Cycloid and Catenary

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

OM=arcPM

Take  angle PCM = theta   and the point  P(x,y), then ;

 x=ON=OM-MN=OM-PK

= a theta - a sin theta
(because  OM=arcPM=a theta)

=a(theta-sin theta)   and;

y=PN=KM=CM-CK

=a-a.cos theta

=a(1-cos theta)

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;

x=a(theta+sin theta)

y=a(1-cos theta)

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 .