Ratio of masses and radii of two circular rings are $1 : 2$ and $2 : 1$ respectively then ratio of moment of inertia will be

The centre of mass of two masses $m$ and $m'$ moves by distance $\frac {x}{5}$ when mass $m$ is moved by distance $x$ and $m'$ is kept fixed. The ratio $\frac {m'}{m}$ is

The instantaneous angular position of a point on a rotating wheel is given by the equation $\theta (t) = 2t^3 -6t^2$. The torque on the wheel becomes zero at $t$ $=$ ........ $\sec.$

Three thin metal rods, each of mass $M$ and length $L$ are welded to form an equilateral triangle. The moment of inertia of the composite structure about an axis passing through the centre of mass of the structure and perpendicular to its plane is

A circular disc is rolling on a horizontal plane. Its total kinetic energy is $300\, J$. ........ $J$ is its translational $K.E.$

A sphere of diameter $r$ is cut from a sphere of radius $r$ such that the centre of mass of the remaining mass be at maximum distance from original centre; then the distance is