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 $R / h$ will be ________.

Answer 1

Rolling is rotation about a point of contact.

Applying angular momentum equation about $P$,
$Mv( R + h )= I _{ P } \omega$
As sphere rolls,
$v=\omega R$ and
$I_{P}=\frac{2}{5} m R^{2}+m R^{2}=\frac{7}{5} m R^{2}$
$\therefore M \omega R ( R + h )=\frac{7}{5} MR ^{2} \omega$
$R + h =\frac{7}{5} R$
$h =\frac{2}{5} R$
$\frac{ R }{ h }=\frac{5}{2}=2.5$