Solid cylinders of radii $r_1, r_2$ and $r_3$ roll down an inclined plane from the same place simultaneously. If $r_1 > r_2 > r_3$,which one would reach the bottom first?

$A$ hollow sphere of mass $m$ filled with a non-viscous liquid of same mass $m$ is released on a slope inclined at angle $\theta$ with the horizontal. The friction between the sphere and the slope is sufficient to prevent sliding,and frictional forces between the inner surface of the sphere and the liquid are negligible. After descending a certain height,the ratio of translational and rotational kinetic energies is found to be $x:y$. Find the numerical value of the expression $(x+y)_{min}$.

How can a solid sphere and a hollow sphere of the same material and same size be distinguished without weighing them?

$A$ solid sphere and a solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping throughout the motion. The two climb maximum heights $h_{sph}$ and $h_{cyl}$ on the incline. The ratio $\frac{h_{sph}}{h_{cyl}}$ is given by

$A$ solid sphere rolls down without slipping on a $30^{\circ}$ inclined plane. If $g = 10\,m/s^2$,the acceleration of the rolling sphere is

$A$ cylinder of mass $m$ and radius $R$ rolls without slipping down an inclined plane of length $L$ and height $h$. What will be the velocity of its center of mass when the cylinder reaches the bottom?