question_answer

An L-shaped thin uniform rod of total length 2l is free to rotate in a vertical plane about a horizontal axis at P as shown in the figure. The bar is released from rest. Neglect air and contact friction. The angular velocity at the instant it has rotated through \[90{}^\circ \] and reached the dotted position shown is

A) zero      

B)               \[\sqrt{\frac{6g}{5l}}\]

C) \[\sqrt{\frac{3g}{5l}}\]               

D) none

Correct Answer: B

Solution :

Moment of inertia of system of rods about P: \[l=\left( \frac{m{{l}^{2}}}{3} \right)\,\,+\,\,\left[ \frac{m{{l}^{2}}}{l2}+\frac{m5{{l}^{2}}}{4} \right]\,\,=\,\,\frac{5}{3}\,m{{l}^{2}}\]   \[Loss\,\,in\,\,PE\,\,=\,\,gain\,\,in\,\,KE\] \[mg=\frac{l}{2}+mg\frac{l}{2}=\frac{1}{2}l{{\omega }^{2}}\,\,\Rightarrow \,\,\omega =\sqrt{\frac{6g}{5l}}\]