# Physical significance of the terms of acceleration in polar coordinates

How do I get an idea, or a 'feel' of the components of the acceleration in polar coordinates which constitute the component in the eθ direction?

from what I know, $\vec a= (\ddot{r}−r\dot{θ}^2) \hat e_r + (r\ddot{θ}+ 2\dot{r}\dot{θ}) \hat e_θ$, where $\hat e_r$ and $\hat e_θ$ are unit vectors in the radial direction and the direction of increase of the polar angle, θ.)

The two components in $\hat e_r$ direction - $\ddot{r}$ and $r\dot{θ}^2$ - are the usual acceleration along radius vector and the centrifugal force experienced. But what is the significance of the other two terms?. Is there any day-to-day or a common situation where we experience the Coriolis force and the other term?

I can memorize the formula and use it, but I will truly 'understand' its significance only if I can 'feel' the terms.

• thanks zero, for editing the question! i'm kinda new here and have little knowledge of how to put the symbols, so please excuse me :) – Sakazuki Akainu Mar 22 '17 at 20:17

## 2 Answers

$\ddot{r} \hat e_r$: usual radial acceleration

$-r\dot{θ}^2 \hat e_r$: centripetal acceleration

$r\ddot{θ}\hat e_θ$: This is the Euler acceleration. It is an acceleration due to a change of angular velocity (at fixed $r$). Example taken from the linked wikipedia article: on a merry-go-round this is the force that pushes you to the back of the horse when the ride starts (angular velocity increasing) and to the front of the horse when the ride stops (angular velocity decreasing).

$2\dot{r}\dot{θ} \hat e_θ$: Coriolis acceleration

I can only answer part of your query: the Coriolis force does not enter into this. The form you have comes directly from expressing $\ddot{x}$ and $\ddot{y}$ in terms of derivatives aof $r$ and $\theta$ through the geometrical change \begin{align} x(t)&=r(t)\cos(\theta(t))\, ,\qquad y(t)=r(t)\sin(\theta(t)) \end{align} followed by converting $$\hat x=\cos(\theta(t))\hat r- \sin(\theta(t))\hat \theta\, ,\quad \hat y=\sin(\theta(t))\hat r+ \cos(\theta(t))\hat \theta\, .$$ Thus the various bits and pieces in polar coordinates come purely from the change of coordinate system, not the inertial or non-inertial nature of the reference frame, in which you can set whatever coordinate systems you want.

Because the orientation of $\hat r$ and $\hat \theta$ depend on the position in space, it is not easy to get intuition into their components.