Linear Frequency: 5 Important Facts You Should Know


The periodic motion is the repetitive and continuous movement of an object in a regular interval. The simple harmonic motion is a kind of periodic motion.

The linear frequency is the number of vibrations, repetitions, or oscillations completed by an object or body in a unit second. The reciprocal of the time period gives frequency. The word linear frequency is used to differentiate it from spatial frequency and angular frequency. 

The time that an object takes to cover a complete cycle is the time period. For example: if a tuning fork vibrates 5 times in a second, then the linear frequency would be 5 Hz. Frequency plays a vital role in physics to tell about the rate of oscillation and vibrations like audio signals, radio waves, and light.

Linear frequency equation 

The linear frequency is conventionally the time taken to complete one oscillation or vibration. The time period specifies the time consumed by an object to complete one cycle. The frequency and period are interrelated as:

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

For the wave equation the linear frequency is calculated from the equation:

[latex]f = \frac{c}{\lambda }[/latex]

Here,

c equals the speed of the wave

λ is the wavelength. 

The linear frequency is calculated from the relation:

ω = 2πf

Linear frequency units

For oscillations, waves, and simple harmonic motion, frequency is the number of vibrations in one second. The unit of frequency is named after a famous physicist Heinrich Hertz. Its S.I. unit is Hertz (Hz). Before that, the unit for the frequency used to be cps, that is, cycles per second. Since the unit for the period is second for all the systems, for rotating and circulatory devices, frequency is termed as revolutions per minute, abbreviated as rpm. Here 60 rpm equals one hertz. For linear frequency the other unit is:

[latex]s^{-1}[/latex]

Linear frequency symbol 

The frequency tells about the vibrations or cycles completed in unit time. It is vital to specify the nature and character of many important physical concepts like oscillation, periodic motion, waves, light, and changing current and voltage. 

The standard symbol that is used to represent the linear frequency is f. Generally for oscillation, and SHM f is used. But for light and waves, there is another symbol used to denote frequency. It is the Greek symbol ν.  

Angular Frequency vs Linear Frequency 

Linear Frequency vs Angular frequency

The angular frequency of the oscillation and its linear frequency are two different concepts. For an oscillatory object, the angular frequency tells the phase change that is angular displacement. At the same time, the frequency tells about the oscillation completed in unit time.

It gives the relation between the frequency and angular frequency of the oscillation. 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.

For example, a ball is oscillating and completing 5 revolutions in 1 second. Then the frequency would be 5 Hertz, and the angular frequency would be 10π rad/s. 

Angular FrequencyLinear Frequency
The change in the angle of the oscillating body or simply angular displacement is known as angular frequency. The oscillations or vibrations completed in one second is the frequency of the object or particle. 
Angular frequency is represented by the symbol ω. The linear frequency is represented by ν. 
Its standard unit is radian per second (rad/s). The SI unit of frequency is Hertz (Hz).

The formula for angular frequency is ω =2πf or [latex]\omega = \frac{2 \pi}{T}[/latex]
The equation for the linear frequency is:
[latex]f = \frac{1}{T}[/latex]
Difference between angular frequency and linear frequency

How To Find Linear Frequency 

To find the linear frequency of any vibratory object, or oscillatory body the formulas that are used are: 

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

f is frequency and T is period

Or

[latex]f = \frac{c}{\lambda }[/latex]

c is the speed of the wave, and λ is the wavelength. 

Image Credit: Stündle (modification Ideophagous), Pendulum-no-textCC BY-SA 4.0

For example, a pendulum is oscillating with a period of 0.5 seconds. Then the frequency of the pendulum would be:

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

[latex]f = \frac{1}{0.5}[/latex]

[latex]f = \frac{10}{5}[/latex]

[latex]f = 2 Hz[/latex]

In the second case, if the wave speed is given as 320 m/s, and the respective wavelength is given as 8 m then the frequency would be: 

[latex]f = \frac{c}{\lambda }[/latex]

[latex]f = \frac{320}{8 }[/latex]

[latex]f = 4 Hz[/latex]

While calculating the frequency the important thing is to keep all the quantities in their standard units like second for Time, for length it should be a metre. 

Now suppose a ball makes 360 cycles in one minute then frequency would be:

[latex]f = \frac{360}{1 min}[/latex]

[latex]f = \frac{360}{60 s}[/latex]

[latex]f = 6 Hz[/latex]

Frequently Asked Questions (FAQs)

Explain frequency in simple terms. 

In physics, linear frequency is defined as the number of oscillations or vibrations completed in one second. 

The frequency in the simplest concept is the number of occurrences of any phenomenon. Suppose a ball bounces 8 times, then the frequency of the bouncing ball is 8 Hz. 

Is angular frequency the same as linear frequency? 

No, angular frequency and linear frequency are two different physics concepts. 

The angular frequency tells about the object’s angular displacement or phase change. At the same time, linear frequency tells about the number of oscillations or vibrations that an object covers in unit time.

How are the linear frequency and angular frequency related?

The angular frequency and linear frequency are related by the formula:

ω = 2πf

ω is the angular frequency 

f is a linear frequency 

Is the symbol for frequency ν or f? 

Both ν and f are used to represent the frequency. For the wave system, ν is used, whereas, for other oscillatory bodies like pendulum and spring, the symbol f is used. 

What is the general rule of frequency? 

Frequency gives the idea of cycles completed in one second by an oscillatory body. 

The general formula or rule for determining the value of frequency is given as:

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

T represents the period. It tells about the time taken to complete one complete oscillation. The unit for frequency is Hertz (Hz).

How to calculate frequency?

To calculate the frequency, the formula used is f = 1/T, and f = c/λ

Firstly to calculate the frequency, all the values of time or wavelength and velocity should be in their standard units after conversion substitutes the values in their respective equations and does the simplifications. 

For example, a body completes one oscillation in ½ minutes. Then the first step would be to convert the period into seconds, that is:

[latex]T = \frac{1}{2} min[/latex]

[latex]T = \frac{1}{2} \times 60 seconds[/latex]

[latex]T = 30 seconds[/latex]

Now substituting the value of T in the formula we get:

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

[latex]f = \frac{1}{30}[/latex]

[latex]f = 0.03 Hz[/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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