Chapter 9, Problem 68GP

* A horizontal, circular platform can rotate around a vertical axis at its center with negligible friction. You decide to use the rotating platform to design a procedure that will allow you to determine the unknown rotational inertias of different objects, for example, a bowl. You know the rotational inertia of the platform $I_{\text{P}}$. Which of me following procedures would best serve you task? (More than one answer could be correct.) Comment on the procedures that are not suitable and explain what is wrong with them.

a. Exert a constant torque $\tau$ on the empty platform and measure its angular acceleration $\alpha \text{P}$. Then place the bowl on the platform at its center and repeat the previous experiment, obtaining $\alpha \text{PB}$. Calculate the rotational inertia of the bowl using the expression $I_{\text{B}}=\frac{\tau }{\alpha \text{PB}-\alpha \text{P}}$.

b. Perform the same experiment as in (a) but use the expression $I_{\text{B}}=I_{\text{P}}\left(\frac{\alpha \text{p}}{\alpha \text{PB}}-1\right)$.

c. Spin the empty platform and measure its angular velocity ${\omega }_{\text{i}}$. Whle the platform is rotating, place the bowl carefully on the platform at its center. Wait until the bowl rotates together with the platform. Measure the final angular velocity of the bowl-platform system ${\omega }_{\text{f}}$. Calculate the rotational inertia of the bowl using the expression $I_{\text{B}}=I_{\text{P}}\left(\frac{{\omega }_{\text{i}}}{{\omega }_{\text{f}}}-1\right)$.

d. Perform the same experiment as in (c) but use the expression $I_{\text{B}}=I_{\text{P}}\left(\frac{{\omega }_{\text{i}}{}^2}{{\omega }_{\text{f}}{}^2}-1\right)$.

e. Measure the mass of me platform $m\text{P}$. the mass of the bowl $m\text{B}$. the radius of the platform $r\text{P}$, and the radius of the bowl $r\text{B}$. Calculate the rotational inertia of me bowl using the expression $I_{\text{B}}=I_{\text{P}}•\frac{m_{\text{B}}{r}_{\text{B}}{}^2}{m_{\text{p}}{r}_{\text{p}}{}^2}$.