Question

A spherical ball of radius $R$ is moving with velocity $V$ on a smooth horizontal plane as shown. It has a ridge at point $\text{O}$. Calculate the angular velocity of sphere about point $\text{O}$.

• A. $\frac{V}{2R}$
• B. $\frac{7V}{22R}$
• C. $\frac{5V}{8R}$
• D. $\frac{8V}{51R}$

Answer 1

The correct option is C $\frac{5V}{8R}$

Initial angular momentum about O is,

$L_i=MV(R-\frac{R}{8})$

$L_i=\frac{7MVR}{8}$

After collision with $\text{O}$,

$L_f=I\omega$

$\because I=(\frac{2}{5}M{R}^2+M{R}^2)=\frac{7}{5}M{R}^2$

$\therefore L_f=\frac{7}{5}M{R}^2\omega$

Applying angular momentum conservation,

$L_i=L_f$

$\therefore MV\left(\frac{7R}{8}\right)=\frac{7}{5}M{R}^2\omega$

$\Rightarrow \omega =\frac{7V}{8}\times \frac{5}{7R}$

$\therefore \omega =\frac{5V}{8R}$

Hence, option $\left(\text{C}\right)$ is the right choice.

Why this question? This question involves concepts of toppling and conservation of angular moment simultaneously.