A cylinder of mass `m` and radius `R` is kept on a rough surface after giving its centre a horizontal speed `v_(0)`. Find the speed of the centre of the cylinder when it stops slipping.

Explanation

F.B.D. of the cylinder is as follows :

The torque of all the forces about point of contact `P` is zero.
`sumtau_(p)=0`
`:.` Angular momentum remains conserved about axis passing through point `P`
`L_(i)=mv_(CM)r_(bot)+I_(CM)omega`
`=mv_(0)*R+(mR^(2))/(2)*0=mv_(0)R` (clockwise)
`L_(f)=mv_(CM)*r_(bot)+I_(CM)*omega`
`=m.v.R+(mR^(2))/(2)*omega` (since in pure rolling `v=omegaR`)
`=mvR+(mR^(2))/(2)*(v)/(R )=(3)/(2)mvR`
`L_(i)=L_(f)`
or `mv_(0)R=(3)/(2)mvR`
`=(2v_(0))/(3)`