Angular frequency is a fundamental concept in physics, particularly in studying wave motion and oscillations. It measures the angular displacement of a particle per unit time. In this article, we will learn about the meaning and definition of angular frequency, the formula of angular frequency, the SI unit of angular frequency, applications of angular frequency, and differences between angular frequency and angular velocity.
Table of Content
What is Angular Frequency?
Angular frequency is denoted by ω (omega). It is a scalar quantity that shows how fast the phase of a sine wave changes. Its magnitude is the angular velocity, which is measured in radians/sec. Angular frequency describes how fast an object oscillates in circular motion. It’s the speed of one complete revolution or cycle per unit of time.
It is somehow related to frequency, which is the number of cycles or oscillations per unit of time. One (1) hertz (Hz) is expected to be nearly equal to approximately 6.28 rad/s. Angular frequency is useful in different studies, including physics, engineering, and mathematics, where it describes periodic motion and wave phenomena.
Angular Frequency Definition
Angular frequency is a measure of the changing angular displacement with time, and it is denoted as omega (ω). It is the frequency of a rotational motion.
Formula of Angular Frequency
The formula for angular frequency, denoted by the symbol ω (omega), is a fundamental concept in physics and engineering. It represents the rate of change of angular displacement of an object rotating about a fixed axis. The formula for angular frequency is given by:
ω = 2π/T
Where,
- ω (omega) is the angular frequency,
- π (pi) is a mathematical constant approximately equal to 3.14159,
- T is the period of rotation, representing the time taken for one complete cycle of rotation.
The angular frequency is measured in radians per second (rad/s) and represents the rotational speed or frequency of the object’s motion. It plays a crucial role in various fields, including mechanics, electromagnetism, and wave theory.
SI Unit: The SI unit of angular frequency is radians per second (rad/s).
Dimension: The dimension of angular frequency is T-1, where T is time.
Angular Displacement
Angular displacement, denoted by θ, is a fundamental concept in rotational motion that changes an object’s angular position. Angular displacement is a vector quantity, meaning it has both magnitude and direction. It is an angle used to identify how an object rotates from its initial position to its ending position.
Angular displacement is usually measured using a radian scale or a degree scale with one complete turn equal to 2π radians or 360 degrees. For calculating angular displacement, you multiply the angular velocity (ω) by the time (t) over which rotation happens, using the formula:
θ = ω × t
Understanding angular displacement is key to understanding rotational kinematics, dynamics, and angular acceleration, as it provides insights into an object’s location and orientation in rotation motion.
Derivation of Angular Frequency
Angular frequency can be derived from the definition of frequency. Frequency is defined as the number of cycles per unit time. In a circular motion, the angular displacement is the angle subtended by the particle’s radius vector at the circle’s centre in one cycle. Therefore, the angular displacement in one cycle is equal to 2π radians.
The time taken to complete one cycle is the reciprocal of the frequency, i.e., 1/f. Therefore, the angular frequency is:
ω = dθ/dt = (2π radians) / (1/f) = 2πf rad/s
Measurements of Angular Frequency
The various measurements of angular frequency are listed below along with their, symbol, formula and meaning.
|
Measurement of Angular Frequency |
Symbol |
Formula |
Meaning |
|---|---|---|---|
|
Frequency Measurement |
f |
f = 1/T |
The number of cycles or oscillations per unit time. |
|
Period Measurement |
T |
T = 1/f |
The time taken for one complete cycle or oscillation. |
|
Rotational Speed Measurement |
Ω |
|
The rate of rotation of an object, measured in revolutions per unit time. |
|
Doppler Shift Measurement |
Δf |
|
The change in frequency of a wave due to relative motion between the source and observer, where 𝑓0f0 is the rest frequency, 𝑣v is the relative velocity, and 𝑐c is the speed of light. |
|
Zeeman Effect Measurement |
Δλ |
|
The shift in the wavelength of spectral lines in the presence of a magnetic field 𝐵B, where 𝑒e is the elementary charge, 𝑚𝑒me is the electron rest mass, and 𝑐c is the speed of light. |
Use of Angular Frequency in Physics
Angular frequency is used in various physics concepts. Some of these concepts are described below in detail.
Simple Harmonic Motion
In SHM, angular frequency describes the rate at which an object oscillates back and forth around a stable equilibrium position. The angular frequency is related to the oscillation frequency by a factor of 2π. The formula for angular frequency in SHM is:
ω = √(k/m)
Where k is the spring constant, and m is the object’s mass.
Wave Motion
Angular frequency implies the speed of the wave’s phase change in a given period of time. It is related to the wave’s frequency by a factor of 2π. The formula for angular frequency in wave motion is:
ω = 2πf
Where f is the wave’s frequency.
Electromagnetic Waves
In electromagnetic waves, angular frequency generally refers to the pace of change of those electric and magnetic fields throughout time. The angular frequency is defined by a relation between 2π and the frequency of waves. Here, we have an equation for the angular frequency in electromagnetic waves of the following form:
ω = 2πf
With f being the wave frequency.
Quantum Mechanics
In quantum mechanics, the angular frequency is the pace at which the phase of the wave function evolves at a constant rate. The angular frequency is proportional to energy, as described in the formula:
E = ℏω
Which is equal to (ℏ —representing the reduced Planck’s constant) and where E represents energy.
Mechanical Vibrations
In mechanical vibrations, angular frequency describes a system’s rate of oscillation. However, the relationship is with the vibration’s frequency, which is determined by a multiplier of 2π. The frequency of angular motion is:
ω = √(k/m)
Where the opposite is k, the spring constant, and m, the system mass.
Application of Angular Frequency
Angular frequency has various applications in real life, including:
- In electrical engineering, angular frequency is used to calculate the impedance of a circuit.
- In mechanical engineering, angular frequency is used to calculate the natural frequency of a system.
- In physics, angular frequency is used to calculate the energy of a system.
- In music, angular frequency is used to calculate the frequency of a musical note.
Difference Between Angular Frequency and Angular Velocity
The difference between angular frequency and angular velocity is as follows:
|
Angular Frequency |
Angular Velocity |
|---|---|
|
It measure the rate of change of angular displacement or phase of a sinusoidal waveform or sine function |
Angular Velocity represent the rate of change of angular displacement with respect to time |
|
Measures the rate at which an object completes one full revolution or cycle per unit time |
Represents the rotational speed of an object in circular motion |
|
Used in the context of periodic motion and wave phenomena |
Used to describe the rotational motion of an object |
|
Used in various fields like physics, engineering, and mathematics |
Commonly used in dynamics, kinematics, and fluid mechanics |
|
Essential in describing oscillations, waves, and circular motion |
Essential in rotational dynamics and kinematics |
Difference Between Frequency and Angular Frequency
The difference between Frequency and angular frequency is as follows:
|
Aspects |
Frequency |
Angular Frequency |
|---|---|---|
|
Definition |
Number of cycles per second |
Rate of change of phase angle per second |
|
Symbol |
f |
ω |
|
Units |
Hertz (Hz) |
Radians per second (rad/s) |
|
Physical Meaning |
Represents the number of complete cycles in a given time period |
Indicates the rate of change of the phase angle in a periodic motion |
|
Example |
A tuning fork vibrating at 440 Hz |
Earth rotating at 7.27 x 10-5 rad/s |
Also, Check
Solved Examples on Angular Frequency
Example 1: A tuning fork vibrates at a frequency of 440 Hz. Calculate the angular frequency of the tuning fork.
Solution:
To convert frequency to angular frequency, we use the formula:
𝜔 = 2𝜋𝑓
Where: 𝜔 is the angular frequency in rad/s
f is the frequency in Hz
Plugging in the values:
𝜔 = 2𝜋 × 440 = 2,765 rad s
Therefore, the angular frequency of the tuning fork vibrating at 440 Hz is 2,765 rad/s.
Example 2: A pendulum has a period of 2 seconds. Calculate the angular frequency of the pendulum.
Solution:
To convert period to angular frequency, we use the formula:
𝜔 = 2𝜋/𝑇
Where 𝜔 is the angular frequency in rad/s
T is the period in seconds
Plugging in the values:
𝜔 = 2𝜋/2 = 𝜋 rad s
Therefore, the angular frequency of the pendulum with a period of 2 seconds is π rad/s.
Example 3: A flywheel is rotating at 1,200 revolutions per minute (rpm). Calculate the angular frequency of the flywheel.
Solution:
To convert rotational speed in rpm to angular frequency, we use the formula:
𝜔 = 2𝜋𝑛/60
Where: 𝜔 is the angular frequency in rad/s
n is the rotational speed in rpm
Plugging in the values:
𝜔 = 2𝜋×1,200/60 = 125.6 rad s
Therefore, the angular frequency of the flywheel rotating at 1,200 rpm is 125.6 rad/s.
Practice Questions on Angular Frequency
1. A wave has a frequency of 500 Hz. Calculate the angular frequency of the wave.
2. A pendulum has a period of 1.5 seconds. Determine the angular frequency of the pendulum.
3. A wheel is spinning at 300 revolutions per minute. Find the angular frequency of the wheel.
4. Given a spring constant of 50 N/m and a mass of 2 kg, calculate the angular frequency of the oscillating mass.
FAQs on Angular frequency
What symbol is angular frequency?
The symbol for angular frequency is ω (omega).
What is the angular frequency in SHM?
In a SHM case, angular frequency (ω) means how fast an object moves and alternates around its central point (or the point of equilibrium).
What is the difference between angular frequency and frequency of oscillation?
The angular frequency (ω) is the rate of change of the angle form of a sinusoidal waveform with respect to time, while a given time is measured by the number of complete waves or oscillations referred to as frequency (f).
What is the SI unit of angular frequency in physics?
The SI unit of angular frequency in physics is radians per second (rad/s).
What is the relation between angular frequency and wavelength?
Angular frequency (ω) and wavelength (λ) are connected through the formula v = ωλ, which is widely used in physics to describe wave speed.
What is frequency?
Frequency measures how often something happens in a specific time frame, like how many waves pass a point in a second.
What is the time period?
The time period is the duration it takes for one complete cycle of a repeating event, such as the time for one wave to pass by.
What is formula of angular frequency?
Angular frequency, symbolized by ω, is the rate of change of an angle with respect to time and is calculated as ω = 2πf, where f represents frequency.