Maximum Rate of Change of Y Coordinate of Astroid
Theorem
Let $C_1$ and $C_2$ be the epicycle and deferent respectively of an astroid $H$.
Let $C_2$ be embedded in a cartesian plane with its center $O$ located at the origin.
Let the center $C$ of $C_1$ move at a constant angular velocity $\omega$ around the center of $C_2$.
Let $P$ be the point on the circumference of $C_1$ whose locus is $H$.
Let $C_1$ be positioned at time $t = 0$ so that $P$ its point of tangency to $C_2$, located on the $x$-axis.
Let $\theta$ be the angle made by $OC$ to the $x$-axis at time $t$.
Then the maximum rate of change of the $y$ coordinate of $P$ in the first quadrant occurs when $P$ is at the point where:
- $x = a \paren {\dfrac 1 3}^{3/2}$
- $y = a \paren {\dfrac 2 3}^{3/2}$
Proof
The rate of change of $\theta$ is given by:
- $\omega = \dfrac {\d \theta} {\d t}$
From Equation of Astroid: Parametric Form, the point $P = \tuple {x, y}$ is described by the parametric equation:
- $\begin {cases} x & = a \cos^3 \theta \\ y & = a \sin^3 \theta \end{cases}$
The rate of change of $y$ is given by:
\(\ds \dfrac {\d y} {\d t}\) | \(=\) | \(\ds \dfrac {\d y} {\d \theta} \dfrac {\d \theta} {\d t}\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 a \omega \sin^2 \theta \cos \theta\) | Power Rule for Derivatives, Derivative of Sine Function, Chain Rule for Derivatives |
By Derivative at Maximum or Minimum, when $\dfrac {\d y} {\d t}$ is at a maximum:
- $\dfrac {\d^2 y} {\d t^2} = 0$
Thus:
\(\ds \dfrac {\d^2 y} {\d t^2}\) | \(=\) | \(\ds \map {\dfrac \d {\d \theta} } {3 a \omega \sin^2 \theta \cos \theta} \dfrac {\d \theta} {\d t}\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds 3 a \omega^2 \paren {2 \sin \theta \cos^2 \theta - \sin^3 \theta}\) | Product Rule for Derivatives and others |
Hence:
\(\ds \dfrac {\d^2 y} {\d t^2}\) | \(=\) | \(\ds 0\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds 3 a \omega^2 \paren {2 \sin \theta \cos^2 \theta - \sin^3 \theta}\) | \(=\) | \(\ds 0\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds 2 \sin \theta \cos^2 \theta\) | \(=\) | \(\ds \sin^3 \theta\) |
We can assume $\sin \theta \ne 0$ because in that case $\theta = 0$ and so $\dfrac {\d y} {\d t} = 0$.
Thus when $\sin \theta = 0$, $y$ is not a maximum.
So we can divide by $\sin \theta$ to give:
\(\ds 2 \cos^2 \theta\) | \(=\) | \(\ds \sin^2 \theta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tan^2 \theta\) | \(=\) | \(\ds 2\) | Tangent is Sine divided by Cosine | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \tan \theta\) | \(=\) | \(\ds \sqrt 2\) |
We have:
\(\ds x\) | \(=\) | \(\ds a \cos^3 \theta\) | ||||||||||||
\(\ds y\) | \(=\) | \(\ds a \sin^3 \theta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac y x\) | \(=\) | \(\ds \tan^3 \theta\) | Tangent is Sine divided by Cosine | ||||||||||
\(\ds \) | \(=\) | \(\ds 2^{3/2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds 2^{3/2} x\) |
From Equation of Astroid: Cartesian Form:
- $x^{2/3} + y^{2/3} = a^{2/3}$
Hence:
\(\ds x^{2/3} + \paren {2^{3/2} x}^{2/3}\) | \(=\) | \(\ds a^{2/3}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x^{2/3} \paren {1 + 2}\) | \(=\) | \(\ds a^{2/3}\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \paren {\frac {a^{2/3} } 3}^{3/2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a \paren {\frac 1 3}^{3/2}\) |
Similarly:
\(\ds \frac y x\) | \(=\) | \(\ds 2^{3/2}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \frac 1 {2^{3/2} } y\) | |||||||||||
\(\ds \paren {\frac 1 {2^{3/2} } y}^{2/3} + y^{2/3}\) | \(=\) | \(\ds a^{2/3}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^{2/3} \paren {1 + \frac 1 2}\) | \(=\) | \(\ds a^{2/3}\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y^{2/3}\) | \(=\) | \(\ds \frac 2 3 a^{2/3}\) | simplifying | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds y\) | \(=\) | \(\ds \paren {2 \frac {a^{2/3} } 3}^{3/2}\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds a \paren {\frac 2 3}^{3/2}\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $10$