You have heard of velocity and you have heard of angles, but have you heard of angular velocity? Angular velocity describes how fast an object moves in terms of angles instead of in terms of distances. This is a different way of looking at the movement of objects, but it can be very convenient in some cases, and with some simple formulas, we can actually relate 'normal' velocity to angular velocity. Let's dive in!
Definition of Angular Velocity
Similar to how we first learn about position and displacement before learning about velocity, we must first define angular position in order to talk about angular velocity.
Angular Position
The angular position of an object with respect to a point and a reference line is the angle between that reference line and the line that goes through both the point and the object.
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.
If someone is walking directly toward you, her angular position with respect to you does not change (regardless of the reference line you choose).
Angular Velocity
The angular velocity of an object with respect to a point is a measure of how fast that object moves through the point's view, in the sense of how fast the angular position of the object changes.
The angular velocity of an object with respect to you corresponds to how fast you have to turn your head to keep looking directly at the object.
Notice how there is no mention of a reference line in this definition of angular velocity because we don't need one.
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 \(y=\dfrac{\pi}{180}x\).
Remember that degrees might be intuitive and it's fine to use degrees to express angles, but in calculations (for example those of angular velocities), you should always use radians.
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 crucial to use radians in angular velocity units when dealing with equations. If you are given an angular velocity expressed in degrees per unit of time, the very first thing you should do is to convert it to radians per unit of time!
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).
Your field of vision is a certain angle (which is roughly \(180º\) or \(\pi\,\mathrm{rad}\)), so an object's angular velocity determines completely how fast it moves through your field of vision. The appearance of the radius in the formula of angular velocity is the reason that far-away objects move much more slowly through your field of vision than objects that are close to you.
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 velocityof 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:
\[4.1\,\mathrm{rad/h}=\dfrac{235º}{\mathrm{h}}\]
Angular Velocity - Key takeaways
- The angular velocity of an object with respect to a point is a measure of how fast that object moves through the point's view, in the sense of how fast the angular position of the object changes.
- The units of angular velocity are that of inverse time.
- In writing down angular velocity, we may use degrees per unit of time or radians per unit of time.
- In doing calculations with angles, we always use radians.
- Angular velocity \(\omega\) is calculated from (linear) velocity \(v\) and radius \(r\) as \(\omega=\dfrac{v}{r}\).
- This is logical because the faster something goes and the closer it is to us, the faster it moves through our field of vision.
- We can calculate linear velocity from angular velocity and radius by \(v=\omega r\).
- The angular velocity of the Earth's rotation around its axis is\(\dfrac{2\pi}{24}\dfrac{\mathrm{rad}}{\mathrm{h}}\).
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