A homogeneous rod AB of length L and mass M is pivoted at the centre O in such a way that it can rotate freely in the vertical plane. The rod is initially in the horizontal position. An insect S of the same mass M falls vertically with speed V on the point C mid-way between O and B. The initial angular velocity $\omega$ in terms of V and L is

Q. A homogeneous rod AB of length $L$ and mass $M$ is pivoted at the center O in such a way that it can rotate freely in the vertical plane. The rod is initially in the horizontal position. An insect S of the same mass $M$ falls vertically with speed $V$ on the point C mid-way between O and B and sticks to it. The initial angular velocity $\omega$ in terms of $V$ and $L$ is :

Q. A homogeneous rod $XY$ of length $L$ and mass $M$ is pivoted at the centre $C$ such that it can rotate freely in the vertical plane. Initially, the rod is in the horizontal position. A blob of wax of same mass $M$ as that of the rod falls vertically with the speed $V$ and sticks to the rod midway between points $C$ and $Y$. If the rod rotates with angular speed $\omega$ what will be angular speed in terms of $V$ and $L$?

Q. A uniform rod of mass $m$ and length $L$ can rotate freely on a smooth horizontal plane about a vertical axis hinged at point $O$. A point mass having same mass $m$ coming with an initial speed $u$ perpendicular to the rod, strikes the rod in-elastically at its free end. Find out the angular velocity of the rod just after collision?

Q. Moment of inertia of uniform rod of mass 'M' and length 'L' about an axis through its centre and perpendicular to its length is given by $\frac{M{L}^2}{12}$. Now consider one such rod 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 horizontally at a speed 'v' strikes and embedded in one end of the rod. The angular velocity of the rod just after the collision will be