A solid sphere, a hollow sphere and a disc,all having same mass and radius, are placed at the top of 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

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the disc

the solid sphere

the hollow sphere

all will take same time

answer is D.

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Detailed Solution

As the objects are sliding and rolling down the inclined plane, acceleration of their centres is the same and is given by $\large a = \frac{{mg\sin \theta - \mu mg\cos \theta }}{m}\Rightarrow a=g(sin\theta-\mu cos\theta)$
$\large \therefore s = 0 \times t + \frac{1}{2}a{t^2} \Rightarrow t = \sqrt {\frac{{2s}}{a}} = \sqrt {\frac{{2s}}{{g(\sin \theta - \mu \cos \theta )}}}$ so option -(4) is right.

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