Matematika :

Apr 24, 2011

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.


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*21/2. As it moves from the 9:00 position to the 3:00 position the distance is 8- 4*21/2, assuming the r=1.

Michael Shackleford, A.S.A.


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:
Taking the derivitives:
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:
21/2 * (1+cos(t))1/2.
Multiply by (1-cos(t))1/2 / (1-cos(t))1/2 and the integral becomes:
21/2 * sin(t) / (1-cos(t))1/2.
Let u=cos(t) and the integral becomes:
21/2 * (1-u)-1/2.
Integrating this you get:
21/2 * 2 * (1-u)1/2 .
The bounds are 0 to 2*pi, so the total arc length is:
21/2 * 2 * (21/2+21/2) =
21/2 * 2 * 2 * 21/2 = 8

Michael Shackleford, A.S.A.

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