Frequency is a fundamental concept in physics that measures how often a repeating event occurs per unit time. It is measured in Hertz (Hz), where 1 Hz equals one cycle per second. Understanding frequency formulas is essential for students studying waves, oscillations, electromagnetism, and statistics.
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Complete Table of Frequency Formulas
| Type of Frequency | Formula | Variables | Unit | Application |
|---|---|---|---|---|
| Basic Frequency | f = 1/T | f = frequency, T = time period | Hz (Hertz) | General wave motion, oscillations |
| Wave Frequency | f = v/λ | f = frequency, v = wave velocity, λ = wavelength | Hz | Sound waves, light waves, electromagnetic waves |
| Angular Frequency | ω = 2πf or ω = 2π/T | ω = angular frequency, f = frequency, T = time period | rad/s | Circular motion, AC circuits, SHM |
| Resonant Frequency (LC Circuit) | f₀ = 1/(2π√LC) | f₀ = resonant frequency, L = inductance, C = capacitance | Hz | AC circuits, radio tuning, oscillators |
| Resonant Frequency (Mechanical) | f₀ = (1/2π)√(k/m) | k = spring constant, m = mass | Hz | Vibrating systems, springs, pendulums |
| Threshold Frequency | f₀ = W/h or f₀ = φ/h | W or φ = work function, h = Planck’s constant | Hz | Photoelectric effect |
| Frequency from Energy | f = E/h | E = energy, h = Planck’s constant (6.626 × 10⁻³⁴ J·s) | Hz | Quantum mechanics, photon energy |
| Collision Frequency | Z = (n₁n₂σ√(8kT/πμ)) | n = number density, σ = collision cross-section, k = Boltzmann constant, T = temperature, μ = reduced mass | collisions/s | Kinetic theory of gases, chemical reactions |
| Beat Frequency | f_beat = |f₁ – f₂| | f₁, f₂ = frequencies of two waves | Hz | Sound interference, tuning musical instruments |
| Doppler Shifted Frequency | f’ = f(v ± v₀)/(v ∓ vₛ) | f’ = observed frequency, f = source frequency, v = wave velocity, v₀ = observer velocity, vₛ = source velocity | Hz | Moving sources/observers, astronomy, radar |
| Cutoff Frequency (RC Circuit) | f_c = 1/(2πRC) | R = resistance, C = capacitance | Hz | Filters, signal processing |
| Cutoff Frequency (RL Circuit) | f_c = R/(2πL) | R = resistance, L = inductance | Hz | Filters, AC circuits |
| Natural Frequency | f_n = (1/2π)√(g/L) | g = acceleration due to gravity, L = length | Hz | Simple pendulum |
| Frequency in SHM | f = (1/2π)√(a/x) | a = acceleration, x = displacement | Hz | Simple harmonic motion |
| Relative Frequency | Relative Frequency = (Frequency of event)/(Total number of observations) | Dimensionless | – | Statistics, probability |
| Cumulative Frequency | CF = Sum of all previous frequencies including current class | No specific formula (summation process) | – | Statistics, data analysis |
| Frequency Density | Frequency Density = Frequency/Class Width | – | per unit | Histograms with unequal class intervals |
Dimensional Formula of Frequency
Dimensional Formula: [M⁰L⁰T⁻¹]
Derivation:
- Frequency (f) = 1/Time Period (T)
- Dimension of Time = [T]
- Therefore, dimension of frequency = [T⁻¹]
- Complete form: [M⁰L⁰T⁻¹]
This shows that frequency depends only on time and is independent of mass and length.
Detailed Explanation of Key Formulas
1. Basic Frequency Formula
f = 1/T
The most fundamental relationship: frequency is the reciprocal of the time period. If a wave completes one cycle in 0.5 seconds, its frequency is 2 Hz.
2. Angular Frequency Formula
ω = 2πf
Angular frequency measures how fast something rotates or oscillates in radians per second. It’s particularly useful in analyzing AC circuits and rotational motion.
3. Resonant Frequency Formula
f₀ = 1/(2π√LC)
Critical for understanding LC circuits. At resonant frequency, the circuit naturally oscillates with maximum amplitude. Used in radio receivers to tune to specific stations.
4. Threshold Frequency Formula
f₀ = φ/h
In the photoelectric effect, this is the minimum frequency of light required to eject electrons from a metal surface. Below this frequency, no electrons are emitted regardless of light intensity.
5. Collision Frequency Formula
Z = n²σ√(8kT/πμ)
Describes how often gas molecules collide. Essential in chemical kinetics and understanding reaction rates in gases.
6. Cumulative Frequency
Not a mathematical formula but a statistical process where you add frequencies progressively. Used to create cumulative frequency distributions and ogive curves.
Frequently Asked Questions about Frequency Formulas
Q. What is the formula of frequency?
The basic formula of frequency is f = 1/T, where f is frequency and T is the time period. Alternatively, for waves, f = v/λ, where v is wave velocity and λ is wavelength.
Q. What is the difference between frequency and angular frequency?
Frequency (f) measures cycles per second in Hz, while angular frequency (ω) measures radians per second. They’re related by ω = 2πf. Angular frequency is more convenient for circular motion and AC circuit analysis.
Q. How do you calculate resonant frequency?
For an LC circuit, use f₀ = 1/(2π√LC). For a mechanical spring-mass system, use f₀ = (1/2π)√(k/m). Resonant frequency is where the system naturally oscillates with maximum amplitude.
Q. What is the dimensional formula of frequency and how is it derived?
The dimensional formula is [M⁰L⁰T⁻¹] or simply [T⁻¹]. Since frequency = 1/time period, and time has dimension [T], frequency has dimension [T⁻¹].
Q. What is threshold frequency in the photoelectric effect?
Threshold frequency (f₀) is the minimum frequency of incident light required to eject electrons from a metal surface, calculated as f₀ = φ/h, where φ is the work function and h is Planck’s constant.
Q. How is cumulative frequency calculated?
Cumulative frequency is calculated by adding each frequency to the sum of all previous frequencies. For example, if frequencies are 5, 3, 7, then cumulative frequencies are 5, 8, 15.
Q. What is the relationship between frequency and wavelength?
They are inversely proportional: f = v/λ or λ = v/f, where v is wave velocity. Higher frequency means shorter wavelength for a given wave speed.
Q. How do you calculate beat frequency?
Beat frequency is the absolute difference between two frequencies: f_beat = |f₁ – f₂|. This creates the “beating” sound heard when two similar frequencies interfere.
Q. What is collision frequency in kinetic theory?
Collision frequency measures how often molecules collide per unit time, calculated using Z = n²σ√(8kT/πμ). It increases with temperature and molecular concentration.
Q. How is frequency related to energy in quantum mechanics?
Einstein’s relation states E = hf, where E is photon energy, h is Planck’s constant, and f is frequency. Rearranging gives f = E/h. Higher frequency means higher energy.
Practical Applications
Physics & Engineering:
- Wave analysis (sound, light, radio)
- Circuit design (filters, oscillators)
- Resonance phenomena
Chemistry:
- Spectroscopy
- Reaction kinetics
- Molecular collisions
Statistics:
- Data analysis
- Probability distributions
- Frequency tables and histograms
Important Constants
- Planck’s constant (h): 6.626 × 10⁻³⁴ J·s
- Boltzmann constant (k): 1.381 × 10⁻²³ J/K
- Speed of light (c): 3 × 10⁸ m/s
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