Angular Frequency And Frequency: 3 Important Explanations

The angular frequency and frequency are the quantities that measure the oscillation per unit of time. The article discusses the relationship between angular frequency and frequency.

Angular frequency describes the angular displacement of the body per unit of time. In a relationship, the frequency describes the number of oscillations of the body per unit of time. The angular frequency measures a similar characteristic as frequency, and both quantities are scalars that only have magnitude but not direction.

The oscillating body or oscillator means the body is performing the periodic motion by undergoing the one cycle; when it is passed through a range of positions from its mean position and returns to its mean position again.

The quantities of the oscillating body, such as the angular frequency denoted by the omega symbol (ω) and frequency represented by (f), describe the body’s oscillation rate or how much it oscillates from its mean position. But these quantities are based on the types of oscillation. When the oscillation is linear, we examine its frequency. Whereas, when it is angular, we examine its angular frequency.

Since the frequency calculates the number of entire body oscillations per unit of time, the measuring unit of frequency is represented in the vibration per second or, precisely, cycles per second. Simply, its measuring unit is Hertz(Hz) which is equal to one cycle per second.

To determine the frequency of oscillation, we first need to find out its time period. The time period is also the quantity of oscillating body that shows the total time used by the body to achieve one oscillation. Comparing the definitions of both time period and frequency, these quantities of oscillation are reciprocal to each other.

i.e., $f = \frac{1}{T}$ ……….. (#)

For example, in a sinusoidal waveform, the time taken by the wave to complete one oscillation is ½ second, then its frequency is 2 cycles per second or Hertz.

But when the body is oscillating angularly, its displacement from the mean position is measured by the angular frequency. The body travels in a circular path, covering a particular angle is acknowledged as its angular displacement. Since the angular displacement includes an angle, the angular frequency of the oscillating body is expressed in radian per second (rad s-1) or revolution per minute (rpm).

For example, when discussing the rotation of the merry-go-round in the kid’s park, we expressed its angular frequency in radians per minute. But when talking about the angular frequency of the moon rotating around the earth, it makes more sense to express it in radians per day.

Relationship between Frequency and Angular Frequency

The frequency and angular frequency of the oscillating body are related to each other because both quantities are used to define the body’s oscillation rate.

The angular frequency (ω) formula of the oscillating body is the product of the frequency (f), and the angle through the body oscillates. i.e., $\omega = 2\pi f$. That means the angular frequency is analogous to frequency by the constant factor 2π.

The simple harmonic motion (SHM) of the system illustrates that the angular frequency ω and frequency f have identical dimensions. Hence, both quantities are measured by the same unit of the inverse of time. i.e., s-1. This fact does agree with the measuring unit of angular frequency. Still, it compares to the laws of physics and eliminates the difference in the relation between angular frequency and frequency. i.e., $\omega = 2\pi f$.

Like frequency (f) of the oscillation body, its angular frequency (ω) is also related to the time period (T). When the body rotates in an orbital or simply circular path, its time period estimates the total time needed by the body to finish one revolution.

As f =1/T, the relation between angular frequency and frequency becomes $\omega = \frac{2\pi }{T}$. ……(*)

What is 2π in Angular Frequency and Frequency?

When we express rate of oscillation in terms of a time period, the constant factor 2π relates the angular frequency to frequency.

While describing the angular frequency, we explain the body’s rotation in radians per second unit. The body needs to rotate 360° to complete one oscillation. Since 360° = 2π. That is why the constant factor 2π comes into play while relating angular frequency with frequency during oscillation.

During the oscillation of the body from its mean position, we simply see how much an angle of oscillation through which the body oscillates changes in one second. For example, if an angle through which the body oscillates goes from 0 radians to  radians (360°) in one second, we can determine its angular frequency by dividing the change in angle 2π by time period T one second as per formula (*)

If $\omega = \frac{2\pi }{T}$ and $f = \frac{1}{T}$

Then, $\omega = 2\pi f$

That makes the angular frequency of the oscillation body higher than its regular frequency by the factor of 2π.

So, if 1Hz = 10 rad/sec, then $1 radian = \frac{360}{10} = 36 ^{\circ}$.