question_answer

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)   \[\frac{8}{3}M{{R}^{2}}+2M{{b}^{2}}\]

B)   \[\frac{8}{5}M{{R}^{2}}+2M{{b}^{2}}\]                 

C)   \[\frac{32}{15}M{{R}^{2}}+2M{{b}^{2}}\]

D)  \[32M{{R}^{2}}+4M{{b}^{2}}\]                

Correct Answer: C

Solution :

Moment of inertia of a hollow sphere of radius ( R about the diameter passing through' D is \[{{I}_{A}}=\frac{2}{3}M{{R}^{2}}\] ?.(i) Moment of inertia of solid sphere about diameter               \[{{I}_{B}}=\frac{2}{5}M{{R}^{2}}\] ...(ii) \[\therefore \] Moment of inertia of whole system about side    \[AD={{I}_{A}}+{{I}_{D}}+{{I}_{B}}+{{I}_{C}}\] \[=\frac{2}{3}M{{R}^{2}}+\frac{2}{5}M{{R}^{2}}+\left( M{{b}^{2}}+\frac{2}{3}M{{R}^{2}} \right)+\left( M{{b}^{2}}+\frac{2}{5}M{{R}^{2}} \right)\]\[=\frac{32}{15}M{{R}^{2}}+2M{{b}^{2}}\]