$A$ solid sphere and a disc,having the same mass and radius,roll down from rest from the same height on a smooth inclined plane. The ratio of the time taken by them is:

Two identical uniform discs roll without slipping on two different surfaces $AB$ and $CD$ (see figure) starting at $A$ and $C$ with linear speeds $v_1$ and $v_2$,respectively,and always remain in contact with the surfaces. If they reach $B$ and $D$ with the same linear speed and $v_1 = 3 \ m/s$,then $v_2$ in $m/s$ is $(g = 10 \ m/s^2)$.

$A$ circular disc reaches from top to bottom of an inclined plane of length $l$. When it slips down the plane,it takes $t \ s$. When it rolls down the plane,it takes $\left(\frac{\alpha}{2}\right)^{1/2} t \ s$,where $\alpha$ is:

Derive expressions for the kinetic energy and velocity of a body rolling without sliding down an inclined plane of inclination $\theta$.

$A$ solid cylinder and a solid sphere of same mass and radius roll without slipping on a rough inclined plane. The force of friction is ..........

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