,

so

From conservation of energy: ,

, so

However, from an intermediate point ,

So

with

define

So the amount of time is the same from any point.

1. A cycloid is the curve defined by the path of a point on the edge of circular wheel as the wheel rolls along a straight line.

2. Cycloidal curves is the curve of fastest descent under gravity

3. the period of a ball rolling back and forth inside this curve does not depend on the ballâ€™s starting position.

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