Math Problem : Cycloid Problem

If you drew a dot on the edge of a wheel and traced the path of the dot as the wheel rolled one complete revolution along a line, then the path formed would be called a cycloid (shown above), combining both forward and circular motion. What is the length of the path formed by one complete revolution? Assume the wheel has a radius of 1.



Answer

Problem 3 Answer

The answer is 8r, where r equals the radius of the wheel.The distance as the dot moves from the 3:00 to 9:00 postion counterclockwise is 4*2^1/2. As it moves from the 9:00 position to the 3:00 position the distance is 8- 4*2^1/2, assuming the r=1.

Michael Shackleford, A.S.A.

Solution

Problem 3 Solution

Contrary to the diagram in the problem, place the center of the wheel at (0,0) and draw the point at (0,1).Let t be the distance the center of the wheel has moved from (0,0). Then:
x=t+sin(t)
y=cos(t)
Taking the derivitives:
dx/dt=1+cos(t)
dy/dt=-sin(t)
The change in arc length can be defined as ( (dx/dt)² + (dy/dt)² ) ^1/2.
So the total arc length is the integral from 0 to 2pi of ( (dx/dt)² + (dy/dt)² ) ^1/2.
After a few steps this integral becomes:
2^1/2 * (1+cos(t))^1/2.
Multiply by (1-cos(t))^1/2 / (1-cos(t))^1/2 and the integral becomes:
2^1/2 * sin(t) / (1-cos(t))^1/2.
Let u=cos(t) and the integral becomes:
2^1/2 * (1-u)^-1/2.
Integrating this you get:
2^1/2 * 2 * (1-u)^1/2 .
The bounds are 0 to 2*pi, so the total arc length is:
2^1/2 * 2 * (2^1/2+2^1/2) =
2^1/2 * 2 * 2 * 2^1/2 = 8

Michael Shackleford, A.S.A.