Angular Speed

Imagine for a moment that you are laying on your bed and looking up at your ceiling fan (or just check out the one in the image below). For most fans, if you pull the chain, the speed of the fan will change. How would you describe this speed? Do you think of it in miles per hour, the way you may think of a car’s speed? Probably not, since the fan is not moving across the ceiling.

It’s more likely that you think of the fan’s speed in terms of how many complete circular turns it’s making in the few seconds that you are looking at it. When you turn the fan’s speed up, the fan makes a lot more rotations in the same amount of time. This concept of speed–traveling in circles instead of in a line–is called angular speed.

Fig. 1 - Ceiling fan, used to demonstrate angular speed.

Angular Speed and Frequency

As the name would suggest, angular speed has to do with how fast the angle is changing because angles sweep open in the shape of a circle.

Angular speed is often also called angular velocity, although there is a small but important difference between the two. See the section below titled "Difference Between Angular Speed and Velocity" for further explanation.

In general, frequency is defined as how often something happens over a particular period of time, usually one second in Physics. This would be measured in cycles per second, or Hertz (Hz). When talking about circular motion, frequency refers to the number of revolutions or rotations per second. This differs slightly from the definition of angular frequency.

This means that angular frequency is related to frequency by a scale factor of \(2\pi\) since one revolution is equal to \(2\pi\) radians.

Functionally, the definition of angular speed and angular frequency represent the same thing and will, therefore, have the same formula (shown below in the next section).

For further explanation on circular motion, see our article titled "Circular Motion and Gravitation".

Angular Speed Formula

Any speed, or velocity, is measured by the ratio of how much an object displaces to the amount of time that displacement takes. For linear speed or velocity, that is usually found by dividing distance over time. But with angular speed, you do not care so much about HOW FAR an object goes, but more about how much of a rotation it makes.

The angle measure is measured in radians (not degrees). So, it helps to recall that one complete revolution (which is equal to 360 degrees) is \(2\pi\) radians.

Fig. 2 - Circle rotating through angle of \(\Theta\) radians.

Angular speed unit

Because all speeds are defined by the ratio of an object’s displacement over time, then the unit of measure for speed is always

\[\text{speed}=\frac{\text{unit of displacement}}{\text{unit of time}}.\]

So for angular speed or velocity, the units will be radians per second or rad/sec.

Difference Between Angular Speed and Velocity

The difference between angular speed and angular velocity is the same as the difference between linear speed and linear velocity. For both, speed is not considered a vector. It is simply a magnitude that is not associated with a direction. But velocity is a vector. It has both magnitude and direction.

For circular motion, the direction you are concerned with clockwise and counterclockwise. If these are considered to be directions, that also means that things can move in a negative direction. For angular speed, there is no negative value. It simply represents how fast an object is rotating through the circle in any direction. But angular velocity is positive if the object is moving along the axis of rotation. The opposite direction from the axis of rotation will be negative.

Often, the axis of rotation is determined by the Right Hand Rule. See our article on "Torque and Angular Acceleration" to read more about the Right Hand Rule.

Difference Between Angular Speed and Linear Speed

Think back to the initial example of the ceiling fan as your concept of angular speed. Look at one fan blade. Does the entire blade have the same angular speed? Or is the end of the blade faster or slower than the bracket holding the blade to the motor? The entire blade completes the same angular rotation each second because the entire blade completes each revolution as a unit. So, the angular speed is the same for the entire length of the fan blade.

But each end of the fan blade is not traveling the same distance during each revolution. The bracket that connects the fan blade to the motor is making a rather small circle, which will have a small circumference or distance. The end of the fan blade is making a pretty large circle. Its circumference will be bigger, and therefore the end of the fan blade will travel a longer distance in that same amount of time. Thus, the end of the fan blade has a faster linear speed than the bracket despite having the same angular speed. Recall that linear speed is defined as

\[\text{speed}=\frac{\text{distance}}{\text{time}}.\]

But there is a relationship between angular speed and linear speed. Imagine riding a bicycle. In order to travel at a faster linear speed, you have to pedal faster, thus increasing your angular speed as well. The main factor that determines the linear speed of a rotating object is the size of the circle it is making.

Let's look at an example of how to use this formula and the previous formula for angular speed.