A disc is rotating with angular velocity $\vec{\omega}$. A force $\vec{F}$ acts at a point whose position vector with respect to the axis of rotation is $\vec{r}$. The power associated with torque due to the force is given by ……….

A small object of uniform density rolls up a curved surface with an initial velocity $v$. It reaches up to a maximum height of $\frac{3 \mathrm{v}^2}{4 \mathrm{~g}}$ with respect to the initial position. The object is

A solid sphere of mass $m$ and radius $R$ is rotating about its diameter. A solid cylinder of the same mass and same radius is also rotating about its geometrical axis with an angular speed twice that of the sphere. The ratio of their kinetic energies of rotation $E_{sphere}/E_{cylinder}$ will be

A rod of length $1\,meter$ is standing vertically, when its other end touches the ground without slipping then the speed of other end will be

One end of a straight uniform $1\; \mathrm{m}$ long bar is pivoted on horizontal table. It is released from rest when it makes an angle $30^{\circ}$ from the horizontal (see figure). Its angular speed when it hits the table is given as $\sqrt{\mathrm{n}}\; \mathrm{s}^{-1},$ where $\mathrm{n}$ is an integer. The value of $n$ is