$A$ disc of radius $R$ is rolling down an inclined plane whose angle of inclination is $\theta$. Its acceleration would be

Three bodies: a ring $(R)$,a solid cylinder $(C)$,and a solid sphere $(S)$ having the same mass and same radius roll down an inclined plane without slipping. They start from rest. If $v_{R}$,$v_{C}$,and $v_{S}$ are the velocities of the respective bodies on reaching the bottom of the plane,then:

$A$ 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?

$A$ solid sphere of mass $1 kg$ and radius $1 m$ rolls without slipping on a fixed inclined plane with an angle of inclination $\theta = 30^{\circ}$ from the horizontal. Two forces of magnitude $1 N$ each,parallel to the incline,act on the sphere,both at a distance $r = 0.5 m$ from the center of the sphere,as shown in the figure. The acceleration of the sphere down the plane is . . . $m s^{-2}$. (Take $g = 10 m s^{-2}$.)

$A$ hollow spherical ball of uniform density rolls up a curved surface with an initial velocity $3\, m/s$ (as shown in figure). The maximum height with respect to the initial position covered by it will be $...........cm$.

$A$ solid sphere,a disc,and a solid cylinder start rolling down an inclined plane from rest. All three objects are made of the same material and have the same mass. Then,