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/s^{2}.

(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?

(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/s^{2}, the stopping time is:

ω = ω_{o} + α t
→ t = (ω - ω_{o})/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 - θ_{o} = ω_{o} t + ½
α t^{2}

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

Convert to revolutions: 3.08 rad/(2 π rad/rev) = 0.49 revolutions

(d) To find the distance traveled while the wheel was slowing down, multiply the angular displacement (in radians) by r: s = rθ = 4 * 3.08 = 12.3 m