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?
(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
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