A stationary horizontal disc is free to rotate about its axis. When a torque is applied on it, its kinetic energy as a function of $\theta ,$ where $\theta $ is the angle by which it has rotated, is given as $k\theta ^2.$ If its moment of inertia is $I$ then the angular acceleration of the disc is
$\frac {K}{I}\,\theta $
$\frac {K}{2I}\,\theta $
$\frac {K}{4I}\,\theta $
$\frac {2K}{I}\,\theta $
A disc is rolling without slipping on a straight surface. The ratio of its translational kinetic energy to its total kinetic energy is
The speed of a homogeneous solid sphere after rolling down an inclined plane of vertical height $h$, from rest without sliding, is
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
Two point masses of $0.3\ kg$ and $0.7\ kg$ are fixed at the ends of a rod of length $1.4\ m$ and of negligible mass. The rod is set rotating about an axis perpendicular to its length with a uniform angular speed. The point on the rod through which the axis should pass in order that the work required for rotation of the rod is minimum is located at a distance of
A solid sphere of mass $500\ gm$ and radius $10\ cm$ rolls without slipping with the velocity $20\ cm/s$. The total kinetic energy of the sphere will be ........ $J$