Q

# An impulse is applied to a ring of mass m along the line passing through its centre.

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

$\tau =I\alpha\\\tau=fr=\mu mgr\\\Rightarrow \alpha =\frac{\mu g}{r}\\\omega _0=\omega _0+\alpha t=\frac{\mu gt}{r}\\ring\ stop\ sliding\ when\ v=\omega r\\v_0-\mu gt=\frac{\mu gt}{r}r\\v_0=2\mu gt\\v_0=p/m\ t=\frac{p}{2\mu mg}\\v=v_0-\mu gt=2\mu g\frac{p}{2\mu mg}-\mu g\frac{p}{2\mu mg}=\frac{p}{2m}$