question_answer

A light rod of length l has two masses \[{{m}_{1}}\]and \[{{m}_{2}}\] attached to its two ends. The moment of inertia of the system about an axis perpendicular to the rod and passing through the centre of mass is

A)  \[\frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{I}^{2}}\]               

B)  \[\frac{{{m}_{1}}+{{m}_{2}}}{{{m}_{1}}{{m}_{2}}}{{I}^{2}}\]

C)  \[({{m}_{1}}+{{m}_{2}}){{I}^{2}}\]           

D)  \[\sqrt{{{m}_{1}}{{m}_{2}}}{{I}^{2}}\]

Correct Answer: A

Solution :

\[{{X}_{1}}\frac{{{m}_{2}}I}{{{m}_{1}}+{{m}_{2}}}and\,{{X}_{2}}=\frac{{{m}_{1}}I}{{{m}_{1}}+{{m}_{2}}}\] \[I={{m}_{1}}{{X}_{1}}^{2}+{{m}_{2}}{{X}_{2}}^{2}=\frac{{{m}_{1}}{{m}_{2}}}{{{m}_{1}}+{{m}_{2}}}{{I}^{2}}\]