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 \[90{}^\circ \]. 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

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 the 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 of complete structure through its middle point O is \[I=\frac{1}{3}\left( \frac{M}{2} \right){{\left( \frac{L}{2} \right)}^{2}}+\frac{1}{3}\left( \frac{M}{2} \right){{\left( \frac{L}{2} \right)}^{2}}\] \[=\frac{1}{3}\left( \frac{M{{L}^{2}}}{8}+\frac{M{{L}^{2}}}{8} \right)=\frac{M{{L}^{2}}}{12}\]