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 diameter is :

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$\frac{2 M}{5} \frac{(R_{2}^{5} - R_{1}^{5})}{(R_{2}^{3} - R_{1}^{3})}$

$\frac{2 M}{3} \frac{(R_{2}^{5} - R_{1}^{5})}{(R_{2}^{3} - R_{1}^{3})}$

$\frac{4 M}{7} \frac{(R_{2}^{5} - R_{1}^{5})}{(R_{2}^{3} - R_{1}^{3})}$

$\frac{4 M}{5} \frac{(R_{2}^{5} - R_{1}^{5})}{(R_{2}^{3} - R_{1}^{3})}$

answer is A.

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Detailed Solution

$ρ = \frac{M}{\frac{4 π}{3} ({R_{2}}^{3} - {R_{1}}^{3})} I_{s h e l l} = \frac{2 M_{2} {R_{2}}^{2}}{5} - \frac{2 M_{1} {R_{1}}^{2}}{5} - - - (1) M_{2} = ρ \frac{4 π {R_{2}}^{3}}{3} = \frac{M}{\frac{4 π}{3} ({R_{2}}^{3} - {R_{1}}^{3})} \frac{4 π {R_{2}}^{3}}{3} = \frac{M {R_{2}}^{3}}{({R_{2}}^{3} - {R_{1}}^{3})} l l y M_{1} = \frac{M {R_{1}}^{3}}{({R_{2}}^{3} - {R_{1}}^{3})} s u b s t i t u t i n g e q n (1) a a n d (2) w e g e t I_{s h e l l} = \frac{2 M ({R_{2}}^{5} - {R_{1}}^{5})}{5 ({R_{2}}^{3} - {R_{1}}^{3})}$