Frequency Formula
- Introduction
- Understanding the Concept of Waves
- How to Find Frequency Using Formula
- Applications of Frequency
- Solved Examples
- Practice Problems
- Frequently Asked Questions
Introduction
Frequency is how many times something happens in a certain amount of time. It's like counting how many times your dog barks in a minute. Like if we know the speed of a wave, we can find its frequency. Imagine a wave like a roller coaster; how fast it goes up and down tells us its frequency.
Another important thing is period. It's just the time it takes for one cycle to complete. So, if your dog barks once every `5` seconds, the period between barks is `5` seconds. It's like the rhythm of the barking.
This stuff is super important because waves are everywhere! Light is a wave, and so is the sound of your fan. Waves are basically just vibrations moving through stuff, carrying energy. And when we talk about how often waves pass by in a second, that's the frequency, measured in Hertz.
Understanding the Concept of Waves
To understand more about frequency and how the frequency formula works, it's important to learn some basic words and ideas about waves.
`1`. Wave: A wave is like a moving disturbance that carries energy through space. It doesn't carry stuff, just energy. Examples include sound, light, and radio waves.
`2`. Wave Cycle: When we talk about one cycle of a wave, we mean the whole wave from one point to the same point in the next wave.
`3`. Wavelength: Wavelength is the distance between two points in a wave that are the same, like from one peak to the next. We measure it in meters.
`4`. Wave Velocity: Wave velocity, or how fast a wave travels through something, is measured in meters per second.
`5`. Time Period: Time period means how long it takes for one complete wave cycle to happen. We measure it in seconds.
`6`. Angular Frequency: Angular frequency, written as the Greek letter omega `(ω)`, shows how fast a wave repeats itself. It helps relate frequency and the time period of a wave.
How to Find Frequency Using Formula
Frequency is a term used to describe how often something happens over a set period of time. When we talk about frequency in the context of waves, we're looking at how many complete cycles of the wave occur in one second. This is measured in Hertz `(Hz)`. For example, if a wave completes `10` cycles in one second, its frequency is `10 Hz`.
We can calculate frequency using specific formulas that take into account other related quantities like time period, wave speed, and wavelength. These formulas help us understand the behavior of waves and how they propagate through space.
Formula `1`: Frequency formula based on time period `(T)`:
`f = \frac{1}{T}`
This formula calculates frequency `(f)` by taking the reciprocal of the time period `(T)`, where `f` is measured in Hertz `(Hz)` and `T` is measured in seconds `(s)`.
Formula `2`: Frequency formula based on wave speed `(v)` and wavelength `(\lambda)`:
`f = \frac{v}{\lambda}`
This formula determines frequency `(f)` by dividing the wave speed `(v)` by the wavelength `( \lambda )`, where `f` is measured in Hertz `(Hz)`, `v` is measured in meters per second `(m/s)`, and `\lambda` is measured in meters `(m)`.
Formula `3`: Frequency formula based on angular frequency `(\omega)`:
`f = \frac{\omega}{2\pi}`
This formula computes frequency `(f)` by dividing the angular frequency `(ω)` by `2\pi`, where angular frequency represents the rate of change of the phase of the wave with respect to time.
Applications of Frequency
Frequency has numerous practical applications across various fields. Here are few examples:
`1`. Communication Systems: Frequency is fundamental in communication systems, such as radio, television, and cellular networks. Different frequencies are allocated for different purposes, allowing for the transmission of signals over long distances without interference. For instance, in radio broadcasting, each station is assigned a specific frequency on which to broadcast its signal.
`2`. Medical Imaging: In medical imaging techniques like ultrasound and MRI (Magnetic Resonance Imaging), frequency is utilized to generate images of the body's internal structures. Ultrasound machines, for example, emit high-frequency sound waves into the body and then capture the echoes to create images of organs, tissues, and blood flow.
`3`. Music and Audio Engineering: In music and audio engineering, frequency is crucial for producing and reproducing sound. Understanding frequency helps in tasks like equalization, where specific frequency ranges can be adjusted to enhance or attenuate certain aspects of audio signals. This is important in mixing music tracks, designing sound systems, and optimizing acoustics in concert halls or recording studios.
`4`. Wireless Technology: Wireless technologies such as Wi-Fi, Bluetooth, and NFC (Near Field Communication) rely on frequency to transmit data wirelessly between devices. Each technology operates within specific frequency bands allocated by regulatory authorities to avoid interference and ensure efficient communication.
`5`. Environmental Monitoring: Frequency analysis is used in environmental monitoring to study natural phenomena like earthquakes, tsunamis, and weather patterns. Seismographs, for example, measure the frequency of seismic waves generated by earthquakes to determine their magnitude and location. Similarly, weather radar systems analyze the frequency of radio waves reflected from precipitation to track storms and monitor rainfall patterns.
`6`. Industrial Applications: Frequency is employed in various industrial applications, including manufacturing, robotics, and automation. For instance, in robotics, servo motors use pulse-width modulation (PWM) to control rotation speed and position, which is determined by the frequency of the PWM signal. In manufacturing processes, frequency control is vital for tasks like welding, cutting, and material handling, where precise control of machinery is essential for efficiency and quality control.
Solved Examples
Example `1`: Find the frequency of a lightwave when the wavelength of the light is `2` meters.
Solution:
Given: Wavelength `(\lambda)` `= 2` meters (m)
To find: Frequency `(f)`
Using Frequency formula:
`f = \frac{v}{\lambda}`
where `v` is the speed of light (approximately ` 3 \times 10^8 ` meters per second `(m/s)`)
`f = \frac{3 \times 10^8 \ \text{m/s}}{2 \ \text{m}}`
`f = 1.5 \times 10^8 \ \text{Hz}`
Frequency is \(1.5 \times 10^8 \, \text{Hz} \).
Example `2`: Determine the frequency of a pendulum that takes ` 10 ` seconds to complete one cycle.
Solution:
Given: Time `= 10` s
Using Frequency formula:
`f = \frac{1}{T}`
`f = \frac{1}{10}`
`f = 0.1`
Frequency is \(0.1 \, \text{Hz} \).
Example `3`: Calculate the angular velocity of the wave given that the frequency is `50` Hertz.
Solution:
Given: Frequency `(f) =` \(50 \, \text{Hz} \)
To find: Angular Frequency `(\omega)`
Using Frequency formula:
`f = \frac{\omega}{2 \pi}`
`\omega = 2\pi \times f`
`\omega = 2\pi \times 50`
`\omega = 100\pi`
Angular velocity is \(100\pi\).
Example `4`: A sound wave has a frequency of `40` Hertz. Calculate its time period.
Solution:
Given: \(f = 40 \, \text{Hz} \)
To find: Time period `(T)`
Using Frequency formula:
`f = \frac{1}{T}`
`T = \frac{1}{f}`
`T = \frac{1}{40}`
`T = 0.025`
Time period is `0.025` second.
Example `5`: Find the frequency of a lightwave when the wavelength of the light is `200` nm.
Solution:
Given: Wavelength `(\lambda) =` \(200\, \text{nm}\) `=` \(200 \times 10^{-9} \, \text{m} \)
`=` \(2 \times 10^{-7} \, \text{m} \)
To find: Frequency `(f)`
Using Frequency formula:
`f = \frac{v}{\lambda}`
where, ` v ` `=` speed of light `=` ` 3 \times 10^8 \ \text{m/s} `
`f = \frac{3 \times 10^8 \ \text{m/s}}{2 \times 10^{-7} \ \text{m}}`
`f = 15 \times 10^{14} \ \text{sec}^{-1}`
Frequency is \(15 \times 10^{14} \, \text{Hz} \).
Practice Problems
Q`1`: A light wave has a wavelength of `0.2` meters `(m)`. What is its frequency?
- \(1.5 \, \text{Hz} \)
- \(0.5 \times 10^8 \, \text{Hz} \)
- \(15 \times 10^8 \, \text{Hz} \)
- \(20 \times 10^8 \, \text{Hz} \)
Answer: c
Q`2`: If the frequency of a wave is `100` Hz, what is its time period?
- ` 0.01 ` ` s `
- ` 0.02 ` ` s `
- ` 0.03 ` ` s `
- ` 0.04 ` ` s `
Answer: a
Q`3`: Find the frequency of a pendulum that takes `20` seconds to complete one cycle.
- \(0.05 \, \text{Hz} \)
- \(0.1 \, \text{Hz} \)
- \(0.005 \, \text{Hz} \)
- \(0.2 \, \text{Hz} \)
Answer: a
Q`4`: Calculate the frequency of the lightwave when the wavelength of the light is `600` nm.
- \(6 \times 10^{14} \, \text{Hz} \)
- \(5 \times 10^{14} \, \text{Hz} \)
- \(15 \times 10^{14} \, \text{Hz} \)
- \(10 \times 10^{14} \, \text{Hz} \)
Answer: b
Q`5`: Calculate the angular velocity of the wave given that the frequency is `65` Hertz.
- \(100\pi\)
- \(120\pi\)
- \(130\pi\)
- \(90\pi\)
Answer: c
Frequently Asked Questions
Q`1`. What is frequency?
Answer: Frequency is the number of occurrences of a repeating event per unit of time. In the context of waves, it represents how many cycles of the wave occur in one second and is typically measured in Hertz `(Hz)`.
Q`2`. How to calculate frequency?
Answer: Frequency `(f)` can be calculated using the formula `f = \frac{1}{T}`, where `T` is the time period of the wave. Alternatively, if you know the wavelength `(\lambda)` and the wave velocity `(v)`, you can use the formula `f = \frac{v}{\lambda}`.
Q`3`. What is the relationship between frequency and wavelength?
Answer: Frequency and wavelength are inversely proportional to each other. This means that as the frequency of a wave increases, its wavelength decreases, and vice versa. Mathematically, this relationship is described by the formula `f = \frac{v}{\lambda}`, where `v` is the velocity of the wave.
Q`4`. What is the time period of a wave?
Answer: The time period `(T)` of a wave is the time it takes for one complete cycle of the wave to occur. It is the reciprocal of frequency `(f)`, so `T = \frac{1}{f}`.
Q`5`. How does frequency affect sound perception?
Answer: In sound waves, frequency determines the pitch of the sound. Higher frequencies correspond to higher pitches, while lower frequencies correspond to lower pitches. For example, a high-frequency sound wave produces a high-pitched sound, like a whistle, while a low-frequency sound wave produces a low-pitched sound, like a drumbeat.