Question

A solid sphere, a solid cylinder, a hollow sphere and a hollow cylinder (all of the same mass ) simultaneously start rolling down from top of an inclined plane, then the time to reach bottom of inclined plane is :

• A. minimum for solid cylinder, maximum for hollow sphere.
• B. minimum for hollow sphere, maximum for solid cylinder.
• C. minimum for solid sphere, maximum for hollow cylinder.
• D. minimum for hollow cylinder, maximum for solid sphere.

Answer 1

Answer: (C)

minimum for solid sphere, maximum for hollow cylinder.

Acceleration of the body rolling down the inclined plane $a=\frac{g\sin \theta }{1+K}$

where $\theta$ is the angle of inclination.

We know, $K_{solidsphere}=\frac{2}{5}$, $K_{solidcylinder}=\frac{1}{2}$, $K_{hollowsphere}=\frac{2}{3}$ and $K_{hollowcylinder}=1$

$⟹$ $a_{solidsphere}>a_{solidcylinder}>a_{hollowsphere}>a_{hollowcylinder}$

Now let the distance covered by each body along the inclined plane be $S$.

Using $S=ut+\frac{1}{2}a{t}^2$

where $u=0$ (as each body starts from rest)

$\therefore$ $S=0+\frac{1}{2}a{t}^2$

$⟹$ $t=\sqrt{\frac{2S}{a}}$

Thus we get $t_{solidsphere}<t_{solidcylinder}<t_{hollowsphere}<t_{hollowcylinder}$

Hence solid sphere takes minimum time and hollow cylinder takes maximum time to reach the bottom.