Question

The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere,
of same mass ${}^{\prime }{m}^{\prime }$ and radius ${}^{\prime }{R}^{\prime }$ are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of the inclined plane is ______.
[Mark the body as per their respective numbering given in the question]

Answer 1

Answer: 4

$Mg\sin \theta R=\left(m{k}^2+m{R}^2\right)\alpha$
$\alpha =\frac{\mathrm{Rg}\sin \theta }{k^2+R^2}\Rightarrow a=\frac{g\sin \theta }{1+\frac{k^2}{R^2}}$
$t=\sqrt{\frac{2s}{a}}=\sqrt{\frac{2s}{g\sin \theta }\left(1+\frac{k^2}{R^2}\right)}$
for least time, $k$ should be least & we know $k$ is least for solid sphere.