Length of Arc of Cycloid/Proof 1
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Theorem
Let $C$ be a cycloid generated by the equations:
- $x = a \paren {\theta - \sin \theta}$
- $y = a \paren {1 - \cos \theta}$
Then the length of one arc of the cycloid is $8 a$.
Proof
Let $L$ be the length of one arc of the cycloid.
From Arc Length for Parametric Equations:
- $\ds L = \int_0^{2 \pi} \sqrt {\paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2} \rd \theta$
where, from Equation of Cycloid:
- $x = a \paren {\theta - \sin \theta}$
- $y = a \paren {1 - \cos \theta}$
we have:
\(\ds \frac {\d x} {\d \theta}\) | \(=\) | \(\ds a \paren {1 - \cos \theta}\) | ||||||||||||
\(\ds \frac {\d y} {\d \theta}\) | \(=\) | \(\ds a \sin \theta\) |
Thus:
\(\ds \paren {\frac {\d x} {\d \theta} }^2 + \paren {\frac {\d y} {\d \theta} }^2\) | \(=\) | \(\ds a^2 \paren {\paren {1 - \cos \theta}^2 + \sin^2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds a^2 \paren {1 - 2 \cos \theta + \cos^2 \theta + \sin^2 \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 2 a^2 \paren {1 - \cos \theta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 4 a^2 \map {\sin^2} {\theta / 2}\) | Half Angle Formula for Sine |
Thus:
\(\ds L\) | \(=\) | \(\ds \int_0^{2 \pi} 2 a \map \sin {\theta / 2} \rd \theta\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \bigintlimits {-4 a \map \cos {\theta / 2} } 0 {2 \pi}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds 8 a\) |
So $L = 8 a$ where $a$ is the radius of the generating circle.
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Example $2$