Two cylinders $A$ and $B$ are released on a fixed rough inclined plane of angle $\theta$ with the horizontal from height $‘h’$. $A$ is a hollow cylinder and $B$ is a solid cylinder. Both perform pure rolling. Find the distance between the two cylinders when one of them reaches the bottom. Assume that the axis of both cylinders is parallel to the edge $CD$ always

A spool of mass $M=3 \mathrm{kg}$ and radius $R = 20 \mathrm{cm}$ has an axle of radius $r=10 \mathrm{cm}$ around which a string is wrapped. The moment of inertia about an axis perpendicular to the plane of the spool and passing through the center is $\frac{M{R}^2}{2}$. Coefficient of friction between the surface and the spool is $0.4$. Find maximum value of the tension $T$ (in $\mathrm{N}$) that can be applied so that the spool rolls without slipping. $\left(g=10 \mathrm{m} {\mathrm{s}}^{-2}\right)$

A wheel starts rolling with an angular velocity and initial linear velocity up an inclined rough plane of an angle $\theta$ with horizontal and coefficient of friction $\mu$. The linear deceleration is given by:

A wheel of radius $r$ starts rolling with an angular velocity $\omega$ and initial linear velocity $V_0$ up and inclined smooth plane. The wheel will stop going up in time:

A solid sphere of radius $R$ starts rotating on rough horizontal surface with translational velocity $V_0$ and initial angular velocity ${\omega }_0=\frac{2V_0}{3\mathrm{R}}$. The sphere starts pure rolling $3R$ after some time. Find the angle by which sphere rotates up to the instant at which pure rolling starts, if $V$ is the translational velocity at pure rolling. Assume uniformly accelerated motion upto start of pure rolling:

A solid sphere of radius $R$ starts rotating on rough horizontal surface with translational velocity $V_0$ and initial angular velocity ${\omega }_0=\frac{2V_0}{3\mathrm{R}}$. The sphere starts pure rolling $3R$ after some time. Assume uniformly accelerated motion upto start of pure rolling:

If $S$ is the distance covered by the sphere in rotational motion up to the instant at which pure rolling starts, find the velocity (translational) of the sphere during the pure rolling.

A plate of mass $m$ is placed on a frictionless surface. The plate is connected to a block of mass $M$ through a rope, over a massless pulley. A cylinder of mass $m$ is placed on the plate which rolls without slipping. Find the frictional force acting on the cylinder.

 A solid sphere of mass $m$ and radius $r$ is gently placed on a conveyer belt moving with constant velocity $V$. If the coefficient of friction between the belt and sphere is $\frac{2}{7}$, the distance travelled by the centre of the sphere before it starts pure rolling is

The string of a step rolling wheel is pulled by applying force $F$ with different lines of action in two situations as shown. The wheel starts rolling without slipping due to application of the force.