Question

A solid sphere is placed on a smooth horizontal plane. A horizontal impluse $I$ is applied at a distance $h$ above the central line as shown in the figure. Soon after giving the impulse the sphere starts rolling.
The ratio $h/R$ would be

• A. $\frac{1}{2}$
• B. $\frac{2}{5}$
• C. $\frac{1}{4}$
• D. $\frac{1}{5}$

Answer 1

The correct option is B $\frac{2}{5}$

After application of impulse let velocity $=v$

$I=mv\Rightarrow v=I/m$

Angular impulse about C.M

$gmv=I\omega \Rightarrow hmv=\frac{2}{5}m{R}^2\omega \Rightarrow \omega =\frac{5}{2}\frac{hv}{R^2}$

For pure rolling

$v=r\omega \Rightarrow \omega =v/r\Rightarrow \frac{v}{R}=\frac{5}{2}\frac{hv}{R^2}\Rightarrow \frac{h}{R}=\frac{2}{5}$