### Sample problem

You are on a ferris wheel that rotates 1 revolution every 8 seconds. The ferris wheel operator brings the wheel to a stop, and puts on a brake that produces a constant acceleration of -0.1 radians/s2.

(a) If your seat on the ferris wheel is 4 m from the center, what is your speed when the wheel is turning at the rate of 1 revolution every 8 seconds?

(b) How long does it take before the ferris wheel comes to a stop?

(c) How many revolutions does the wheel make while it is coming to a stop?

(d) How far do you travel while the wheel is slowing down?

#### Solution

(a) The wheel initially rotates 1 revolution every 8 seconds, or 0.125 rev/s. Converting to radians/s gives:   ω = 0.125 rev/s * 2π rad/rev = 0.785 rad/s

Your speed is the angular velocity multiplied by your distance from the center of the wheel:   v = r ω = 4 * 0.785 = 3.14 m/s

(b) From the initial angular velocity, final angular velocity of zero, and angular acceleration of -0.1 rad/s2, the stopping time is:

ω = ωo + α t  →  t = (ω - ωo)/α = (0 - 0.785)/(-0.1) = 7.85 s.

(c) To find the number of revolutions the wheel undergoes as it slows to a stop:

θ - θo = ωo t + ½ α t2

θ = (0.785 * 7.85) + ½(-0.1) * (7.85)2 = 3.08 radians