Rate of Change of Cartesian Coordinates of Cycloid
Theorem
Let a circle $C$ of radius $a$ roll without slipping along the x-axis of a cartesian plane at a constant speed such that the center moves with a velocity $\mathbf v_0$ in the direction of increasing $x$.
Consider a point $P$ on the circumference of this circle.
Let $\tuple {x, y}$ be the coordinates of $P$ as it travels over the plane.
Then the rate of change of $x$ and $y$ can be expresssed as:
\(\ds \frac {\d x} {\d t}\) | \(=\) | \(\ds \mathbf v_0 \paren {1 - \cos \theta}\) | ||||||||||||
\(\ds \frac {\d y} {\d t}\) | \(=\) | \(\ds \mathbf v_0 \sin \theta\) |
where $\theta$ is the angle turned by $C$ after time $t$.
Proof
Let the center of $C$ be $O$.
Without loss of generality, let $P$ be at the origin at time $t = t_0$.
By definition, $P$ traces out a cycloid.
From Equation of Cycloid, $P = \tuple {x, y}$ is described by:
- $(1): \quad \begin{cases} x & = a \paren {\theta - \sin \theta} \\
y & = a \paren {1 - \cos \theta} \end{cases}$
Let $\tuple {x_c, y_c}$ be the coordinates of $O$ at time $t$.
We have that:
- $y_c$ is constant: $y_c = a$
From Body under Constant Acceleration: Velocity after Time
- $x_c = \mathbf v_0 t$
as the acceleration of $O$ is zero.
But $x_c$ is equal to the length of the arc of $C$ that has rolled along the $x$-axis.
- $x_c = a \theta$
and so:
\(\text {(2)}: \quad\) | \(\ds \theta\) | \(=\) | \(\ds \frac {\mathbf v_0 t} a\) | |||||||||||
\(\text {(3)}: \quad\) | \(\ds \leadsto \ \ \) | \(\ds \frac {\d \theta} {\d t}\) | \(=\) | \(\ds \frac {\mathbf v_0} a\) |
Thus:
\(\ds x\) | \(=\) | \(\ds a \paren {\frac {\mathbf v_0 t} a - \sin \theta}\) | substituting for $\theta$ from $(2)$ | |||||||||||
\(\ds x\) | \(=\) | \(\ds \mathbf v_0 t - a \sin \theta\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d x} {\d t}\) | \(=\) | \(\ds \mathbf v_0 - a \cos \theta \frac {\d \theta} {\d t}\) | Chain Rule for Derivatives | ||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf v_0 - a \cos \theta \frac {\mathbf v_0} a\) | from $(3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf v_0 \paren {1 - \cos \theta}\) |
and:
\(\ds y\) | \(=\) | \(\ds a \paren {1 - \cos \theta}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac {\d y} {\d t}\) | \(=\) | \(\ds a \sin \theta \frac {\d \theta} {\d t}\) | Chain Rule for Derivatives | ||||||||||
\(\ds \) | \(=\) | \(\ds a \sin \theta \frac {\mathbf v_0} a\) | from $(3)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \mathbf v_0 \sin \theta\) |
$\blacksquare$
Sources
- 1992: George F. Simmons: Calculus Gems ... (previous) ... (next): Chapter $\text {B}.21$: The Cycloid: Problem $4 \ \text{(a)}$