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

Answer 1

Answer: (B)

$J/2m$

Let $v$ and $v_o$ be the velocity of CM after pure rolling and before pure rolling.

$J=\Delta p=m{v}_o$

Also, $v=Rw$

Now $f=ma⟹a=\frac{f}{m}$

$fR=I\alpha$

$fR=\left(m{R}^2\right)\alpha ⟹mR\alpha =f$

$v=v_o-at⟹v=v_o-\frac{f}{m}t$ ............(1)

$w=0+\alpha t=\frac{f}{mR}t$ ............(2)

Thus $v=v_o-wR$

$v=v_o-v⟹v=\frac{v_o}{2}=\frac{J}{2m}$