Question: The Wave Swinger ride at the fair has two circular rows of chairs suspended by chains of equal lengths. The outer row is along the outer edge of the top of the ride and the inner row has a smaller radius. Do riders in both rows swing out at the same angle when the ride is rotating?

Answer: When the ride is in motion, the passengers in the chairs swing outward. At one particular angle the chain provides the necessary horizontal force for circular motion and the vertical force needed to balance the gravitational force. Riders in both rows move with the same angular velocity, but those in the outer row move in a larger radius and thus have a larger angular acceleration. Because of this larger angular acceleration, a larger centripetal force is needed. Thus for riders in the outer row, the horizontal component of the chain force is larger relative to their gravitational force than it is for those on the inner row. The result is that riders in the outer row swing out to a larger angle from the vertical direction than do the riders in the inner row as you can see in the figure below.