Question

The following bodies,
(1) a ring
(2) a disc
(3) a solid cylinder
(4) a solid sphere,
of same mass $'m'$ and radius $'R'$ 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

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