As I understood, the x-t graph for uniform circular motion is sinusoidal, v-t is cosinusoidal, a-t is also cosinusoidal. Is there any futher derivative that is constant throughout each revolution?
2 Answers
Derivatives of sines and cosines alternate between cosines and sines for all higher order derivatives. So, no, higher order derivatives of what you have described will not end up being constant.
By definition, what is constant in uniform circular motion is the magnitude of these vectors though. Circular motion means a constant position magnitude relative to the center of the circle, and the "uniform" specification does indicate motion at a constant speed. From there it follows that the acceleration also has a constant magnitude.
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1$\begingroup$ it really depends on the coordinate system, if one uses polar coordinates with the origin at the center of the circle, all derivatives will be trivially constant. $\endgroup$– paulinaCommented Sep 4 at 12:00
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$\begingroup$ @paulina I have focused on the quantities asked about in the OP as well as quantities that do not depend on the coordinate system. $\endgroup$ Commented Sep 4 at 12:07
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$\begingroup$ I know, I was just reformulating your last statement in a somewhat snarky way. $\endgroup$– paulinaCommented Sep 4 at 14:40
You should start thinking of a motion in 2D as a vector quantity.
Given a point taken as a origin $O$, position vector of point $P$ can be thought as the vector going from the origin to the point $P$, $\mathbf{r}_P(t)$.
You can represent the position of the point $P$ in the 2-dimensional plane using different sets of coordinates. As an example:
- Cartesian coordinates, $x$, $y$;
- polar coordinates, $r$, $\theta$, representing the distance from the origin and the angle measured w.r.t. a reference direction;
Using both coordinates, the position of point $P$ can be written as
$$\begin{aligned} \mathbf{r}_P(t) & = R \cos \omega t \, \mathbf{\hat{x}} + R \sin \omega t \, \mathbf{\hat{y}} = \\ & = R \, \mathbf{\hat{r}}(t) \end{aligned} \ ,$$
being the unit vectors of the Cartesian coordinates $\mathbf{\hat{x}}$, $\mathbf{\hat{y}}$ constant, while the unit vectors of the polar coordinates $\mathbf{\hat{r}}(t)$, $\boldsymbol{\hat{\theta}}(t)$ time-dependent as they follow the motion of the point.
What is constant
For the particular case of uniform circular motion,
- the radial component is constant
- the tangential component of the velocity is constant $v_{\theta} = \mathbf{v} \cdot \boldsymbol{\hat{\theta}} = \omega R$ is constant
- the angular velocity $\boldsymbol{\omega}$
What is not constant
- the Cartesian coordinates are not constant $x(t) = R \cos \omega t$, $y(t) = R \sin \omega t$
- the velocity is not constant, $$\mathbf{v}(t) = R \omega \boldsymbol{\hat{\theta}}(t) = - R \omega \sin \omega t \mathbf{\hat{x}} + R \omega \cos \omega t \mathbf{\hat{y}}$$