The angular frequency, ω, of an object undergoing periodic motion, such as a ball at the end of a rope being swung around in a circle, measures the rate at which the ball sweeps through a full 360 degrees, or 2π radians. The easiest way to understand how to calculate angular frequency is to construct the formula and see how it works in practice.
Angular Frequency Formula
The formula for angular frequency is the oscillation frequency f (often in units of Hertz, or oscillations per second), multiplied by the angle through which the object moves. The angular frequency formula for an object which completes a full oscillation or rotation is:
A more general formula is simply:
where θ is the angle through which the object moved, and t is the time it took to travel through θ.
Remember: a frequency is a rate, therefore the dimensions of this quantity are radians per unit time. The units will depend on the specific problem at hand. If you are taking about the rotation of a merry-go-round, you may want to talk about angular frequency in radians per minute, but the angular frequency of the Moon around the Earth might make more sense in radians per day.
Angular frequency is the rate at which an object moves through some number of radians. If you know the time it took for the object to move through an angle, the angular frequency is the angle in radians divided by the time it took.
Angular Frequency Formula Using Period
To fully understand this quantity, it helps to start with a more natural quantity, period, and work backwards. The period (T) of an oscillating object is the amount of time it takes to complete one oscillation. For example, there are 365 days in a year because that is how long it takes for the Earth to travel around the Sun once. This is the period for the motion of the Earth around the Sun.
But if you want to know the rate at which the rotations are occurring, you need to find the angular frequency. The frequency of rotation, or how many rotations take place in a certain amount of time, can be calculated by:
For the Earth, one revolution around the sun takes 365 days, so f = 1/365 days.
So what is the angular frequency? One rotation of the Earth sweeps through 2π radians, so the angular frequency ω = 2π/365. In words, the Earth moves through 2π radians in 365 days.
An Example Calculation
Try another example calculating angular frequency in another situation to get used to the concepts. A ride on a Ferris wheel might be a few minutes long, during which time you reach the top of the ride several times. Let’s say you are sitting at the top of the Ferris wheel, and you notice that the wheel moved one quarter of a rotation in 15 seconds. What is its angular frequency? There are two approaches you can use to calculate this quantity.
First, if ¼ rotation takes 15 seconds, a full rotation takes 4 × 15 = 60 seconds. Therefore, the frequency of rotation is f = 1/60 s −1, and the angular frequency is:
Similarly, you moved through π/2 radians in 15 seconds, so again, using our understanding of what an angular frequency is:
Both approaches give the same answer, so looks like our understanding of angular frequency makes sense!
One Last Thing…
Angular frequency is a scalar quantity, meaning it is just a magnitude. However, sometimes we talk about angular velocity, which is a vector. Therefore, the angular velocity formula is the same as the angular frequency equation, which determines the magnitude of the vector.
Then, the direction of the angular velocity vector can be determined by using the right hand rule. The right hand rule allows us to apply the convention that physicists and engineers use for specifying the “direction” of a spinning object.
- Young, H. D., Freedman, R. A., (2012) University Physics. San Francisco, CA: Addison-Wesley.
- Wolfram ScienceWorld: Angular Frequency
- The indicator of the musical equipment. image by Andrey Khritin from Fotolia.com