# 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.

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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 how fast an object rotates through a central angle with respect to time.

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.

Angular frequency is the degree of rotation in radians an object makes per one second.

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 formula for finding angular speed or velocity $$\omega$$ is the ratio of the angular displacement $$\Theta$$ to the time $$t$$ in seconds: $\omega=\frac{\Theta}{t}.$

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.

The formula for finding the linear speed $$v$$ of an object rotating through a circle with radius $$r$$ and angular speed $$\omega$$ is

$v=r\omega.$

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

You are riding your bicycle that has wheels with a $$33\, \text{cm}$$ radius. Your wheels make $$50$$ complete revolutions in 7 seconds. Find the wheels' angular speed and your bicycle's linear speed.

Solution

First, there are a couple of conversions we will need to make in order to use the formulas. First, the angular displacement is supposed to be measured in radians rather than revolutions. The bike makes $$50$$ revolutions, and there are $$2\pi$$ radians per revolution. So, altogether, the bike has an angular displacement of

$50(2\pi)=100\pi \, \text{radians}.$

Next, generally speed is measure in meters per second, at least in most physics problems. So, that is the unit of measure you will use here. This means that the radius of the bicycle's wheels need to be converted from centimeters to meters. The radius of each wheel is $$33\, \text{cm}$$, which is $$0.33$$ meters.

Now, on to finding the speeds. It makes the most sense to start with angular speed since you will need that value in the formula for linear speed. Recall that the formula for angular speed is

$\omega=\frac{\Theta}{t}.$

You know that $$\Theta=100\pi$$ radians and $$t=7$$ seconds. So, the angular speed of the bicycle is

\begin{align}\omega &=\frac{100\pi \text{ rad}}{7 \text{ sec}}\\ &\approx44.88 \,\text{rad/sec}.\end{align}

Next, you need to use that speed in order to find the linear speed. The formula you need is

$v=r\omega.$

For the sake of accuracy, you should probably use the exact form of the angular speed that you just found. So, the linear speed of the bicycle wheel is

\begin{align}v&=(0.33\,\text{m})\left(\frac{100}{7}\pi\,\text{rad/sec}\right)\\ &\approx 14.81 \,\text{m/s}.\end{align}

## Angular Speed - Key takeaways

• Angular speed is how fast an object rotates through a central angle with respect to time.
• The formula for finding angular speed or velocity $$\omega$$ is the ratio of the angular displacement $$\Theta$$ to the time $$t$$ in seconds: $\omega=\frac{\Theta}{t}.$
• The unit of measure for angular speed will be radians per second or rad/sec.
• Angular speed is not a vector and thus does not have direction. Angular velocity is a vector and has both magnitude and direction.
• Linear speed is related to angular speed through the formula$v=r\omega$ where linear speed is $$v$$, the radius of the object's revolution is $$r$$, and the object's angular speed is $$\omega$$.

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What is the formula for angular speed?

The formula for finding angular speed or velocity 𝝎 is the ratio of the angular displacement 𝜽 to the time t in seconds: 𝝎=𝜽/t.

What is angular speed?

Angular speed is how fast an object rotates through a central angle with respect to time. Basically, angular speed is 𝝎=𝜽/t.

What are the units of angular speed?

The angular displacement is measured in radians, and the time is measured in seconds. So, the units for angular speed are radians per second.

How to calculate angular speed?

Divide the angular displacement 𝜽 by the amount of time t the displacement took. This will give you speed in radians per second.

How to convert linear speed to angular speed?

The formula for relating the linear speed v of an object rotating through a circle with radius r and angular speed ω is v=rω.

## Test your knowledge with multiple choice flashcards

Two friends decide to race one lap around a track. One friend stands in lane one, and the other friend stands parallel to them in lane eight. They both run one full lap in their own lanes and finish at the same time. Which statement(s) is/are true?

True or False: Frequency and angular frequency are both measured in Hertz (Hz).

True or False: Angular speed can be negative.

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