After fixing a flat tire on a bicycle you give the wheel a spin. (a)if its initial angular speed was 6.35 rad/s and it rotated 14.2 revolutions before coming to a rest, what was its average angular acceleration? (b)for what length of time did the wheel rotate?

Part (a)
ω₀ = 6.35 rad/s, initial angular speed
θ = 14.2 rev= 14.2*2π rad = 89.22 rad, angular distance traveled.

Let α =  the angular acceleration.
Because the wheel comes to rest,
ω₀² + 2αθ = 0
(6.35 rad/s)² + 2*(α rad/s²)*(89.22 rad) = 0
40.3225 + 178.44α = 0
α = - 0.226 rad/s²

  -0.226 rad/s²

Part (b)
Let t =  the time for the wheel to come to rest.
Then
ω₀ + αt = 0
6.35 - 0.226t = 0
t = 6.35/0.226 = 28.1 s

28.1 s