Angular frequency calculator
50 Hz mains corresponds to an angular frequency of 314.159265359 rad/s, with a period of 0.02 s. Enter any positive frequency in hertz to get ω in radians per second, the form used in AC circuits and oscillation equations. Free in your browser.
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Worked example
- Frequency
- 50
- Angular frequency
- 314.159265359
- Period
- 0.02
- Frequency unit
- Hz
- Angular frequency unit
- rad/s
The angular frequency calculator takes an ordinary oscillation frequency in hertz — cycles per second — and returns the angular frequency in radians per second using the defining relation ω = 2πf. Angular frequency is the form that actually appears in the equations of harmonic motion, AC circuit analysis, wave propagation and control theory, because differentiating sines and cosines only stays clean when the argument is measured in radians.
Why angular frequency exists at all
Hertz counts complete cycles each second, which is how instruments are calibrated and how specifications are written. But the mathematics of oscillation is written in radians, because a sinusoid is fundamentally a projection of circular motion. One full cycle corresponds to 2π radians of phase, so a system oscillating at f hertz sweeps its phase at 2πf radians per second.
That quantity — the angular frequency, written ω — is what appears inside the sine: x(t) = A·sin(ωt + φ). Using ω instead of f removes a factor of 2π from every derivative, every impedance formula and every resonance condition. The conversion itself is a single multiplication, but getting it right matters: a missing 2π is one of the most common errors in signal processing homework and in production firmware alike.
How the computation works
Given a frequency f, the result is ω = 2πf computed in double precision and rounded to twelve significant decimal places so the output is byte-stable across machines and safe to compare in tests. Validation is deliberately strict. Zero and negative frequencies are rejected: an oscillation at zero hertz does not oscillate, and a negative count of cycles per second has no physical meaning in this context.
Non-numeric input, empty values, NaN and Infinity are rejected for the same reason.
Where it fits in real work
Electrical engineers move between f and ω constantly: inductive reactance is ωL, capacitive reactance is 1/ωC, and filter cutoffs are quoted in radians per second internally even when datasheets say hertz. Physicists need ω for the simple harmonic oscillator, where the equation of motion ẍ = −ω²x only takes that clean form in angular units. Audio and vibration engineers convert machinery RPM-derived frequencies into rad/s before feeding simulation models.
What you can do with it
AC circuit analysis
Convert the 50 Hz or 60 Hz mains frequency to ω = 2πf before computing inductive and capacitive reactances in impedance calculations.
Harmonic oscillator modelling
Feed the natural frequency of a spring-mass system into ω for the equation of motion and for phase-accurate time-domain simulation.
Vibration and audio pipelines
Turn measured tone or machinery frequencies in hertz into radians per second for spectral models and control-system transfer functions.
FAQ
What is the formula?
ω = 2πf, where f is the frequency in hertz and ω is the angular frequency in radians per second.
Why was my input rejected?
The frequency must be a positive, finite number.
How precise is the result?
The multiplication is done in double precision and rounded to twelve significant decimal places, so the output is stable across machines and safe to snapshot in tests.
What is the angular frequency of 50 Hz mains?
It is 314.159265359 rad/s, since one cycle is 2π radians and there are 50 cycles each second. The period that comes with it is 0.02 s, the time for one full cycle.
Why use radians per second instead of hertz?
Because the equations of oscillation, AC circuits and waves are written with angular frequency, and reactance, resonance and the sine inside a waveform all use it. Working in radians per second removes a factor of 2π from every derivative.
Related tools
For developers — API access
Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.
API endpoint
Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.
Call it from your stack
curl -X POST https://api.kit.forhosting.com/phys/angular-frequency \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"frequency":50}'const res = await fetch("https://api.kit.forhosting.com/phys/angular-frequency", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"frequency": 50
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/phys/angular-frequency",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"frequency": 50
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/phys/angular-frequency", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"frequency":50}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"frequency":50}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/phys/angular-frequency", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"frequency": 50
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "phys.angular_frequency",
"status": "queued",
"_links": {
"result": "/tasks/tsk_…/result"
}
}The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.
Pricing
Published price — no tokens, no invented credits. A failed task is never charged.
Errors
| HTTP | Code | Meaning |
|---|---|---|
401 | unauthorized | Missing or invalid API key. |
402 | insufficient_balance | Your balance doesn't cover the task price. |
404 | unknown_type | That task type doesn't exist. |
429 | rate_limited | Too many requests. Use the webhook instead of polling. |
Technical notes
Send a positive frequency, get back ω along with the period T = 1/f, in one deterministic call with no rounding surprises.
Why angular frequency exists at all
This endpoint performs that one multiplication with full floating-point care and rejects any input that is not a positive, finite number of hertz.
How the computation works
Alongside ω you also receive the period T = 1/f in seconds, because the two quantities travel together in almost every practical problem — a motor speed, a tone, a mains supply. The accepted aliases — frequency, f, freq and hz — exist so payloads from different tools map onto the same field without pre-processing, and they all follow the identical validation path.
Where it fits in real work
In all of these pipelines the conversion is trivial but repeated thousands of times, which is exactly where a deterministic API earns its keep: one call, $0.002 per request, no local constants drift, no copy-pasted π with too few digits. The same code that runs on the edge runs free in your browser on this page, so you can verify a value by hand before you wire the endpoint into an automated workflow.
Technical questions
What does it cost?
$0.002 per request. It is also free to run in your browser on this page.
Which field names are accepted?
frequency, f, freq and hz are all accepted as aliases for the same value.