A solid disc and a ring, both of radius `10 cm` are placed on a horizontal table simultaneously, with initial angular speed equal to `10 pi rad//s`. Which of the two will start to roll earlier ? The coefficient of kinetic friction is `mu_(k) = 0.2`.

Explanation

As the force of friction produces an acceleration a in the centre of mass.
Therefore equations of motion is `mu_(k) mg=ma`
`a=u_(x)g, v=v+at=0+mu_(k) "gt"`
The torque due to friction = HimgR which produces an angular retardation `alpha`
`alpha=(-mu_(k)mgR)/(I)`
As `omega= omega_(0)+alphat`
`= omega_(0)-(mu_(k) mg Rt)/(I) " "...(i)`
As `u=mu_(k) "gt"`
and `v=R omega`
`:. mu_(k) "gt"= R omega" "`
`mu_(k)"gt"= R omega_(0)-(mu_(k)mg R^(2)t)/(I)`
`mu_(k)"gt"=R omega_(0)-(mu_(k) mg R^(2)t)/(I)`
`t=(omega_(0)R)/(2mu_(k)g)" "...(iii)`
For a disc, `I=(1)/(2) mR^(2)`
`mu_(k) "gt")=Romega_(0)-(mu_(k) mgR^(2)t)/(I)`
`mu_(k)"gt"=R omega_(0)-2mu_(k)"gt"`
`2mu_(k) "gt"=R omega_(0)`
`t=(Romega_(0))/(3mu_(k)g) " "......(iv)`
From (iii) and (iv) we can say that the disc will begin to roll earlier than the ring.