Calculating Angular Frequency
You have learned about the frequency velocity of an object and its different motions. We do have an idea of the frequency. Whenever an object rotates along its axis, it makes an angle. For example, take a string and tie a rubber ball at its one end and make the ball swing in a circle on its axis. (just imagine that the string doesn’t exist and the ball is rotating in a circle, making an angle along its imaginary axis) You will notice that it moves at a certain speed completing 360°. This is the situation when we need angular frequency for computation. Let us do a detailed analysis of Angular Frequency.
What is Angular Frequency?
Before understanding angular frequency, let us go through the meaning of time period and angular displacement.
The time period is the time taken by an object to complete one oscillation.
Angular displacement is the shortest angle made by an object from the first position to the second position of stop while rotating in a circle.
Definition of Angular Frequency – It is the angular displacement of an element of a wave per unit of time or in other words, you can say that it is the rate at which the change in rotation takes place or the rate at which change in the sinusoidal waves occurs. For example, if you say that the object has a high angular frequency, it turns very speedily. Angular frequency is the magnitude of the angular velocity. Thus, it is the scalar quantity, i.e. it does not have direction. Angular frequency helps find the rate of rotation of a body in periodic motion.
Different names- Angular speed, radial frequency, circular frequency, orbital frequency, radian frequency and pulsatance.
Derivation of Formula
Imagine that a rubber attached to a string makes a complete circle, i.e. 2π in a time T. We know that angular frequency is the radian of the angle through which the body moves per unit time.
- Thus we can write angular frequency as,
- ω =2π /T, where ω is the angular frequency.
- Also, we know that frequency is 1/T. Thus, angular frequency in terms of frequency (f) can be written as, ω =2πf
- Unit The SI unit of angular frequency in radian per second.
Example
Calculating Angular Frequency
Angular frequency (\( \omega \)) refers to the rate at which an object oscillates or rotates per unit of time. It is a scalar quantity and is directly related to the frequency of oscillation, representing how many radians the object covers per second.
The general approach to calculating angular frequency includes:
- Identifying the period of oscillation \( T \), or the regular frequency \( f \) of the motion.
- Knowing the relationship between the period or frequency and angular frequency.
- Applying the appropriate formula for angular frequency to calculate the result.
Angular Frequency Formula
The general formula for angular frequency is:
\[ \omega = \frac{{2\pi}}{{T}} \quad \text{or} \quad \omega = 2\pi f \]Where:
- \( T \) is the period of the motion (in seconds, s).
- \( f \) is the regular frequency of the oscillation (in Hertz, Hz).
- \( \omega \) is the angular frequency (in radians per second, rad/s).
Example:
If the period of a motion is 4 seconds, the angular frequency is:
- Step 1: Use the formula \( \omega = \frac{{2\pi}}{{T}} \) with \( T = 4 \, \text{s} \).
- Step 2: Calculate \( \omega = \frac{{2\pi}}{{4}} = \pi \, \text{rad/s} \).
Angular Frequency from Frequency
If the frequency of the motion is given, angular frequency can be calculated using the formula \( \omega = 2\pi f \).
Example:
If the frequency of an oscillating object is \( f = 2 \, \text{Hz} \), the angular frequency is:
- Step 1: Use the formula \( \omega = 2\pi f \) with \( f = 2 \, \text{Hz} \).
- Step 2: Calculate \( \omega = 2\pi \times 2 = 4\pi \, \text{rad/s} \).
Real-life Applications of Angular Frequency
Calculating angular frequency has many practical applications, such as:
- Determining the angular frequency of mechanical systems like springs and pendulums.
- Calculating the angular frequency of light waves in physics experiments.
- Measuring the angular frequency in rotating machinery and engines.
Common Units of Angular Frequency
SI Unit: The standard unit of angular frequency is radians per second (\( \text{rad/s} \)).
Angular frequency can also be expressed in other units, such as revolutions per minute (RPM), but radians per second is the standard unit for most calculations.
Common Operations with Angular Frequency
Simple Harmonic Motion (SHM): Angular frequency is used to describe oscillations and vibrations in SHM, such as the motion of a mass on a spring.
Rotational Motion: Angular frequency is key in understanding rotational dynamics, including the behavior of gears, wheels, and rotating objects.
Wave Motion: The angular frequency is used to describe oscillations in waves, such as sound waves, electromagnetic waves, and other periodic waves.
| Problem Type | Description | Steps to Solve | Example |
|---|---|---|---|
| Calculating Angular Frequency from Period | Finding the angular frequency when the period of the motion is given. |
|
If the period of the motion is 4 seconds, the angular frequency is \( \omega = \frac{{2\pi}}{{4}} = \pi \, \text{rad/s} \). |
| Calculating Angular Frequency from Frequency | Finding the angular frequency when the regular frequency is given. |
|
If the frequency is \( f = 2 \, \text{Hz} \), the angular frequency is \( \omega = 2\pi \times 2 = 4\pi \, \text{rad/s} \). |
| Calculating Angular Frequency from Wave Speed | Finding the angular frequency when the wave speed and wavelength are given. |
|
If the wave speed is \( v = 6 \, \text{m/s} \) and the wavelength is \( \lambda = 3 \, \text{m} \), the angular frequency is \( \omega = \frac{{2\pi \times 6}}{{3}} = 4\pi \, \text{rad/s} \). |
| Real-life Applications | Applying angular frequency to solve practical problems in oscillatory motion and waves. |
|
If a pendulum completes one oscillation every 2 seconds, the angular frequency is \( \omega = \frac{{2\pi}}{{2}} = \pi \, \text{rad/s} \). |
