2D Rotational Kinematics

Alexei Gilchrist

The description of 2D rotational motion is developed in analogue to linear kinematics.

1 General motion

The position of a particle can be written in polar coordinates where an angle \(\theta\) takes an analogous role to linear position in angular motion:

\[\begin{align*}x &= r \cos\theta \\ y &= r \sin\theta\end{align*}\]
and the inverse transformations are \(r = \sqrt{x^2+y^2}\) and \(\theta = \tan^{-1}(y/x)\).

The angular velocity is then \(\omega\equiv\dot{\theta}\) which we can relate to the linear coordinates by differentiating the expressions above

\[\begin{align*}\dot{x} &= \dot{r}\cos\theta - r\sin\theta \,\dot{\theta} \\ \dot{y} &= \dot{r}\sin\theta + r\cos\theta \,\dot{\theta}.\end{align*}\]
Similarly the angular acceleration is \(\alpha \equiv \dot{\omega} = \ddot{\theta}\), and differentiating the above equations again (note that the last terms are the product of three functions of time) we get
\[\begin{align*}\ddot{x} &= \ddot{r}\cos\theta -2\dot{r}\dot{\theta}\sin\theta -r\cos\theta\,\dot{\theta}^2 - r \sin\theta\,\ddot{\theta} \\ \ddot{y} &= \ddot{r}\sin\theta +2\dot{r}\dot{\theta}\cos\theta -r\sin\theta\,\dot{\theta}^2 + r \cos\theta\,\ddot{\theta}\end{align*}\]

In 2D the angular-acceleration, -velocity and -position (\(\alpha\), \(\omega\), and \(\theta\)) will play analogous roles to the acceleration, velocity and position in linear motion. For instance the equations of motion for constant acceleration are the same for both sets, just different symbols. In 3D things are not so simple but that is a subject for other articles.

2 Motion in a circle

Specialising to the case of motion in a circle, the radius will be fixed so that \(\dot{r}=0\). The position variables stay the same

\[\begin{align*}x &= r \cos\theta \\ y &= r \sin\theta\end{align*}\]
but the velocities simplify
\[\begin{align}\dot{x} &= - r\sin\theta \,\dot{\theta} = -\dot{\theta} y \label{eq:velocityincircles1}\\ \dot{y} &= r\cos\theta \,\dot{\theta} = \dot{\theta} x \label{eq:velocityincircles2}\end{align}\]
so as a vector we could write
\[\begin{equation*}\dot{\vec{r}} \equiv \vec{v} = r \dot{\theta} \begin{pmatrix}-\sin\theta \\ \cos\theta\end{pmatrix}\end{equation*}\]
which has a magnitude \(v = r\dot{\theta}\) since \(\cos^2\theta+\sin^2\theta=1\). The acceleration also simplifies a bit:
\[\begin{align*}\ddot{x} &= -r(\cos\theta\,\dot{\theta}^2 +\sin\theta\,\ddot{\theta} )\\ \ddot{y} &= -r(\sin\theta\,\dot{\theta}^2 - \cos\theta\,\ddot{\theta})\end{align*}\]

If the motion in a circle is at a constance angular velocity \(\dot{\theta}=\omega=\mathrm{const.}\) (i.e. \(\ddot{\theta}=0\)), then the acceleration simplifies further

\[\begin{align*}\ddot{x} &= -r\cos\theta\,\omega^2 = -\omega^2 x\\ \ddot{y} &= -r\sin\theta\,\omega^2 = -\omega^2 y\end{align*}\]
Note that this is directed in the opposite direction to \(\vec{r}\). These two equations are independent of each other and in fact describe the equations of motion of a harmonic oscillator with the solutions:
\[\begin{align*}x(t) &= A_x\cos(\omega t+\phi_x) \\ y(t) &= A_y\cos(\omega t+\phi_y).\end{align*}\]
where \(A_x\), \(A_y\), \(\phi_x\), and \(\phi_y\) are determined by the initial conditions. Note that the angular frequency of an oscillator is often denoted by the same symbol \(\omega\) that is commonly used to denote angular velocity. Confusingly, in the case of constant motion in a circle they are the same quantity! In general they refer to different things though, so take care about which one you are referring to.

Writing the acceleration for constant motion in a circle as a vector, we have

\[\begin{equation*}\ddot{\vec{r}} \equiv \vec{a} = -\omega^2 \vec{r}\end{equation*}\]
whose magnitude is \(a = \omega^2 r = v^2/r\), and it’s directed towards the centre of the circle.

There is a neat pictorial way of seeing the changes that occur when an object moves in a circle. Imagine some point in the object is located at \(\vec{r}\) and it undergoes a small change \(\Delta \theta\):

A small rotation \(\Delta\theta\) will change \(x\) by \(-y \Delta\theta\), and \(y\) by \(x \Delta\theta\).

From the diagram above and using simple trigonometry we can see

\[\begin{align*}\Delta x &= -r \Delta\theta\sin\theta = -r \Delta\theta \frac{y}{r} = -y \Delta\theta \\ \Delta y &= -\Delta\theta\cos\theta = r \Delta\theta \frac{x}{r} = x\Delta\theta.\end{align*}\]
If the change \(\Delta\theta\) is due to a small change \(\Delta t\) in time then we simply recover equations \eqref{eq:velocityincircles1} and \eqref{eq:velocityincircles2}.

More formally, imagine that \(x\) and \(y\) change due to small changes \(\Delta \theta\) and \(\Delta r\) so that \(x \rightarrow x+\Delta x\) and \(y \rightarrow y+\Delta y\). Taking \(x\) and \(y\) to be functions of \(\theta\) and \(r\): \(x\equiv x(\theta,r)\), \(y\equiv y(\theta,r)\), the changes \(\Delta x\) and \(\Delta y\) are related to changes \(\Delta \theta\) and \(\Delta r\) by

\[\begin{align*}\Delta x &= \frac{\partial x}{\partial r}\Delta r + \frac{\partial x}{\partial \theta}\Delta \theta = \cos\theta \Delta r - r \sin\theta \Delta\theta \\ \Delta y &= \frac{\partial y}{\partial r}\Delta r + \frac{\partial y}{\partial \theta}\Delta \theta = \sin\theta \Delta r + r \cos\theta \Delta\theta.\end{align*}\]
If \(\Delta r=0\) we recover the equations for changes in circular motion \(\Delta x=-y \Delta\theta\) and \(\Delta y = x \Delta\theta\). This formal way of looking at changes is one well worth becoming familiar with, it’s normally called the total derivative.

© Copyright 2021 Alexei Gilchrist