry. She has two arms. Doh!$$I_e = I_b + 2*I_a + 2*I_w$$

anonymous · Answer

That did it, 0.96 was the answer. Thank you very much!

anonymous · Answer

Great. Good luck.