A pulley is accelerated uniformly from rest at a rate of 8 rad/s 2 . After 20 s the acceleration ...

(a) Angular velocity after acceleration period,ω2 = ω1+ αt = 0 + (8)(20) = 160 rad/s

(b) ω3 = ω2 + αt from which,

angular acceleration, α = (ω3 - ω2)/t = (0 - 160)/40 = - 4 rad/s2

i.e. angular deceleration is 4 rad/s2

(c) Initial angle turned through, θ1 = [(ω1 + ω2)/2]*t = [(0 + 160)/2]*(20) = 1600 rad =1600/2π rev

 At constant speed, angle turned through, θ2 = 160 rad/s × (2 × 60)s = 19200 rad = 19200/2π rev

Angle turned through during deceleration, θ3 = [(160+ 0)/2]*(40) = 3200 rad = 3200/2π rev

Hence, total number of revolutions made by the pulley = θ1+ θ2 + θ3

=1600/2π +19200/2π + 3200/2π

=24000/2π = 12000/π rev or 3820 rev