Angular Velocity

Angular Position

This isn't the most intuitive definition, so see the illustration below for a clear picture of what is meant.

We see that absolute distances don't matter to the angular position, but only ratios of distances: we can rescale this whole picture and the angular position of the object would not change.

Angular Velocity

Demonstration of the angular velocity of a smiley with respect to its center, adapted from image by Sbyrnes321 Public domain.

Units of Angular Velocity

From the definition, we see that angular velocity is measured in an angle per unit of time. As angles are unitless, the units of angular velocity are the inverses of the units of time. Thus, the standard unit to measure angular velocities is \(s^{-1}\). As an angle always comes with its unitless measure, e.g. degrees or radians, an angular velocity can be written down in the following ways:

\[\omega=\dfrac{xº}{s}=\dfrac{y\,\mathrm{rad}}{s}=y\dfrac{\mathrm{rad}}{s}\]

Here, we have the familiar conversion between degrees and radians as \(\dfrac{x}{360}=\dfrac{y}{2\pi}\), or \(y=\dfrac{\pi}{180}x\).

Formula for Angular Velocity

Let's look at a situation that is not too complicated, so suppose a particle is moving in circles around us. This circle has a radius \(r\) (which is the distance from us to the particle) and the particle has a speed \(v\). Obviously, the angular position of this particle changes with time due to its circular speed, and the angular velocity \(\omega\) is now given by

\[\omega=\dfrac{v}{r}\]

It is now time to examine if this equation makes sense. First of all, the angular velocity doubles if the particle's speed doubles, which is expected. However, the angular velocity also doubles if the particle's radius is halved. This is true because the particle will only have to cover half the original distance to make one full round of its trajectory, so it will also only need half the time (because we assume a constant speed when halving the radius).

Angular Velocity to Linear Velocity

Using the formula above, we can also calculate an object's linear velocity \(v\) from its angular velocity \(\omega\) and its radius \(r\) as follows:

\[v=\omega r\]

This formula for linear velocity is just a manipulation of the previous formula, so we already know that this formula is logical. Again, make sure to use radians in calculations, so also while using this formula.

In general, we can state that the linear velocity of an object is directly related to its angular velocity through the radius of the circular trajectory it's following.

Angular Velocity of Earth

Rotation of the Earth around its axis, sped up, Wikimedia Commons CC BY-SA 3.0.

A nice example of angular velocity is the Earth itself. We know that the Earth makes a full rotation of \(360º\) every 24 hours, so the angular velocity$\omega$of an object on the equator of the Earth with respect to the middle of the Earth is given by

\[\omega=\dfrac{360º}{24\,\mathrm{h}}\]

\[\omega=\dfrac{2\pi}{24}\dfrac{\mathrm{rad}}{\mathrm h}\]

Note how we immediately converted to radians for our calculation.

The Earth's radius is \(r=6378\,\mathrm{km}\), so we can now calculate the linear velocity \(v\) of an object on the equator of the Earth using the formula we introduced earlier:

\[v=\omega r\]

\[v=\dfrac{2\pi}{24}\dfrac{\mathrm{rad}}{\mathrm h}·6378\,\mathrm{km}\]

\[v=1670\,\dfrac{\mathrm{km}}{\mathrm h}=464\,\dfrac{\mathrm{m}}{\mathrm s}\]

Angular Velocity of Cars on a Round-About

Suppose a round-about in Dallas is a perfect circle centered in downtown with a radius of \(r=11\,\mathrm{mi}\) and the speed limit on this round-about is \(45\,\mathrm{mi/h}\). The angular velocity of a car driving on this road at the speed limit with respect to downtown is then calculated as follows:

\[\omega=\dfrac{v}{r}\]

\[\omega=\dfrac{45\,\mathrm{mi/h}}{11\,\mathrm{mi}}\]

\[\omega=4.1\,\mathrm{h}^{-1}\]

\[\omega=4.1\,\mathrm{rad/h}\]

If we want to, we can convert this to degrees:

\[4.1\,\mathrm{rad/h}=\dfrac{235º}{\mathrm{h}}\]