### Cycloid and Catenary

This is a Cycloid Pendulum

If there are equation

**x=a(theta - sine theta)**and,

**y=a(1-cos theta)**Also

*be a fixed straight line called*

**OX**

**x-axis**And a circle of radius

*roll, without sliding, along this line,*

**a**Then the cycloid is the curve traced out by a point

*on the circumstances of the circle. Also if*

**P***be the center of circle. The point moves*

**C***to*

**O***such that*

**P**

**OM=arcPM**Take

*and the point*

**angle PCM = theta***, then ;*

**P(x,y)**

**x=ON=OM-MN=OM-PK**

**= a theta - a sin theta**(because

*)*

**OM=arcPM=a theta***and;*

**=a(theta-sin theta)**

**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

*describes the curve*

**P***when*

**ODA***i.e.*

**y=0***If the motion is continued, we get an infinity number of such curves.*

**theta=0 or 2pi.**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)**__is such a curve in in which a uniform chain hangs freely under gravity. If the curve be measured from__

**Centenary***to any point and*

**A***. Tangent at*

**arc AP=s***makes an angle*

**P***with*

**sai***. Then;*

**x-axis***where*

**s=a.tan sai,**

**a=constant .**
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