Question

A solid sphere, a hollow sphere and a disc, all having same mass and radius, are placed at the top on an incline and released. The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling. Least time will be taken in reaching the bottom by
(a) the solid sphere
(b) the hollow sphere
(c) the disc
(d) all will take same time.

Answer 1

(d) all will take same time

Let θ be the inclination angle.



From the free body diagram, we have:
$$\begin{gathered} N=mg\cos \theta ...\left(i\right) \\ ma=mg\sin \theta -f_r...\left(ii\right) \\ \mathrm{Putting}{f}_r=\mu N\mathrm{in}\left(ii\right)weget, \\ a=g\left(\sin \theta -\mu \cos \theta \right) \end{gathered}$$

The friction coefficients between the objects and the incline are same and not sufficient to allow pure rolling; therefore, all the bodies come down with the same acceleration.