question_answer

Four spheres each having mass m and radius rare placed with their centres on the four corners of a square of side a. Then the moment of inertia of the system about an axis along one of the sides of the square is

A)  \[\frac{8}{5}\,m{{r}^{2}}\]

B)  \[\frac{8}{5}\,m{{r}^{2}}+m{{a}^{2}}\]

C)  \[\frac{8}{5}\,m{{r}^{2}}+2\,\,m{{a}^{2}}\]

D)  \[\frac{4}{5}\,m{{r}^{2}}+4\,\,m{{a}^{2}}\]

Correct Answer: C

Solution :

Moment of inertia of the system \[{{I}_{AB}}={{I}_{A{{B}_{1}}}}+{{I}_{A{{B}_{2}}}}+{{I}_{A{{B}_{3}}}}+{{I}_{A{{B}_{4}}}}\] \[=\frac{2}{5}m{{r}^{2}}+\frac{2}{5}m{{r}^{2}}+\left( \frac{2}{5}m{{r}^{2}}+m{{a}^{2}} \right)\] \[+\left( \frac{2}{5}m{{r}^{2}}+m{{a}^{2}} \right)\] \[=\frac{8}{5}m{{r}^{2}}+2m{{a}^{2}}\]