Circular motion velocity calculator
A 2 m radius covered in a 4 s lap gives a speed of π m/s, or one circumference of 4π m every period. Enter the radius and the period to get the speed, the circumference and the angular velocity together. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 2
- Period
- 4
- Velocity
- 3.1415926536 m/s
- Circumference
- 12.5663706144
- Angular velocity
- 1.5707963268
- Precision
- 10
This circular motion velocity calculator finds the tangential speed of an object that moves in a uniform circle when you know the path radius and the period of one full revolution. The closed form is v equals two pi times r over T, which is simply the circumference of the circle divided by the time needed to travel once around it. When radius is given in metres and period in seconds, the result is reported in metres per second; any consistent length and time units work the same way. The engine also returns the circumference and the angular velocity omega equals two pi over T so you can check related uniform-circular-motion identities without a second tool. Everything is fully deterministic arithmetic: no network calls, no sampling, and no date-dependent shortcuts.
What tangential speed means and what you get
In introductory mechanics, an object in uniform circular motion travels at constant speed while its velocity vector keeps turning toward the centre of the path. The tangential speed is that constant magnitude of the linear velocity: it is how many metres of arc the object covers per second along the circle. Because one full lap covers the circumference two pi r, and the time for one lap is the period T, the identity v equals two pi r over T follows immediately.
Angular velocity omega is the rate of change of angle and equals two pi over T for a full turn every T seconds, so the linear and angular pictures are linked by v equals omega times r. This calculator answers the finite, closed-form case used in homework and first-order engineering checks when the radius and period are known and motion is treated as uniform. Zero radius is valid and yields zero speed when the period is non-zero.
Where circular tangential speed fits in study, labs, and software
Students meet tangential speed when a textbook introduces uniform circular motion, when a lab measures the period of a mass on a string or a turntable radius, or when a contest problem gives r and T and asks for v without calculus. Teachers can pre-generate answer keys for modest radius and period values without trusting opaque calculator apps that silently switch units or drop the factor of two pi.
Developers call the same contract from scripts to seed deterministic test vectors for game physics demos of circular tracks, to normalize user-entered radius and period before a visualization, or to satisfy programming-challenge statements that demand v rather than a hand-waved sketch. First-order engineering checks use the mean definition when only a path radius and a stopwatch reading of one lap are known: average tangential speed equals circumference over period for that interval, before drag, drive torque, or non-circular path corrections are modeled.
What you can do with it
Homework and lab checks
Compute tangential speed from measured path radius and revolution period before grading worksheets or comparing to sensor data.
Tutoring and teaching software
Normalize user-entered radius and period into tangential speed for interactive circular-motion lessons and step helpers.
FAQ
What does the circular motion velocity calculator cost?
It is free in your browser on this page.
What happens if the period is zero?
Tangential speed is undefined when the period equals zero because division by zero has no finite value.
Can the radius be zero?
Yes. A zero radius with a non-zero period yields zero tangential speed and zero circumference. Negative radius is rejected.
Does this tool handle non-uniform or elliptical motion?
No. It assumes uniform circular motion with constant radius and constant period. Variable speed along an arc or orbital eccentricity needs a different model.
How is angular velocity related to tangential speed?
Angular velocity is two pi divided by the period, the angle turned per second. Multiply it by the radius and you get the tangential speed, which is why the tool shows both numbers side by side.
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/mech/circular-velocity \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":2,"period":4}'const res = await fetch("https://api.kit.forhosting.com/mech/circular-velocity", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 2,
"period": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/circular-velocity",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 2,
"period": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/circular-velocity", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":2,"period":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":2,"period":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/circular-velocity", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"radius": 2,
"period": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.circular_velocity",
"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.
Limits
max_abs_radius | 1000000000000000 |
max_abs_period | 1000000000000000 |
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
What tangential speed means and what the response contains
The response returns the normalized inputs radius and period, the computed tangential velocity, the circumference, the angular velocity, the precision used, a method tag circular_tangential_velocity, a short formula reminder, and a unit hint m/s for SI inputs. There is no numerical ODE integration of non-uniform paths and no Monte Carlo sampling of orbits: the engine either evaluates the division inside the supported numeric range or rejects the input with invalid_input. Rejected cases include a missing radius or period, non-finite values, magnitudes above the documented ceilings, a negative radius, and any period that equals zero, because division by zero is undefined and there is no finite tangential speed when a revolution takes no time.
How radius and period are read and how v stays reproducible
Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces and non-numeric tokens are rejected. The preferred field for the path radius is radius, but aliases r, R, and path_radius are accepted so spreadsheets, query strings, and typed forms can reuse the same contract. Radius must be finite and non-negative; a negative value fails with invalid_input because a geometric circle radius cannot be negative, and magnitudes above the documented ceiling are also rejected. The preferred field for one revolution is period, with aliases T, t, revolution_period, and time_period. Period must be finite and non-zero: a literal zero fails with invalid_input and a message that names the field and explains that v equals two pi r over T is undefined when the period equals zero, so a bad cell does not become a silent infinity or NaN. An optional precision field, with aliases decimals and dp, chooses how many decimal places to expose for the numeric outputs, defaulting to ten and capped at fifteen so JSON goldens stay stable. Pair the free browser run on this page with the priced API when you need the identical mathematical result inside an automated pipeline without re-implementing the two-pi-r-over-T formula yourself.
Where circular tangential speed fits in study, labs, and software
Product and education tools use the endpoint when a learner types a radius and a period and the UI must show metres per second along with omega and the path length. On this page the work runs entirely in your browser; through the API each successful lookup costs $0.002, failed validation is not charged, and nothing is stored after the response returns. That split — free interactive exploration and a priced, identical API — matches the rest of the KIT tier-A catalog and lets you move from a classroom demo to an automated pipeline without changing the mathematical result. Pair it with average-acceleration or free-fall-time helpers when you need related kinematics, and with other math endpoints when you are building a small lab of free, deterministic physics calculators.
Integration use cases
Deterministic test fixtures
Generate reproducible velocity, circumference, and angular-velocity values for unit tests, game-physics demos, and CI assertions on v = 2πr/T.
Technical questions
What formula is used for tangential speed?
Tangential speed is v = 2πr / T, where r is the path radius and T is the period of one revolution. With SI inputs the unit is m/s. Equivalently v = C/T with circumference C = 2πr.