Question

An impulse $J$ is applied on a ring of mass $m$ along a line passing through its centre $O$. The ring is placed on a rough horizontal surface. The linear velocity of centre of ring once it starts rolling without slipping is

• A. $J/4m$
• B. $J/2m$
• C. $J/m$
• D. $J/3m$

Answer 1

The correct option is B $J/2m$

Draw a diagram of given problem.

Calculate linear velocity of center of ring.

According to diagram, $v$ be the velocity of $COM$ of ring just after impulse is applied and let $v^{\prime }$ is the velocity when pure rolling starts.
When, impulse $J$ is applied on ring,
Impulse = change in linear momentum
$J=mv$, from which
$v=\frac{J}{m}\left(i\right)$
Also shown in diagram, angular momentum of ring at bottom most will remain conserved.
Therefore,
$L_i=L_f$
$mvr=m{v}^{\prime }r+I\omega$
$mvr=m{v}^{\prime }r+\left(m{r}^2\right)\left(v^{\prime }/r\right)$
$=2m{v}^{\prime }r$
$v^{\prime }=\frac{v}{2}=\frac{J}{2m}$