question_answer

Two identical thin rods AB and CD having mass 'M and length 'L? are fixed as shown. Find the moment of inertia of the system about axis XX?

A) \[\frac{M{{L}^{2}}}{12}\]

B)                    \[\frac{M{{L}^{2}}}{12}\left( 1+{{\sin }^{2}}\theta  \right)\]

C) \[\frac{M{{L}^{2}}}{6}\]

D) zero

Correct Answer: B

Solution :

[b] \[{{I}_{1}}={{\int_{-L/2}^{+L/2}{\frac{M}{L}dy\,\left( y\sin \theta  \right)}}^{2}}\] \[{{I}_{1}}=\frac{M{{L}^{2}}}{12}{{\sin }^{2}}\theta \] \[{{I}_{2}}=\frac{M{{L}^{2}}}{12}\] \[\therefore \,\,\,\,I={{I}_{1}}+{{I}_{2}}=\frac{M{{L}^{2}}}{12}\left( 1+{{\sin }^{2}}\theta  \right)\]