4.1 Polar Coordinates in Two Dimensions

We have studied in great detail the mechanics of objects travelling in straight lines. Now we want to extend this to more general situations where objects can move along curved paths. We will first study motion in 2D as it is much simpler than 3D. As we saw in section 1.4, vectors in 2D can be equivalently described by two cartesian components or their length together with the angle they make with the $x$ -axis. Depending on the type of motion in the problem, it is much simpler to describe the trajectory of an object the latter way. These are known as polar coordinates.

To set up the coordinate system, we define two new coordinate axes relative to the position vector $\boldsymbol{r}$ . The first axis points along the direction of the position vector, and the second points one quarter turn anticlockwise from the first. These axes get two orthonormal vectors to form a basis which we label $\hat{\boldsymbol{r}}$ and $\hat{\boldsymbol{\theta}}$ respectively.

Then a 2D vector $\boldsymbol{A}$ has a cartesian representation $\boldsymbol{A}=A_{x}\hat{\boldsymbol{\imath}}+A_{y}\hat{\boldsymbol{\imath}}$ and a polar representation $\boldsymbol{A}=A_{r}\hat{\boldsymbol{r}}+A_{\theta}\hat{\boldsymbol{\theta}}$ . To find the components in one coordinate system using the components in the other, we have to know the relations between all the basis vectors. Using trigonometry, we can see that

	$\displaystyle\hat{\boldsymbol{r}}$	$\displaystyle=\cos\theta\hat{\boldsymbol{\imath}}+\sin\theta\hat{\boldsymbol{% \jmath}}$		(4.1)
	$\displaystyle\hat{\boldsymbol{\theta}}$	$\displaystyle=-\sin\theta\hat{\boldsymbol{\imath}}+\cos\theta\hat{\boldsymbol{% \jmath}}.$		(4.2)

Substituting this into the polar coordinate representation of $\boldsymbol{A}$ , we get

	$\displaystyle\boldsymbol{A}$	$\displaystyle=A_{r}(\cos\theta\hat{\boldsymbol{\imath}}+\sin\theta\hat{% \boldsymbol{\jmath}})+A_{\theta}(-\sin\theta\hat{\boldsymbol{\imath}}+\cos% \theta\hat{\boldsymbol{\jmath}})$		(4.3)
		$\displaystyle=(A_{r}\cos\theta-A_{\theta}\sin\theta)\hat{\boldsymbol{\imath}}+% (A_{r}\sin\theta+A_{\theta}\cos\theta)\hat{\boldsymbol{\jmath}}.$		(4.4)

Comparing coefficients with the cartesian representation, we see that

	$\displaystyle A_{x}$	$\displaystyle=A_{r}\cos\theta-A_{\theta}\sin\theta$		(4.5)
	$\displaystyle A_{y}$	$\displaystyle=A_{r}\sin\theta+A_{\theta}\cos\theta.$		(4.6)

By doing the same procedure the other way around, we can deduce that

	$\displaystyle A_{r}$	$\displaystyle=A_{x}\cos\theta+A_{y}\sin\theta$		(4.7)
	$\displaystyle A_{\theta}$	$\displaystyle=-A_{x}\sin\theta+A_{y}\cos\theta.$		(4.8)

The definition of the axes in polar coordinates may seem confusing at first as the position vector moves around over time. This means that the axes and unit vectors also change over time, and we will see how this affects calculation of the motion of an object in the next section.

Once thing that doesn’t change is the representation of the position vector itself, which is always given by $\boldsymbol{r}=r\hat{\boldsymbol{r}}$ . The conversion between representations of the trajectory in cartesian and polar coordinates is then given by equations 1.24, 1.25, 1.26, and 1.27.

$\displaystyle x$	$\displaystyle=r\cos\theta$	(4.9)
$\displaystyle y$	$\displaystyle=r\sin\theta$	(4.10)
$\displaystyle r$	$\displaystyle=\sqrt{x^{2}+y^{2}}$	(4.11)
$\displaystyle\tan\theta$	$\displaystyle=\frac{y}{x}.$	(4.12)