*The periodic motion of an object, particle, or quantity at regular intervals about a mean position is known as oscillation. *

**When a body oscillates, it comprises linear as well as angular displacement, this angular displacement is known as angular frequency of oscillation. There are also other terms in physics for angular frequency such as angular speed orbital frequency.**

The angular frequency is the scalar measure of the angular displacement of an oscillating particle. For sinusoidal waves, it is referred to as the rate of phase change. When a ball tied to a rope is rotated in a circular motion, the rate at which it completes one oscillation of 360 degrees is known as the angular frequency.

**Angular Frequency of Oscillation Formula**

The change in angle that occurs in one second is termed angular frequency. Hence the basic formula to derive the angular frequency is;

[latex]\omega =\frac{\Theta }{t}[/latex]

Here;

ω is the angular frequency

Θ is the angle through which an object is displaced.

t is for the time taken.

For the simple harmonic motion or simply oscillation, the formula of angular frequency is derived by multiplying the linear frequency with the angle that is covered by oscillating particles. For one complete cycle, the angle is 2π. Hence the formula for angular frequency becomes;

ω =2πf

Using the relationship between frequency and time period in the above equation the formula becomes;

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

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

Since angular frequency is angular displacement rate, its unit becomes radian per unit time, that is;

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

**Angular Frequency of Oscillation Spring**

In the above spring-mass system, on adding the load, the spring displaces to distance y, and the oscillation stretches it to a further x position.

According to Hooke’s law.

[latex]F=ky[/latex]

From the diagram, we can see that

[latex]W=mg=ky[/latex]

From the free body diagram, we can see that weight is acting downward. Inertia force that is ma is acting upward, and restoring force that is k(x+y) is also acting upward.

We will get:

[latex]ma+kx+y-W=0[/latex]

[latex]ma+kx+ky-W=0[/latex]

We know that W=ky; hence we get:

[latex]ma+kx=0[/latex]

Dividing by m:

[latex]\frac{ma}{m}+ \frac{k}{m}x=0[/latex]

[latex]a+ \frac{k}{m}x=0[/latex]

On comparing it with the SHM equation, we get:

[latex]\omega^{2} =\frac{k}{m}[/latex]

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

This is the angular frequency of the spring oscillation.

**Angular Frequency Of Oscillation Pendulum**

A pendulum is a small bob tied to a thread. It swings to generate oscillation. The amplitude of pendulum oscillation is measured as the maximum displacement that a bob covers starting from the central position. In a simple pendulum, the mass of the string is negligible compared to the mass of the bob.

Through the figure above, we can see the forces acting on the bob of the pendulum. The gravitational weight is acting downwards. The restoring torque acting on the pendulum is the element of the weight of the bob. From the figure, we get the value of torque as;

[latex]\tau =-L(mgsin\Theta)[/latex]

[latex]I\alpha =-L(mgsin\Theta)[/latex]

[latex]I\frac{\mathrm{d^{2}}\Theta }{\mathrm{d} x^{2}} =-L(mgsin\Theta)[/latex]

[latex]m L^{2}\frac{\mathrm{d^{2}}\Theta }{\mathrm{d} x^{2}} =-L(mgsin\Theta)[/latex]

[latex]\frac{\mathrm{d^{2}}\Theta }{\mathrm{d} x^{2}} =-\frac{g}{L}sin\Theta[/latex]

[latex]\frac{\mathrm{d^{2}}\Theta }{\mathrm{d} x^{2}} +\frac{g}{L}sin\Theta=0[/latex]

For every small angle we have;

[latex]sin\Theta\approx \Theta[/latex]

Hence we get;

[latex]\frac{\mathrm{d^{2}}\Theta }{\mathrm{d} x^{2}} +\frac{g}{L}\Theta=0[/latex]

Comparing it with the simple harmonic motion equation:

[latex]\frac{\mathrm{d^{2}}x }{\mathrm{d} t^{2}} +\omega^{2}x =0[/latex]

We get:

[latex]\omega^{2}=\frac{g}{L}[/latex]

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

Here;

g is the acceleration due to gravity, and L is the length of the pendulum.

**Angular Frequency of Oscillation Of The Object**

For an oscillating object, the SHM equation is given as:

[latex]x=Asin(\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

For the oscillating object, the angular frequency is given as;

ω =2πf

It tells about how much angle the object is rotating for displacing.

**How to Find Angular Frequency of Oscillation **

For the different objects and scenarios, a different formula is used to calculate the angular frequency of oscillation.

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

For instance, the amplitude of the oscillation is given to be 0.14m; the phase change is 0. Now to cover 14 cm in 8.5 seconds, the angular frequency is calculated using the formula;

[latex]0.14=0.14sin(8.5\omega)[/latex]

[latex]1=sin(8.5\omega)[/latex]

[latex]sin^{-1}1=8.5\omega[/latex]

[latex]\frac{\pi}{2}=8.5\omega[/latex]

[latex]\omega=\frac{\pi}{16.2} rad\cdot s^{-1}[/latex]

For calculating the angular frequency of the pendulum, the formula used is;

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

For example, if the length of the pendulum is 10 cm, then the angular frequency of oscillation is;

[latex]\omega=\sqrt{\frac{10}{0.10}}[/latex]

[latex]\omega=\sqrt{100}[/latex]

[latex]\omega=10 rad. s^{-1}[/latex]

To calculate the angular frequency of spring, the formula is:

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

If the spring constant is given to 2 N/m and mass is given to be 8 kg, then the angular frequency would be;

[latex]\omega=\sqrt{\frac{2}{8}}[/latex]

[latex]\omega=\sqrt{\frac{1}{4}}[/latex]

[latex]\omega=\frac{1}{2}[/latex]

[latex]\omega=0.5 rad. s^{-1}[/latex]

**Frequently Asked Questions (FAQs)**

**What is the angular frequency of oscillation?**

The repetitive motion of a particle about a fixed point is known as oscillation.

**The angle change of the particle is the angular frequency of oscillation. In physics, it is also termed the rate of change of phase. It is a scalar element as it is just the angular displacement without any direction. The formula for the angular frequency is given as;**

ω =2πf

**How is angular frequency is related to the time period?**

The oscillating objects are comprised of both linear displacements as well as angular displacement.

**The basic formula for angular frequency is given as;**

[latex]\omega=\frac{\Theta }{t}[/latex]

**It shows the relation of time and angular frequency of oscillation. **

**Now the general formula for angular frequency is:**

** ω =2πf**

**Substituting the given relation**

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

**We get;**

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

**This equation relates angular frequency and time period. **

**What is the unit of angular frequency?**

The angular frequency is the change in angle of the oscillating particle in unit time.

**The unit of the angular frequency is given as radian per unit sec such as;**

[latex]1 \omega=1 rad\cdot s^{-1}[/latex]

**When the object covers one complete cycle in one second, then the angular frequency becomes 1. **

**Is angular frequency the same as frequency?**

The number of oscillations the object makes in one second is known to be the frequency.

**No, the frequency and angular frequency are not the same things. Angular frequency is the change in the angle of the oscillating particle in unit time, whereas the frequency is the oscillation made in one second. They both are different terms used for a different concept of physics. **