A spherical solid ball of $1\,kg$ mass and radius $30\,cm$ is rotating about an axis passing through its centre with an angular velocity of $50\,radian/s$ . The kinetic energy of rotation is ......... $J$.
Two uniform circular discs are rotating independently in the same direction around their common axis passing through their centres. The moment of inertia and angular velocity of the first disc are $0.1 \;kg - m ^{2}$ and $10\; rad \,s^{-1}$ respectively while those for the second one are $0.2 \;kg - m ^{2}$ and $5\; rad \,s ^{-1}$ respectively. At some instant they get stuck together and start rotating as a single system about their common axis with some angular speed. The Kinetic energy of the combined system is ...........$J$
$A$ rod hinged at one end is released from the horizontal position as shown in the figure. When it becomes vertical its lower half separates without exerting any reaction at the breaking point. Then the maximum angle $‘\theta ’$ made by the hinged upper half with the vertical is ......... $^o$.
Starting from the rest, at the same time, a ring, a coin and a solid ball of same mass roll down an incline without slipping .The ratio of their translational kinetic energies at the bottom will be