question_answer

Four spheres of diameter la and mass M are placed with their centres on the four comers of a square of side b. Then the moment of inertia of the system about an axis along one of the sides of the square is

A)  \[\frac{4}{5}M{{a}^{2}}+2M{{b}^{2}}\]                 

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

C) \[\frac{8}{5}M{{a}^{2}}\]             

D) \[\frac{4}{5}M{{a}^{2}}+4M{{b}^{2}}\]

Correct Answer: B

Solution :

[b] \[{{I}_{1}}=\frac{2}{5}M{{a}^{2}}=M.I.\]  of sphere 1 about AB axis \[{{I}_{2}}=\frac{2}{5}M{{a}^{2}}=MI\] of sphere 2 about AB axis \[{{I}_{3}}=\frac{2}{5}M{{a}^{2}}+M{{b}^{2}}=M.I.\] of sphere 3 about AB axis \[{{I}_{4}}=\frac{2}{5}M{{a}^{2}}+M{{b}^{2}}=M.I.\] of sphere 4 about AB axis M.I. of system about AB axis \[{{I}_{system}}={{I}_{1}}+{{I}_{2}}+{{I}_{3}}+{{I}_{4}}\] \[=2\left( \frac{2}{5}M{{a}^{2}} \right)=2\left( \frac{2}{5}M{{a}^{2}}+M{{b}^{2}} \right)\] \[=\frac{8}{5}M{{a}^{2}}+2M{{b}^{2}}\]