When a circle with radius a rolls along a straight line, the curve traced by a point on its circumference is called a cycloid.
x = a(t − sin t), y = a(1 − cos t)
Example ① Find the points of the cycloid on the x-axis (values of t for which y=0).
Solution: 1−cos t = 0 → t = 0, 2π, 4π, ...
Then x = 0, 2πa, 4πa, ...
Solution: 1−cos t = 0 → t = 0, 2π, 4π, ...
Then x = 0, 2πa, 4πa, ...
Key Point
- It moves 2πa in one revolution.
- Vertex: at t=π, it is (πa, 2a)
Practice Problems
- Find the highest point of the cycloid x=2(t−sin t), y=2(1−cos t).
Solutions and Explanations
- Solution: (2π, 4) (when t=π)