Consider a rod of mass $$M$$ and length $$L$$ pivoted at its centre free to rotate in a vertical plane. The rod is at rest in the vertical position. A bullet of mass $$M$$ moving horizontal at a speed $$v$$ strikes and gets embedded in one end of the rod. The angular velocity of the rod just after the collision will be

A particle of mass $$m$$ is allowed to fall freely under gravity from a point $$P$$ as shown in the figure. If the vector position $$Q$$ of the particle from the origin is represented by $$\mathbf{r}$$, the magnitude of torque acting on the particle at time $$t$$ with respect to the origin $$O$$ is.