পোলার স্থানাঙ্ক

When a top spins on its axis, a planet about its orbit, or a wheel around its axis, it is cumbersome to use the Cartesian coordinates $x$ and $y$. In Cartesian coordinates, circular motion is described by a confusing set of sines and cosines that are cumbersome to deal with. The reason is that they are not circularly symmetric: we want a set of coordinates that will make it easy to describe a rotation about an axis.

For this reason we use the polar coordinates $r$ and $\theta$, where $r$ is the distance to a center point and $\theta$ is the angle around that point. If we rotate the system by some angle $\theta_0$, we simply add $\theta \to \theta + \theta_0$ and keep $r$ constant. That's an easy transformation! When an object orbits, for instance a planet around a star at constant radius, the equation describing its motion is very simple:

$$r = r_0 \\ \theta = \theta_0 + \omega t$$

$\omega$ is the angular velocity, as we will learn about later.

The connection to Cartesian coordinates is $r = \sqrt{x^2+y^2}$ and $\theta = \tan^{-1} (y/x)$.

Units of polar coordiantes rotation

The radius $r$ is measured in units of length. In the SI system, this has units of meters.

Angles are typically measured in radians. These are natural units where a complete revolution is $2\pi\;\mathrm{rad}$. You might be more used to degrees, where a complete revolution is $360^\circ$. One radian is $360^\circ/2\pi = 57.3^\circ$.

One very useful approximation is small angles. When we use radians, $\sin \theta \approx \theta$ and $\cos \theta \approx 1 - \frac{1}{2} \theta^2$. For instance, the sine of 1 degree is $$sin(1^\circ) = sin(0.017) \approx 0.017$$