If a solid sphere is rolling, the ratio of its rotational energy to the total kinetic energy is given by

A $T-$ shaped object of uniform thickness and same material with dimensions shown in the figure, is lying on a smooth floor. A force $\vec F$ is applied at the point $P$ parallel to $AB,$ such that the object has only the translation motion without rotation. Find the location of $P$ with respect to $C$

A particle of mass $m$ moves in the $XY$ plane with a velocity $V$ along the straight line $AB$ . If the angular momentum of the particle with respect to origin $O$ is $L_A$ when it is at $A$ and $L_B$ when it is at $B$ , then

A cylinder of mass $M$ and radius $r$ is mounted on a frictionless axle over a well. A rope of negligible mass is wrapped around the solid cylinder and a bucket of mass $m$ is suspended from the rope. The linear acceleration of the bucket will be

Two racing cars of masses $m_1$ and $m_2$ are moving in circles of radii $r_1$ and $r_2$ respectively. Their speeds are such that each makes a complete circle in the same time $t$. The ratio of the angular speeds of the first to the second car is

For a rolling body, the velocity of $P_1$ and $P_2$ are ${\vec v_1}$ and ${\vec v_2}$ , respectively