Calculator Input Panel
Choose a method, enter values, and compute angular frequency instantly.
Plotly Graph
The graph updates after each successful calculation and shows how angular frequency changes for the selected model.
Recent Calculation History
| Time | Method | Inputs | ω (rad/s) | f (Hz) | T (s) |
|---|---|---|---|---|---|
| No saved calculations yet. Run the calculator to populate this table. | |||||
Example Data Table
| Case | Input Values | Formula | Angular Frequency |
|---|---|---|---|
| Period Example | T = 0.50 s | ω = 2π/T | 12.566371 rad/s |
| Frequency Example | f = 3 Hz | ω = 2πf | 18.849556 rad/s |
| Mass-Spring Example | k = 200 N/m, m = 2 kg | ω = √(k/m) | 10.000000 rad/s |
| Pendulum Example | L = 1 m, g = 9.80665 m/s² | ω = √(g/L) | 3.131557 rad/s |
| LC Circuit Example | L = 20 mH, C = 100 μF | ω = 1/√(LC) | 707.106781 rad/s |
| Torsion Example | κ = 4 N·m/rad, I = 0.25 kg·m² | ω = √(κ/I) | 4.000000 rad/s |
Formula Used
From Period
ω = 2π / T
Use this when the oscillation period is known directly.
From Frequency
ω = 2πf
Use this when frequency is measured in cycles per second.
Mass-Spring System
ω = √(k / m)
Natural angular frequency depends on stiffness and attached mass.
Simple Pendulum
ω = √(g / L)
Valid for small oscillation angles where motion remains approximately harmonic.
LC Circuit
ω = 1 / √(LC)
This gives the resonant angular frequency of an ideal LC oscillator.
Torsional Oscillator
ω = √(κ / I)
Angular frequency increases with torsion strength and decreases with inertia.
Symbols: ω = angular frequency, T = period, f = frequency, k = spring stiffness, m = mass, g = gravitational acceleration, L = pendulum length, C = capacitance, κ = torsion constant, I = moment of inertia.
Output note: The calculator also reports equivalent frequency, period, degrees per second, and revolutions per second.
How to Use This Calculator
- Select the oscillation model that matches your problem.
- Pick the unit system for each entered quantity.
- Choose output unit, decimal places, graph span, and graph points.
- Enter the required values for the selected method only.
- Click the calculate button to show the result above the form.
- Review the graph, history table, and converted outputs.
- Use the CSV or PDF buttons to export your work.
Frequently Asked Questions
1) What is angular frequency?
Angular frequency measures how fast an oscillator moves through its cycle in radians per second. It is commonly written as ω and links directly to period and frequency.
2) How is angular frequency related to frequency?
The relationship is ω = 2πf. Multiply frequency in hertz by 2π to convert cycles per second into radians per second.
3) Why does a larger mass lower spring angular frequency?
In a mass-spring system, ω = √(k/m). When mass increases and stiffness stays fixed, the ratio k/m becomes smaller, so oscillation slows down.
4) When is the pendulum formula valid?
The simple pendulum formula works well for small angles. At larger amplitudes, the period increases slightly, and the small-angle approximation becomes less accurate.
5) What units should I use for angular frequency?
The standard unit is radians per second. This calculator also shows degrees per second and revolutions per second for easier interpretation.
6) Can I use this calculator for electrical oscillations?
Yes. The LC circuit mode calculates resonant angular frequency for ideal inductance and capacitance values using ω = 1/√(LC).
7) What is the difference between period and angular frequency?
Period is the time for one full cycle. Angular frequency tells how quickly phase changes. They are inversely related through ω = 2π/T.
8) Why include a graph with the result?
The graph helps you see sensitivity. It shows how angular frequency changes when one driving variable varies around your entered value.