question_answer

The moment of inertia of a hollow thick spherical shell of mass M and its inner radius \[{{R}_{1}}\] and outer radius \[{{R}_{2}}\] about its diameter is

A) \[\frac{2M}{5}\frac{(R_{2}^{5}-R_{1}^{5})}{(R_{2}^{5}-R_{1}^{3})}\]

B) \[\frac{2M}{5}\frac{(R_{2}^{5}-R_{1}^{5})}{(R_{2}^{3}-R_{1}^{3})}\]

C) \[\frac{4M}{5}\frac{(R_{2}^{5}-R_{1}^{5})}{(R_{2}^{3}-R_{1}^{3})}\]

D) \[\frac{4M}{3}\frac{(R_{2}^{5}-R_{1}^{5})}{(R_{2}^{3}-R_{1}^{3})}\]

Correct Answer: A

Solution :

[a] \[\rho =\frac{M}{\frac{4}{3}\pi (R_{2}^{3}-R_{1}^{3})}\]            \[{{I}_{shell}}=\frac{2}{5}{{M}_{2}}R_{2}^{2}-\frac{2}{5}{{M}_{1}}R_{1}^{2}\]   ?. (1) \[{{M}_{2}}=\rho \times \frac{4}{3}\pi R_{2}^{3}\] \[=\frac{MR_{2}^{3}}{(R_{2}^{3}-R_{1}^{3})};\,{{M}_{1}}=\frac{MR_{1}^{3}}{R_{2}^{3}-R_{1}^{3}}\] Putting values of \[{{M}_{1}}\]and \[{{M}_{2}}\]in eq. (1), \[{{I}_{shell}}=\frac{2M}{5}\frac{(R_{2}^{5}-R_{1}^{5})}{(R_{2}^{3}-R_{1}^{3})}\]