A spherical shell of radius $\mathrm{R}$ is rolling down an incline of inclination $\mathrm{\theta }$ without slipping. Find minimum value of coefficient of friction -

A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. 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 cylinder of mass $m$ and radius $R$ rolls down inclined plane without slipping. The speed of its $C.M.$ when it reaches the bottom is___

A small roller of diameter $20 \mathrm{cm}$ has an axle of diameter $10 \mathrm{cm}$ (see figure below on the left). It is on a horizontal floor and a meter scale is positioned horizontally on its axle with one edge of the scale on top of the axle (see figure on the right). The scale is now pushed slowly on the axle so that it moves without slipping on the axle, and the roller starts rolling without slipping. After the roller has moved $50 \mathrm{cm},$ the position of the scale will look like (figures are schematic and not drawn to scale)