Sample problem

You are on a ferris wheel that is rotating at the rate of 1 revolution every 8 seconds. The operator of the ferris wheel decides to bring it to a stop, and puts on the brake. The brake produces a constant acceleration of -0.11 radians/s².

(a) If your seat on the ferris wheel is 4.2 m from the center of the wheel, what is your speed when the wheel is turning at a constant rate, before the brake is applied?

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

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

Solution

First, we follow the method. The diagram is shown above.
Let's choose the origin to be the position of your seat (in red above) at the instant the brakes are applied.
Let's choose counter-clockwise to be positive, since the ferris wheel is rotating counter-clockwise.
Everything we were given is summarized in the table above.

(a) The wheel is rotating at a rate of 1 revolution every 8 seconds, or 0.125 rev/s. This is the initial angular velocity. It is often most convenient to work with angular velocity in units of radians/s; doing the conversion gives:

w = 0.125 rev/s * 2p rad/rev = 0.785 rad/s

Your speed is simply this angular velocity multiplied by your distance from the center of the wheel:

v = r w = 4.2 * 0.785 = 3.30 m/s

(b) We've calculated the initial angular velocity, the final angular velocity is zero, and the angular acceleration is -0.11 rad/s². This allows the stopping time to be found:

w = w_o + a t
t =
w - w_o

a
=
0 - 0.785

-0.11
= 7.14 s

(c) One way to find the number of revolutions the wheel undergoes as it slows to a stop is to find the angle it moves through:

q - q_o = w_o t + ½ a t²

q = (0.785 * 7.14) + ½ (-0.11) * (7.14)² = 2.80 radians

This can be converted to revolutions:

2.80 rad

2p rad/rev
= 0.446 revolutions

(d) To figure out the distance you traveled while the wheel was slowing down, the angular displacement (in radians) can be converted to a displacement by multiplying by r:

s = rq = 4.2 * 2.80 = 11.8 m