Question

A solid sphere rests on a horizontal surface. A horizontal impulse is applied at height h from centre. The sphere starts rolling just after the application of impulse. The ratio $\frac{h}{R}$ will be :

• A. $\frac{1}{2}$
• B. $\frac{2}{5}$
• C. $\frac{1}{5}$
• D. $\frac{2}{3}$

Answer 1

Answer: (B)

$\frac{2}{5}$

Applying the equation for rotation about the point of contact P
$J(R+h)=I_P\omega$
$mv(R+h)=I_P\omega$
As sphere rolls, $v=\omega R$
$I_P=\frac{2}{5}m{R}^2+m{R}^2=\frac{7}{5}m{R}^2$
Hence, $mv(R+h)=\frac{7}{5}m{R}^2\times \omega =\frac{7}{5}m{R}^2\times \frac{v}{R}$
$\Rightarrow \frac{R+h}{R}=\frac{7}{5}$
$\Rightarrow 1+\frac{h}{R}=\frac{7}{5}$
$\Rightarrow \frac{h}{R}=\frac{2}{5}$