Angular Frequency Simple Harmonic Motion: 5 Important Facts


The oscillating body’s quantity that fundamentally measures its periodic motion is termed as the angular frequency simple harmonic motion (SHM). The article discusses about the angular frequency in SHM.

The angular frequency simple harmonic motion (SHM) is the characteristic of the oscillating system as a simple pendulum. The SHM involves the “to and fro” oscillation, hence its motion is sinusoidal. The number of oscillations carried by the bob in SHM is called its angular frequency – which measures how many times the bob oscillates in a specific time.

When the body’s motion repeats itself at frequent periods, it is said to be in periodic motion. Since the motion of the swinging bob of the simple pendulum is repetitious or periodic, we called SHM as the simplest form of periodic motion. But what makes an object continuously change its motion at regular intervals? When the bob goes to a higher position when we pull or push, it is a restoring force that is exerted on the bob, bringing it to the initial place and causing oscillation. That also means the bob performs one oscillation when it is returned to its initial position.

When the restoring force on the bob is similar to its displacement from the initial position, the bob’s motion is said to be “simple harmonic motion (SHM)”. The SHM is sinusoidal in time that describes smooth periodic oscillation of the bob. The bob’s sinusoidal nature demonstrates its angular frequency, which measures its oscillation rate. The simple harmonic motions are periodic and oscillatory, but not all oscillatory motions are simple harmonic motions; even though they are periodic. Likewise, the uniform circular motion (UCM) is called the periodic motion, not the oscillatory. 

Angular Frequency Simple Harmonic Motion
Angular Frequency Simple Harmonic Motion

Further, we will explore some differences that we need to understand SHM, like distinguishing between angular frequency and angular velocity in SHM.

Read more about the Simple Pendulum’s Potential to Kinetic energy conversation.

What is Angular Velocity in Simple Harmonic Motion?

The angular velocity of the oscillating body in simple harmonic motion (SHM) is the change in the oscillating body’s angular position per unit time.

In simple harmonic motion (SHM), the angular velocity measures the angular speed of the oscillating or rotating body. i.e., the rate at which the body oscillate or rotates – which explains the body’s rotational movement. Since the angular velocity direction is perpendicular to its angular displacement, it estimates how the body can move around its mean position. Therefore, the angular velocity of the rotating body depends on its rotating movement. That means – quicker the rotating movement of the body, greater its angular velocity.

The angular velocity in SHM can be accomplished by differentiating the angular displacement ([latex]\theta[/latex]) with time.

[latex]\omega = \frac{d\theta}{dt}[/latex] …………(1)

The omega symbol [latex]\omega[/latex] indicates the angular velocity.

As per equation (1), the measuring unit for angular velocity is radians per second. Another unit of angular velocity is RPM, or revolution per minute. Its direction is predicted by the right-hand rule. By convention rule, the clockwise rotation showed negative angular velocity, while the anticlockwise is positive. 

Angular Velocity in Simple Harmonic Motion
Angular Velocity Simple Harmonic Motion (credit: wikipedia)

Usually, the body is assumed to be accelerated when its velocity varies with time. In terms of SHM, the velocity continuously differs over a while. Hence, the body accelerated to oscillates depending on its displacement from the mean position. That is why the pendulum’s bob is accelerated when we push or pulled it from the mean position. But eventually, it stopped, and after some time, it returned to its mean position again.

Read more about the Tension Force in Simple Pendulum

What is Angular Frequency in Oscillation?

The angular frequency is called the radial frequency in oscillation – which measures the angular displacement of the oscillating body per unit of time.

The oscillation relates to the repeated ‘to and fro’ movement of the body about a fixed position between two positions. It is the periodic motion that reproduces itself in a regular interval. For sinusoidal wave motion, the body moves from its mean position, stands the highest position, and returns to its mean position due to restoring force. The oscillating body’s maximum movement or displacement from its mean position is termed as the Amplitude (A). Whereas the magnitude of its angular displacement from the mean position is termed as Angular Frequency.

Angular Frequency in Simple Harmonic Motion
Angular Frequency Simple Harmonic Motion Sinusoidal Waveform (credit: shutterstock)

A set of oscillations we experienced all around us, from the atoms vibrating to the heart beating. Another examples of oscillation in physics are the sine waves in the side-to-side pendulum vibration or the up-and-down spring movement. In oscillation, the angular frequency is the rate of changing the state of a sinusoidal waveform. Consequently, the radians per second is the measuring unit for oscillating the body’s angular frequency. The angular frequency is a scalar, which indicates it is just a magnitude. But when we address the angular velocity in SHM, it is a vector. The angular frequency is identical to the angular velocity that defines the magnitude of its vector quantity.

Read more about the Relation between Simple Harmonic Motion and Uniform Harmonic Motion.

How do you find the Angular Frequency?

Since simple harmonic motion(SHM) is periodic, we can first find out the period and then the angular frequency by timing the complete oscillation of the body.

The Simple Harmonic Oscillation (SHM) assists us in finding out the displacement, velocity, and acceleration of the oscillating body. But first of all, we need to discover the fundamental characteristics of the period motion – like its amplitude and frequency. To define the oscillation frequency, we need to understand the time period quantity. The total time the oscillating body takes to complete one oscillation is called its Time Period (T). Moreover, the number of oscillations made by the body per unit of time is called its Frequency (f) – which measures the oscillation rate.

For a linear SHM, the period and amplitude are non-dependent on the amplitude of oscillation. For example, the guitar’s string oscillates with an equal frequency whether we plucked it hard or easily. That is because the period of oscillation is a constant, whereas a simple harmonic oscillator is utilized as a clock.

The frequency and time period of the oscillating body are reciprocal to each other. 

[latex]f = \frac{1}{T}[/latex]……………. (2)

When the body oscillates angularly, we consider its frequency as angular frequency. Consequently, to understand the rotations rate, we need to obtain the angular frequency.

Simple Harmonic Motion
Angular Frequency Simple Harmonic Motion Oscillating Body (credit: wikiversity)

Frequency to Angular Frequency

Angular frequency measures the same characteristic as normal frequency, but rather than cycles, it employs radians.

The angular frequency of the oscillating body is more significant than the regular frequency by a factor of 2π. The constant factor 2π originates from the basis that one revolution per second being similar to the 2π radians per second. Utterly saying, when the oscillating body dismissed from its mean position makes one revolution per second, it oscillates angularly as 2π radians per second.

The angular frequency formula for the oscillating body which completes one oscillation is computed as:

[latex]\omega = 2\pi f[/latex] ………………(*)

The angular frequency of the oscillating body is always greater than the regular frequency.

Difference between Frequency and Angular Frequency
Angular Frequency Simple Harmonic Motion
Difference between Frequency and Angular Frequency (credit: quora)

The angular frequency can be expressed in terms of time period T since the amount of time in seconds needs the oscillating body to complete one revolution. Therefore, as per equation (2), we can compute the angular frequency formula in terms of the time period as:

[latex]\omega = \frac{2\pi}{T}[/latex] …………..(3)

The angular frequency is the oscillating frequency of the body measured in oscillation per second and multiplied by the angular displacement [latex]\theta[/latex] of the body.

As per the equation (1), we can rewrite the above equations as

[latex]\omega = \frac{\theta }{t}[/latex] ………..(4)

Angular Frequency Spring

Let’s take an example of an oscillating object having mass m attached to the flexible connector such as spring.

We will find the angular frequency in spring by applying Hooke’s law and the concept of SHM. Hooke’s law defines the elastic attributes of any materials only in the area the force and the displacement are proportional. It states the amount of force needed for the elastic material to stretch or compress.

Mathematically,

[latex]F = -kx[/latex] ……… (5)

Where x is displacement and k is spring constant.

As per Newton’s second law of motion, a force is equivalent to mass times acceleration.

[latex]F = ma[/latex] ………..(6)

Since we can relate the angular frequency and mass of the oscillating object to the spring, we can discover displacement, velocity, and acceleration.

First, we formulate the SHM equations for displacement from the mean position as,

[latex]x = Asin\theta[/latex] ………….(7)

Where A is the amplitude of oscillation

Substituting equation (4) into the above equation, we obtained 

[latex]x = Asin\omega t[/latex] …………(8)

We obtained the SHM equations for acceleration of oscillating body as,

[latex]a = -A\omega ^{2}sin\omega t[/latex] ……… (9)

Now implanting both equations in (8) and (9) into force equations (5) and (6) and comparing, we get

[latex]ma = -kx[/latex] 

[latex]m(-A\omega ^{2}sin\omega t) = -k(Asin\omega t)[/latex] 

Divide both sides by [latex]-Asin\omega t[/latex], we get

[latex]m\omega ^{2} = k[/latex] 

We got the angular frequency formula in terms of spring constant and mass of the oscillating body as:

[latex]\omega = \sqrt{\frac{k}{m}}[/latex] …………..(10)

The above equation is the angular frequency formula in SHM when spring is ideal. i.e., no damping.

Finally, comparing equation (10) with equation (*), we can also compute the angular frequency formula for the period in terms of spring constant and mass of the oscillating body as:

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

 [latex]T = {2\pi} \sqrt{\frac{m}{k}}[/latex] ……………….(11)

The above equation is the time period for the oscillating object attached to the spring.

Angular Frequency in Springs
Angular Frequency Simple Harmonic Motion
Spring (credit: hyperphysics)

Is Angular Frequency constant in Simple Harmonic Motion?

The angular frequency, or the magnitude of the vector quantity of the oscillating body, is constant in simple harmonic motion (SHM).

In the uniform circular motion (UCM), both angular frequency and angular velocity are constant. But when the body oscillates angularly concerning a fixed axis, its motion becomes the ‘angular simple harmonic motion’. In a simple pendulum, when a bob is pushed or pulled, it gains constant angular frequency, but its angular velocity changes with time. That is why the angular velocity of the oscillating body in angular SHM is not constant, but its angular frequency is.

Generally, the angular frequency is based upon what forces are acting on the oscillating body. In the case of the simple pendulum or ideal spring, the force does not depend on angular velocity; but on the angular frequency. The angular frequency formula (10) shows that the angular frequency depends on the parameter k used to indicate the stiffness of the spring and mass of the oscillation body. For larger oscillation amplitudes, the value of k also changes, which are drastic enough to damage the spring. Therefore, it is clear that angular frequency will remain constant for a given system even if its angular velocity changes during oscillation.

Spring Simple Harmonic Motion
Angular Frequency Simple Harmonic Motion
(credit: shutterstock)

Let’s see if the angular frequency in SHM is a universal constant or not by taking an example of a simple pendulum, where the restoring force due to bob’s weight produces the SHM.

Similar to equation (11), we can write the time period formula for a simple pendulum as:

[latex]T = 2\pi \sqrt{\frac{g}{l}}[/latex] …………(12)

Where g is the acceleration due to gravity on the bob, and l is the pendulum’s length.

Comparing equation (12) with equation (3), we get

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

We obtained the angular frequency formula of the oscillating body in terms of acceleration due to gravity.

Now, if you measure the spring’s oscillation from the same mean position, the angular frequency will be constant. But if you measure the oscillation from another position of the same spring, you may notice a slight difference in angular frequency value [latex]\omega[/latex] due to small changes in g. That means the angular frequency [latex]\omega[/latex] is constant for the same mean position in SHM, but it is not a universal constant.

Angular Frequency Simple Harmonic Motion
Angular Frequency Simple Harmonic Motion
(credit: shutterstock)

How is Angular Frequency different from Angular Velocity?

The critical difference between the oscillating body’s angular frequency and angular velocity quantity is that one is a scalar, whereas the other is a vector.

The differences between angular frequency and angular velocity are analogues to the difference between speed and velocity in linear motion. The angular velocity is a vector quantity; that’s why the right-hand rule defines its direction. But since the angular frequency quantity is a scalar, we can say it is just the magnitude of the angular velocity. As per equations (1) and (*), both quantities have the same symbol and formula but different meanings. The angular frequency tells us the angular displacement of the oscillating body per unit of time. On the other hand, angular velocity measures the rate or degree of change in its angular rotation.

Difference between Angular Frequency and Angular Velocity
Angular Frequency Simple Harmonic Motion
Angular Frequency Vs Angular Velocity
(credit: wikipedia)

You may have noticed that the motion doesn’t have to be expressed by a standard rotation but just only a movement that periodically returns its position. However, the angular velocity is connected to the motion. Since the angular velocity only comprises the rotational movement of the oscillating body, the angular frequency is more commonly used to represent a wide range of physical problems in oscillation. That is why angular frequency is widely used when we talk about simple harmonic motion.


Manish Naik

Hello, I'm Manish Naik completed my MSc Physics with Solid-State Electronics as a specialization. I have three years of experience in Article Writing on Physics subject. Writing, which aimed to provide accurate information to all readers, from beginners and experts. In my leisure time, I love to spend my time in nature or visiting historical places. I am honoured to be part of LambdaGeeks. Looking forward to connecting you through LinkedIn - https://www.linkedin.com/in/manish-ashok-naik/ Also, for Maharashtra travel guide and heritage conservation articles, visit my website Wandering Maharashtra - https://wanderingmaharashtra.com/travel-blogs/

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