$A$ thin circular ring first slips down a smooth incline then rolls down a rough incline of identical geometry from the same height. The ratio of the time taken in the two motions is ........

$A$ body slides down a smooth inclined plane of inclination $\theta$ and reaches the bottom with velocity $V$. If the same body is a ring which rolls down the same inclined plane,then the linear velocity at the bottom of the plane is:

The acceleration of a body rolling down on an inclined plane does not depend upon

$A$ solid sphere at rest rolls down an inclined plane of vertical height $h$ without sliding. Its speed on reaching the bottom of the plane is ($g=$ acceleration due to gravity).

$A$ solid disc of radius $a$ and mass $m$ rolls down without slipping on an inclined plane making an angle $\theta$ with the horizontal. The acceleration of the disc will be $\frac{2}{b} g \sin \theta$ where $b$ is $........$. (Round off to the Nearest Integer)
($g =$ acceleration due to gravity)
($\theta =$ angle as shown in figure)

$A$ cylinder of mass $10 \; kg$ and radius $15 \; cm$ is rolling perfectly on a plane of inclination $30^{\circ}$. The coefficient of static friction $\mu_{S} = 0.25$.
$(a)$ How much is the force of friction acting on the cylinder?
$(b)$ What is the work done against friction during rolling?
$(c)$ If the inclination $\theta$ of the plane is increased,at what value of $\theta$ does the cylinder begin to skid and not roll perfectly?