This is a Cycloid Pendulum
If there are equation
x=a(theta - sine theta)
and, y=a(1-cos theta)
Also OX be a fixed straight line called x-axis
And a circle of radius a roll, without sliding, along this line,
Then the cycloid is the curve traced out by a point P on the circumstances of the circle. Also if C be the center of circle. The point moves O to P such that
Take angle PCM = theta and the point P(x,y), then ;
= a theta - a sin theta
(because OM=arcPM=a theta)
=a(theta-sin theta) and;
It is clear that in one complete revolution of the circle but point P describes the curve ODA when y=0 i.e. theta=0 or 2pi. If the motion is continued, we get an infinity number of such curves.
This fixed line is called the base and the highest point from the fixed line is called the vertex or cusp.
When the curve is inverted the equation become;
Centenary is such a curve in in which a uniform chain hangs freely under gravity. If the curve be measured from A to any point and arc AP=s . Tangent at P makes an angle sai with x-axis . Then;
s=a.tan sai, where a=constant .