A wheel of moment of inertia $10\ kg-m^2$ is rotating at $10$ rotations per minute. The work done in increasing its speed to $5$ times its initial value, will be.......... $J$

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 at $50\%$ of its energy is used in increasing its temperature. Then the rise in temperature of the sphere is

A hollow sphere is rolling on a plane surface about its axis of symmetry. The ratio of rotational kinetic energy to its total kinetic energy is $\frac{x}{5}$. The value of $x$ is________.

A cord is wound round the circumference of wheel of radius $r$. The axis of the wheel is horizontal and the moment of inertia about it is $I. \,A$ weight $mg$ is attached to the cord at the end. The weight falls from rest. After falling through a distance $ 'h '$, the square of angular velocity of wheel will be ..... .

Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length $L$, and negligible  mass. The rod is to be set rotating about an axis perpendicular to it. The position of point $P$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity $\omega_0$ is minimum, is given by

The moment of inertia of a body about a given axis is $1.2 \;kg m^{2}$. Initially, the body is at rest. In order to produce a rotational kinetic energy of $1500\; joule$, an angular acceleration of $25 \;rad s^{-2}$ must be applied about that axis for a duration of