Three solid cylinders $A, B$ and $C$ each of mass $m$ and radius $R$ are allowed to roll down on three inclined planes $A\text{'}, B\text{'}$ and $C\text{'}$ respectively (each of inclination $\theta$) such that for $A\text{'}$ and $A$ coefficient of friction is zero, for $B\text{'}$ and $B$ coefficient of friction is greater than zero but less than $\frac{\tan \theta }{3}$ and for $C\text{'}$ and $C$ coefficient of friction is more than $\frac{\tan \theta }{3}$. On reaching bottom of the inclined plane

For pure rolling , ${\mu }_{min}=\frac{\tan \theta }{1+{\displaystyle \frac{m{R}^2}{I_c}}}=\frac{\tan \theta }{1+{\displaystyle \frac{m{R}^2}{m{R}^2/2}}}=\frac{\tan \theta }{3}$

Therefore for A , it is pure translational motion .For B , it is rolling with slipping and for C, it is pure rolling .

$C$ has max torque $\Rightarrow Maximum \omega .$ In case of  $B,$energy is  dissipated.

In case of A , acceleration of centre of cylinder ( g sin $\theta )$ is maximum , so time taken is minimum .

$$\begin{gathered} mg\sin \theta -\mu mg\cos \theta =m{a}_B \Rightarrow a_B>\frac{2}{3}g \sin \theta \\ In case of C ,a_c=\frac{g\sin \theta }{1+{\displaystyle \frac{1}{2}}}=\frac{2}{3}g\sin \theta \end{gathered}$$