Question

A thin ring of mass $2kg$ and radius $0.5m$ is rolling without slipping on a horizontal plane with velocity $1m/s$. A small ball of mass $0.1kg$, moving with velocity $20m/s$ in the opposite direction, hits the ring at a height of $0.75m$ and goes vertically up with velocity $10m/s$. Immediately after the collision.

• A. The ring has pure rotation about its stationary
• B. The ring comes to a complete stop
• C. Friction between the ring and the ground is to the left
• D. There is no friction between the ring and the ground

Answer 1

The correct option is C Friction between the ring and the ground is to the left

Let impulse imparted by ball$=J$ from impulse momentum theorem,

$J=△P$

$v^{\prime }=$final ring velocity

$v=$initial ring velocity

$J=M{v}^{\prime }-m(-v)=M{v}^{\prime }+Mv$

$J\times 0.25=M{R}^2\left(w_f\right)-M{R}^2\left(\frac{v}{R}\right)$

$J=4M{R}^2{w}_f+4M{R}^2\frac{V}{R}$

$4M{R}^2{w}_f+4MRv=M{v}^{\prime }+Mv$

$4\times \frac{5}{10}\times \frac{8}{10}{w}_f+4\times \frac{8}{10}\times 1=v^{\prime }+1$

$w_f+2=v^{\prime }+1$

$v^{\prime }=w_f+1$

$2w_f.R+1=v^{\prime }$

$\Rightarrow v^{\prime }>w_{f}f$

Friction on ring due to ground is to the left.