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  4. A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights hsph and hcyl on the incline. The ratio (hsph/hcyl) is given by :

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Q. A solid sphere and solid cylinder of identical radii approach an incline with the same linear velocity (see figure). Both roll without slipping all throughout. The two climb maximum heights $h_{sph}$ and $h_{cyl}$ on the incline. The ratio $\frac{h_{sph}}{h_{cyl}}$ is given by :Physics Question Image

JEE MainJEE Main 2019System of Particles and Rotational Motion

Solution:

for solid sphere
$\frac{1}{2} mv^{2} + \frac{1}{2} . \frac{2}{5} mR^{2} . \frac{v^{2}}{R^{2}} =mgh_{sph} $
for solid cylinder
$ \frac{1}{2}mv^{2} + \frac{1}{2} . \frac{1}{2} mR^{2} . \frac{v^{2}}{R^{2}} = mgh_{cyl} $
$ \Rightarrow \frac{h_{sph}}{h_{cyl}} = \frac{7 /5}{3 /2} = \frac{14}{15} $