Formula used
- ω = 2πf (when frequency is known)
- ω = 2π / T (when period is known)
- ω = √(k/m) (ideal spring–mass system)
- ω = √(g/L) (simple pendulum, small-angle approximation)
- ω = 1 / √(LC) (ideal LC oscillator)
The calculator also derives f = ω/(2π) and T = 2π/ω for quick comparison.
How to use this calculator
- Select the method matching your known values.
- Enter inputs and choose appropriate units.
- Press Calculate to view results above the form.
- Use CSV/PDF buttons to save the computed report.
- Change inputs or method and recalculate as needed.
Example data table
| Scenario | Given | Computed ω (rad/s) | Computed f (Hz) | Computed T (s) |
|---|---|---|---|---|
| From frequency | f = 5 Hz | 31.415927 | 5 | 0.2 |
| From period | T = 0.2 s | 31.415927 | 5 | 0.2 |
| Spring–mass | k = 120 N/m, m = 0.8 kg | 12.247449 | 1.949242 | 0.51301993 |
| Pendulum | L = 0.75 m, g = 9.80665 m/s² | 3.6160107 | 0.57550598 | 1.7376014 |
| LC circuit | L = 10 mH, C = 0.1 µF | 31622.777 | 5032.9212 | 0.00019869177 |
Angular Frequency of Oscillation: A Practical Guide
Angular frequency, written as ω, describes how fast an oscillator cycles in radians per second. Unlike ordinary frequency (cycles per second), ω plugs directly into sine and cosine motion equations. One full cycle equals 2π radians.
1) Core relationship between ω, f, and period
The key links are ω = 2πf and ω = 2π/T. If you time repeats with a stopwatch, period T is often the most reliable input. If you have frequency f from a sensor, multiply by 2π. For quick checks, convert back with f = ω/(2π).
2) Simple harmonic motion and phase
For ideal SHM, displacement is x(t) = A sin(ωt + φ). Here ω sets the time scale: doubling ω halves the time needed to reach the same phase angle. Peak acceleration rises with ω², so small changes can be dramatic.
3) Spring–mass systems
For a mass on a spring, ω = √(k/m). Stiffer springs (higher k) increase ω, while heavier masses reduce it. If your result feels off, check that k is in N/m and m is in kg, not grams. In experiments, effective mass can include attached hardware and fixtures.
4) Small‑angle pendulum behavior
A simple pendulum has ω = √(g/L) when the swing angle is small. Longer lengths lower ω and increase period. Measure L from the pivot to the bob’s center of mass and keep angles under about 10°.
5) LC electrical oscillations
In an LC circuit, ω = 1/√(LC). Inductance must be in henries and capacitance in farads. Because microfarads and millihenries are common, unit prefixes matter: a 1000× unit slip changes ω by √1000 ≈ 31.6.
6) Damping and real‑world corrections
Real systems lose energy, so amplitude decays. Light damping shifts the observed rate: ωd = √(ω02 − β2). If drag is visible, record several cycles and average the period to reduce noise. Avoid timing just one swing; reaction time dominates.
7) Interpreting results and sanity checks
Lab oscillators fall between 1 and 100 rad/s. Use the calculator’s extra outputs (frequency and period) to check intuition: a period of 2 s corresponds to ω ≈ 3.14 rad/s. Review method and units before exporting; save CSV or PDF.
FAQs
1) What is the difference between ω and frequency f?
Frequency f counts cycles per second (Hz). Angular frequency ω measures phase change per second in radians. They are related by ω = 2πf, so one cycle corresponds to 2π radians.
2) Can I use degrees instead of radians?
The formulas here use radians. If your data is in degrees, convert first: radians = degrees × π/180. Using degrees directly will make ω incorrect by a factor of 180/π.
3) Why does my pendulum result differ from a real swing?
The simple formula ω = √(g/L) assumes small angles and low damping. Large amplitudes, air resistance, and a flexible string or rod increase the period. Keep angles small and measure length carefully.
4) How accurate is ω = √(k/m) for springs?
It’s accurate for a linear spring with negligible damping. Errors come from spring mass, friction, and nonlinear stiffness at large deflections. If needed, treat part of the spring mass as “effective” mass and re‑measure the period.
5) What units should I use for LC oscillations?
Use L in henries (H) and C in farads (F). Convert mH to H and µF or nF to F before calculating. Prefix mistakes are the most common source of unrealistic ω values.
6) Which input is best: f, T, or physical parameters?
If you can measure period T directly, it usually gives the cleanest ω via 2π/T. Use f when you have a spectrum or tachometer reading. Use physical parameters when you’re designing before building.