$A$ disc is given an angular velocity $\omega_0$ as shown in the figure and is kept on an inclined plane of inclination $\theta$ having a coefficient of friction $\mu = \tan \theta$. If the initial acceleration of the centre of mass is $a$ and the angular acceleration is $\alpha$,then:

$A$ solid cylinder of mass $M$ and radius $R$ rolls down an inclined plane without slipping. The speed of its centre of mass when it reaches the bottom is ...

$A$ disc of mass $M$ and radius $R$ rolls on a horizontal surface and then rolls up an inclined plane as shown in the figure. If the velocity of the disc is $v,$ the height to which the disc will rise will be

$A$ solid cylinder $30\ cm$ in diameter at the top of an inclined plane $2.0\ m$ high is released and rolls down the incline without loss of energy due to friction. Its linear speed at the bottom is.......... $m/s$.

$A$ 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 $g = 10 m s^{-2}$.)

$A$ uniform solid sphere of mass $m$ and radius $r$ rolls without slipping down an inclined plane,inclined at an angle $45^{\circ}$ to the horizontal. Find the magnitude of the minimum frictional coefficient at which slipping is absent.