$A$ cube of side $a$ is moving with velocity $v$ on a smooth horizontal surface. It hits a linear raised obstacle $O$ on the horizontal surface (as shown in the figure). The angular speed of the block after hitting $O$ will be

$A$ metal sphere of radius $r$ and specific heat $S$ is rotated about an axis passing through its centre at a speed of $f$ rotations per second. It is suddenly stopped,and $50\%$ of its rotational kinetic energy is used in increasing its temperature. The rise in temperature of the sphere is

Fill in the blanks:
$(1)$ If $|\vec{A} \times \vec{B}| = \vec{A} \cdot \vec{B}$,then the angle between $\vec{A}$ and $\vec{B}$ is ............ .
$(2)$ The angle between the angular momentum and linear momentum of a particle in rotational motion is ............ .
$(3)$ $A$ force $F\hat{k}$ acts on a particle having position vector $(2\hat{i} + \hat{j})$. The torque acting on the particle is ............ .

$A$ thin hollow cylinder slides with velocity $v$ without rotating. It then rolls without slipping with the same velocity $v$. Find the ratio of the kinetic energies in the two cases.

$A$ rod of mass $M$ and length $l$ is at rest on a smooth horizontal surface. $A$ particle of the same mass $M$ strikes one end of the rod with velocity $u$ perpendicular to the rod,elastically. Just after the collision,what is the kinetic energy of the upper half part of the rod?

$A$ $2 \, kg$ steel rod of length $0.6 \, m$ is clamped on a table vertically at its lower end and is free to rotate in a vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end,the speed of the free end of the rod when it passes through its lowest position is $\ldots \ldots \ldots \ldots \, ms^{-1}$. (Take $g = 10 \, ms^{-2}$)