question_answer

For spheres each of mass M and radius R are placed with their centres on the four corners 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 \[AD={{I}_{A}}+{{I}_{D}}+{{I}_{B}}+{{I}_{C}}\]                 \[=\frac{2}{3}M{{R}^{2}}+\frac{2}{5}M{{R}^{2}}+\left( M{{b}^{2}}+\frac{2}{3}M{{R}^{2}} \right)\]                                                 \[+\left( M{{b}^{2}}+\frac{2}{5}M{{R}^{2}} \right)\] \[=\frac{32}{15}M{{R}^{2}}+2M{{b}^{2}}\]