A sphere starts rolling down an incline of inclination $\theta$. Find the speed of its center when it has covered a distance $l$.

A hollow sphere is released from the top of an inclined plane of inclination $\theta .$

$\left(a\right)$ What should be the minimum coefficient of friction between the sphere and the plane to prevent sliding?

$\left(b\right)$ Find the kinetic energy of the ball as it moves down a length $l$ on the inclined plane, if the friction coefficient is half the value calculated in part $\left(a\right)$.

The figure shows a rough track, a portion of which is in the form of a cylinder of radius $R$, with what minimum linear speed should a sphere of radius $r$ be set rolling on the horizontal part so that it completely goes round the circle on the cylindrical part?

The following figure shows a small spherical ball of mass $m$ rolling down the loop track. The ball is released on the linear portion at a vertical height $H$ from the lowest point. The circular part shown has a radius $R$.
$\left(a\right)$ Find the kinetic energy of the ball when it is at a point $A$ where the radius makes an angle $\theta$ with the horizontal.

$\left(b\right)$ Find the radial and the tangential accelerations of the center when the ball is at $A$.