The Cycloid Pendulum
The standard equation for the period of a pendulum is only valid for small angles. When the angle is small, the restoring force is approximately in direct proportion to the displacement. The condition is that the sine of the angle is approximately equal to the angle. This direct proportion gives you a simple harmonic oscillator. And the period depends only on the length of the string and the force of gravity. However, in the 1600's Christiaan Huygens noticed that this wasn't quite true. And the larger the angle, the less true this would be. The Cycloid Pendulum is one solution to this problem. By constraining the pendulum to move in a cycloid path, the period will remain constant independent of the maximum angle of the string. Curiously, the curve which constrains the pendulum is also a cycloid with the same generating radius. For a cycloid pendulum, the pendulum length is four times this generating radius. The curve in this demonstration allows a maximum angle of 60 degrees. For a 90 degree curve, it would have needed to be much wider.
