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, 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 r.	Answers:
A) 8r  B) 4r  C) 9r  D) 7r  E) 10r
Hint: (1-cos(t))0.5 = 20.5×sin(t/2)   The answer is 8r.   Imagine the circle resting on coordinate (0,0) and moving east.   Also magine the point on the circle to start at (0,0).   Let t be the angle between the point on the circle and the center of the circle.   The position of the point on the circle, relative to t, is:   x = rt - r×sin(t)   y = r - r×cos(t)   Taking the derivatives:   dx/dt = r - r×cos(t)   dy/dt = r×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:   r×21/2 × (1-cos(t))1/2.   Using the hint:   r×21/2 × 21/2 × integral of sin(t/2) dt from 0 to 2×pi   = 2×r × (-2×cos(t/w) from 2×pi to 0)   = 8r   correct answer 8r
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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, 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 r.