Angular Frequency vs Frequency: 5 Answers You Should Know


A body moving back and forth repetitively is known as an oscillating body, and this motion is oscillation.


When the body oscillates, it displaces linearly, but apart from that, there is also the angular frequency of oscillation. Let’s analyze angular frequency vs frequency. Angular frequency is nothing but the angle change of the body. In the physics concept, angular frequency and frequency are distinct.

The angular frequency and the linear frequency of an oscillation should not be confused. They are two
different concepts with a different use. The angular frequency determines the phase change, whereas the linear frequency gives the specification of the number of oscillations completed in a second.

Angular frequency vs frequency


To examine angular frequency vs frequency we take the example of a swing in the children’s park. It keeps swinging back and forth at regular intervals. Hence it is an oscillatory motion. From the above figure, it is seen that the swing moves ahead at an angle of θ and also moves back at the same angle. The change in angle that occurs in unit time is the angular frequency of oscillation.

ω=2π/T

[latex]\omega = \frac{2\pi}{T}[/latex]

The number of times it moves back and forth in one second would be the frequency of swing.


The oscillation is a simple harmonic motion, and hence its equation is the same as the sinusoidal wave
equation.
For an oscillating object, the SHM equation is given as:

[latex]x = A sin(\omega t+ \phi)[/latex]


Here;
x is the displacement of the object 
A is the amplitude of the oscillation 
φ is the phase change
ω is the angular frequency 


Suppose a small bob is hanging from a string. When we strike the bob, it starts moving and oscillating.
From the above free body diagram, one can see that the bob is displacing at an angle. Hence the angular
frequency is given as:

ω=2π/T

[latex]\omega = \frac{2\pi}{T}[/latex]

At the same time, the frequency of the bob would be the oscillation made in 1 sec. Suppose here it makes 5 oscillations in one second. Then the frequency of the bob is 5.

Angular Frequency Vs Frequency


The figure above clearly shows the difference between the angular frequency of oscillation and the
Frequency. The general formula to calculate frequency, f, of the oscillation is given as:

f=1/T

[latex]f = \frac{1}{T}[/latex]

Here, T is the time period.


The formula for the angular frequency of oscillation is:

ω=2π/T

[latex]\omega = \frac{2\pi}{T}[/latex]


On substituting the value of T from the frequency formula into the angular frequency formula we
get;

ω=2πf

[latex]\omega = 2 \pi f[/latex]


It gives the relation between the frequency and angular frequency of the oscillation. The formula of angular frequency is calculated by multiplying the frequency by the angle that an oscillating object covers. For a full cycle, the angle is 2π.

Image Credit: MormegilPendulumCC BY-SA 3.0

For a simple pendulum, the angular frequency and linear frequency are given as:

[latex]\omega = \sqrt{\frac{g}{L}}[/latex]

And

[latex]f = \frac{1}{2\pi} \sqrt{\frac{g}{L}}[/latex]

For the spring, the formula for angular frequency and linear frequency is given as:

[latex]\omega = \sqrt{\frac{k}{m}}[/latex]

and

[latex]f = \frac{1}{2\pi} \sqrt{\frac{k}{m}}[/latex]


The unit of angular frequency is given as radian per unit time, that is:

1ω=1rad.sec-1

[latex]1 \omega = 1 rad. sec^{-1}[/latex]

The standard unit of the frequency of the oscillation is Hertz, Hz.


Suppose a ball is oscillating and completing 6 oscillations in 1 second. Then the frequency of the oscillatory ball is:

f=6Hz

[latex]f = 6 Hz[/latex]

And therefore, the angular frequency becomes:

ω=2π*6

ω=12π

[latex]\omega = 2 \pi \times 6[/latex]

[latex]\omega = 12 \pi[/latex]

[latex]\omega = 12 \pi[/latex]


Frequently Asked Questions (FAQs)


What is Oscillation?


In physics, when an object displaces from its position with time, it is said to be in motion.


An object moving back and forth repetitively is known as an oscillating body, and this motion is oscillation. All the oscillations are periodic, but the converse may not be true. The earth revolves around the sun in periodic motion but not oscillatory.

What are the terms related to oscillation?


A body moving in a continuous partner about a fixed point is oscillatory.


The oscillatory body has equation as: [latex]x = A sin(\omega \dot t+ \phi)[/latex]
Here the general terms of oscillation are:
x stands for the object’s displacement
A is its amplitude
φ is the phase change
ω is the angular frequency

Frequency is an important term of oscillations that tells the number of oscillations completed in one
second.

How to find the angular frequency of a simple pendulum?


The pendulum is a simple oscillatory body having a mass bob hanging to a string.


When the pendulum oscillates, it changes the angle at regular intervals. Hence the simple pendulum has angular displacement. The angular frequency of oscillation of the pendulum is calculated by the formula:

[latex]\omega = \sqrt{\frac{g}{L}}[/latex]
here,
g is the acceleration due to gravity and
L is the length of the pendulum.


What is the difference between angular frequency and linear frequency?

The angular frequency is far different from the linear frequency. Both have distinct meanings.


When a body oscillates, it undergoes a change in its angle, which is the angular frequency.
The number of times a particle oscillates in one second is the frequency of the object. The
unit of angular frequency in rad/sec. The unit for frequency of Hertz.


What is the relation between linear frequency and angular frequency of oscillation?


When a body moves continuously between two points, it is said to be in oscillatory motion. The frequency and the angular frequency is related to each other from the given
formula:

The formula for frequency and angular frequency is f=1/T and ω=2π/T

The formula for frequency and angular frequency is [latex]f = \frac{1}{T}[/latex] and [latex]\omega = \frac{2 \pi}{T}[/latex]
Substituting the value of T we get:

ω=2πf

[latex]\omega = 2 \pi f[/latex]

Rabiya Khalid

Hi,  I am Rabiya Khalid, currently pursuing my masters in Mathematics. Article writing is my passion and I have been professionally writing for more than a year now. Being a science student, I have a knack for reading and writing about science and everything related to it. If you like what I write you can connect with me on LinkedIn: https://www.linkedin.com/mwlite/in/rabiya-khalid-bba02921a In my free time, I let out my creative side on a canvas. You can check my paintings at: https://www.instagram.com/chronicles_studio/

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