question_answer

A thin rod of length L and mass M is bent at its midpoint into two halves so that the angle between them is \[{{90}^{o}}\]. The moment of inertia of the bent rod about an axis passing through the bending point and perpendicular to the plane defined by the two halves of the rod is [AIPMPT (S) 2008]

A) \[\frac{M{{L}^{2}}}{24}\]

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

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

D) \[\frac{\sqrt{2}M{{L}^{2}}}{24}\]

Correct Answer: B

Solution :

Since rod is bent at the middle, so each part of it will have same length \[\left( \frac{L}{2} \right)\] and mass \[\left( \frac{M}{2} \right)\] as shown.

Moment of inertia of each part through its one end \[=\frac{1}{3}\left( \frac{M}{2} \right){{\left( \frac{L}{2} \right)}^{2}}\]

Hence, net moment of inertia through its middle point O is

\[I=\frac{1}{3}\left( \frac{M}{2} \right){{\left( \frac{L}{2} \right)}^{2}}+\frac{1}{3}\left( \frac{M}{2} \right){{\left( \frac{L}{2} \right)}^{2}}\]

\[=\frac{1}{3}\left[ \frac{M{{L}^{2}}}{8}+\frac{M{{L}^{2}}}{8} \right]=\frac{M{{L}^{2}}}{12}\]