A small bead of mass m = 1 kg is free to move on a circular hoop. The circular hoop has centre at C and radius r = 1 m and it rotates about a fixed vertical axis. The coefficient of friction between bead and hoop is μ = 0.5 . The maximum angular speed of the hoop for which the bead does not have relative motion with respect to the hoop, at the position shown in the figure is: (Take g = 10 m s - 2 )

Step-by-step solution

The maximum angular speed of the hoop corresponds to the situation when the bead is just about to slide upwards. The free-body diagram of the bead is For the bead not to slide upwards m ω 2 ( r sin ⁡ 45 o )   c o s   45 o - m g sin ⁡ 45 o < μ N ...(i) Where N = m g cos ⁡ 45 o + m ω 2 ( r sin ⁡ 45 o )   sin ⁡ 45 o ...(ii) From (i) and (ii) we get, ω = 30 2 rad   s - 1