Polar coordinates

Two ways to describe position

In rectangular coordinates, a point in a plane is described by $(x,y)$. The value $x$ tells how far left or right the point is, and $y$ tells how far up or down it is.

In polar coordinates, the same point is described by $(r, heta)$. The value $r$ is the distance from the origin, and $heta$ is the angle measured from a reference direction, usually the positive x-axis.

Both systems describe the same physical point. They simply use different information.

Why polar coordinates are useful

Polar coordinates are useful when a problem has circular or radial symmetry. Examples include planets orbiting the Sun, objects moving in circles, waves spreading from a source, electric fields around a point charge, and gravitational fields around a spherical mass.

If something naturally depends on distance from a center and angle around that center, polar coordinates may be simpler than rectangular coordinates.

Converting from polar to rectangular

The connection between polar and rectangular coordinates comes from right-triangle trigonometry. If a point is distance $r$ from the origin at angle $heta$, then

$x = rcos heta$

$y = rsin heta$

These formulas are the same component formulas used for vectors. The radius $r$ acts like the vector magnitude, and the x and y coordinates are components.

For example, a point at $r = 10 m$ and $heta = 30^circ$ has

$x = 10cos 30^circ$

$y = 10sin 30^circ$

Converting from rectangular to polar

To convert from $(x,y)$ to $(r, heta)$, use the Pythagorean theorem:

$r = sqrt{x^2 + y^2}$

For the angle,

an heta = rac{y}{x}

so

ight)$$ As with vector directions, quadrant matters. The signs of $x$ and $y$ tell where the point lies. A calculator's inverse tangent may need adjustment, or you can use atan2 if available. ## Non-unique coordinates Polar coordinates are not unique. The angle $ heta$ and $ heta + 2pi$ point in the same direction. A point can also be represented using a negative $r$ with an angle shifted by $pi$. In physics, we usually choose the clearest representation, often with $r geq 0$. This non-uniqueness is not a flaw. It reflects the repeating nature of angles. ## Circular motion For circular motion at fixed radius, polar coordinates are especially natural. If an object moves around a circle of radius $r$, its distance from the center stays constant while $ heta$ changes with time. Instead of tracking both $x(t)$ and $y(t)$ separately, you can focus on the changing angle. Angular speed is often written $$omega = rac{Delta heta}{Delta t}$$ where $ heta$ is measured in radians. The connection between arc length and angle is $$s = r heta$$ so circular motion links linear distance and angular change. ## Fields and radial dependence Many physical quantities depend mainly on radius. The gravitational force from a spherical planet depends on distance from its center. The electric field from a point charge depends on distance from the charge. In such cases, polar or spherical coordinates reveal the natural structure of the problem. ## The big idea Polar coordinates describe position by distance and direction. They are especially useful for circular, rotational, and radial problems. Conversions between polar and rectangular coordinates come directly from trig and the Pythagorean theorem. Choosing the right coordinate system can make a physics problem much easier.