Question

A solid sphere $X$ of mass $M$ and radius $R$ and another sphere $Y$ of mass $2M$ and radius $R/2$ are simultaneously released at rest from the top of an inclined plane as shown in figure. The spheres roll without slipping. Also we may consider the spheres and the surface on which they roll to be perfectly rigid. Match column I and II

Answer 1

$I_{sphereX}>I_{sphereY}$
$\Rightarrow v_{sphereX}<v_{sphereY}$
$a=\frac{g\sin \theta }{1+\frac{k^2}{R^2}}$
$a_X=\frac{g\sin \theta }{1+\frac{\frac{2}{5}{R}^2}{R^2}}=\frac{5}{7}g\sin \theta$
$mgh=\frac{1}{2}m{v}^2+\frac{1}{2}\frac{2}{5}m{R}^2(\frac{v}{R}{)}^2=\frac{1}{2}m{v}^2(1+\frac{2}{5})$
$\therefore$ v is independent of mass and radius of the sphere.
v is same for both.
A->3
KE at bottom is Mgh for sphere X and 2Mgh for sphere Y.
D->2
Velocity at bottom:
$v_X=v_Y$
Time to reach bottom:
$t_X=t_Y$ since $a_X=a_Y$
Angular acceleration:
$a_X=a_Y$
$\Rightarrow R{\alpha }_X=\frac{R}{2}{\alpha }_Y$
$\Rightarrow {\alpha }_Y>{\alpha }_X$