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

Frictional force causes the CM to accelerate from its initial zero velocity. Frictional torque causes retardation in the initial angular speed `ω_(o)`. The equations of motion are `mu_(k)mg=ma and mu_(k)mgR=-Ialpha`, which yield `v=mu_(k)g t,omega=omega_(o)-mu_(k)mgRt//I`. Rolling begins when `v=Romega`. For a ring, `I=mR^(2)`, and rolling begins at `t=omega_(o)R//2mu_(k)g`. For a disc, `I=1//2mR^(2)` and rolling starts at break line `t=Romega_(o)//3mu_(k)g`. Thus, the disc begins to roll earlier than the ring, for the same `R and ω_(o)`. The actual times can be obtained for `R = 10 cm`, `omega_(o)=10pi" rad s"^(-1),mu_(k)=0.2`