$\"\"$

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The cycloid equations are $\"\"$ and $\"\"$ .

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Find the curvature of cycloid.

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Apply curvature formula from 69(a): $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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$\"\"$

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Curvature is $\"\"$ .

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Substitute $\"\"$ , $\"\"$ , $\"\"$ and $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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Top of one of it $\"\"$ s arches $\"\"$ is maximum, this occurs when $\"\"$ is maximum.

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Maximum value of $\"\"$ is $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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$\"\"$

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$\"\"$ .