Plane polar co-ordinate system:
In the study of two-dimensional cases, we observe the motion takes place in a plane. Sometimes the plane polar co-ordinate is very suitable for this purpose. In this plane polar co-ordinate system, the two co-ordinates for a point \( A \) are:
(i) r is the radial distance of the point A from the origin O. This radial distance is always positive and the value varies from zero to infinity.
(ii) \( \theta \) is the angle, which the radial vector \( \vec{r} \) makes with the positive direction of the X-axis. This angle is always measured in the anti-clockwise direction with respect to the X-axis. The value of this angle varies from 0 to \( 2\pi \)
Relation with cartesian co-ordinates:
Let us consider \( x \) and \( y \) be the cartesian co-ordinate of the point A, as shown in Fig. 3.
From the image we can write:
\( x=OB=r\cos\theta \) and
\( y=AB=r\sin\theta \)therefore, \( r=\sqrt{x^2+y^2} \), and \( \theta=\tan^{-1}\frac{y}{x} \).