A solid cylinder having radius R and length L is slipping on a rough horizontal plane. At time t=0 the cylinder has a translational velocity v₀=49 m/s, perpendicular to its axis and a rotational velocity v₀/4R about the centre. The time taken by the cylinder to start rolling is ________ seconds. (coefficient of kinetic friction _K=0.25 and g=9.8 m/s ^2 )

Step-by-step solution

Initial translational velocity, v₀ = 49 m/s. Initial angular velocity, ₀ = v₀ 4R . Velocity of the lowest point of the cylinder is v_ bottom = v₀ - ₀ R = v₀ - v₀ 4 = 3v₀ 4 > 0 . Since the lowest point is slipping forward, kinetic friction f_k = _k m g acts in the backward direction. Translational acceleration, a = - f_k m = - _k g . Angular acceleration, = f_k R I = _k m g R 1 2 m R^2 = 2 _k g R . At time t , the translational velocity is v = v₀ - _k g t . The angular velocity is = ₀ + t = v₀ 4R + 2 _k g R t . For