We can use the following kinematic equations to find the angular acceleration and time it takes for the wheel to come to a stop:
θ = θ_0 + ω_0 t + 1/2 α t^2
ω^2 = ω_0^2 + 2 α (θ - θ_0)
where:
θ = final angle = 60.0 rad
θ_0 = initial angle = 0
ω_0 = initial angular velocity = 18 rad/s
α = angular acceleration (unknown)
t = time it takes for the wheel to come to rest (unknown)
Using the second equation, we can solve for α:
α = (ω^2 - ω_0^2) / 2(θ - θ_0)
= (0 - (18 rad/s)^2) / 2(60.0 rad - 0)
= -2.7 rad/s^2
Therefore, the angular acceleration of the wheel is -2.7 rad/s^2 (negative sign indicates deceleration).
Using the first equation, we can solve for t:
θ = θ_0 + ω_0 t + 1/2 α t^2
60.0 rad = 0 + (18 rad/s) t + 1/2 (-2.7 rad/s^2) t^2
Solving for t using the quadratic formula, we get:
t = 6.57 s
Therefore, it takes 6.57 seconds for the wheel to come to rest.