# Basic question pertaining to Polar Coordinates & how to employ them

• warhammer
In summary, you would integrate the velocity expression like you do usually with respect to time when the said expression involves r cap as well as theta cap.
warhammer
I have a question that might be considered vague or even downright idiotic but just wanted to know that once we find out the velocity & acceleration of a body in angular motion in plane polar coordinates, and are asked to integrate the expressions in order to find position at some specified time 't' , would we proceed to integrate the velocity expression (say) like we do usually with respect to time when the said expression involves r cap as well as theta cap; since the polar unit vectors are not "fixed" unlike the unit vectors of i,j,k in Cartesian system?

The position vector is $\mathbf{r} = r\mathbf{e}_r(\theta)$. You need both $r(t)$ and $\theta(t)$ to fully specify the position. The velocity vector is $$\dot{\mathbf{r}} = \dot r \mathbf{e}_r + r\dot\theta \frac{d}{d\theta}\mathbf{e}_r = \dot r \mathbf{e}_r + r\dot \theta \mathbf{e}_\theta$$ so if $\dot{\mathbf{r}}= v_r(t)\mathbf{e}_r + v_\theta(t) \mathbf{e}_\theta$ then \begin{align*} r &= \int v_r\,dt \\ \theta &= \int \frac{v_\theta}{r}\,dt \end{align*}

(You can always resolve components because $\mathbf{e}_r \cdot \mathbf{e}_r = \mathbf{e}_\theta \cdot \mathbf{e}_\theta = 1$ and $\mathbf{e}_r \cdot \mathbf{e}_\theta = 0$ are constant.)

warhammer and vanhees71
pasmith said:
The position vector is $\mathbf{r} = r\mathbf{e}_r(\theta)$. You need both $r(t)$ and $\theta(t)$ to fully specify the position. The velocity vector is $$\dot{\mathbf{r}} = \dot r \mathbf{e}_r + r\dot\theta \frac{d}{d\theta}\mathbf{e}_r = \dot r \mathbf{e}_r + r\dot \theta \mathbf{e}_\theta$$ so if $\dot{\mathbf{r}}= v_r(t)\mathbf{e}_r + v_\theta(t) \mathbf{e}_\theta$ then \begin{align*} r &= \int v_r\,dt \\ \theta &= \int \frac{v_\theta}{r}\,dt \end{align*}

(You can always resolve components because $\mathbf{e}_r \cdot \mathbf{e}_r = \mathbf{e}_\theta \cdot \mathbf{e}_\theta = 1$ and $\mathbf{e}_r \cdot \mathbf{e}_\theta = 0$ are constant.)
Thank you so much for your help sir

vanhees71 and berkeman

## 1. What are polar coordinates and how are they different from Cartesian coordinates?

Polar coordinates are a coordinate system used to locate points on a plane. They are different from Cartesian coordinates in that they use a distance and angle from a fixed point, known as the pole, to locate a point, as opposed to using x and y coordinates.

## 2. How do you convert between polar and Cartesian coordinates?

To convert from polar to Cartesian coordinates, you can use the following formulas: x = r * cos(theta) and y = r * sin(theta), where r is the distance from the pole and theta is the angle. To convert from Cartesian to polar coordinates, you can use the formulas: r = sqrt(x^2 + y^2) and theta = arctan(y/x).

## 3. What are some applications of polar coordinates?

Polar coordinates are commonly used in mathematics, physics, and engineering to describe circular and rotational motion, as well as in the study of complex numbers. They are also used in navigation and mapping, as well as in polar graphs and equations.

## 4. How do you plot points in polar coordinates?

To plot a point in polar coordinates, you need to know the distance from the pole and the angle. Start by drawing a line from the pole at the given angle, then measure the distance along that line to locate the point. You can also use a protractor to measure the angle and a ruler to measure the distance.

## 5. What are some common mistakes to avoid when using polar coordinates?

Some common mistakes to avoid when using polar coordinates include mixing up the order of the coordinates (r and theta), using the wrong scale for the angle, and forgetting to convert between polar and Cartesian coordinates when necessary. It is also important to pay attention to the quadrant in which the point is located, as this can affect the signs of the coordinates.

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