Question

For spheres each of mass $M$ and radius $R$ are placed with their centres on the four comers $A, B, C$ and $D$ of a square of side $b$. The spheres $A$ and $B$ are hollow and $C$ and $D$ are solids. The moment of inertia of the system about side $AD$ of square is

• A. $\frac{8}{3}MR^{2}+2Mb^{2}$
• B. $\frac{8}{5}MR^{2}+2Mb^{2}$
• C. $\frac{32}{15}MR^{2}+2Mb^{2}$
• D. $32MR^{2}+4Mb^{2}$

Answer 1

Answer: (C)

$\frac{32}{15}MR^{2}+2Mb^{2}$

Answer 2

Moment of inertia of a hollow sphere of radius $R$ about the diameter passing through $D$ is
$I_{A}=\frac{2}{3} M R^{2} \ldots(i)$
Moment of inertia of sol id sphere about diameter
$I_{B}=\frac{2}{5} M R^{2} \ldots(i i)$
$\therefore$ Moment of inertia of whole system about side
$A D=I_{A}+I_{D}+I_{B}+I_{C}$
$=\frac{2}{3} M R^{2}+\frac{2}{5} M R^{2}+\left(M b^{2}+\frac{2}{3} M R^{2}\right)+\left(M b^{2}+\frac{2}{5} M R^{2}\right)$
$=\frac{32}{15} M R^{2}+2 M b^{2}$