Period of circular motion calculator
Divide the circumference by the speed and you have the period: an object on a 2 m circle moving at π m/s completes one lap every 4 seconds, 0.25 laps per second. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Radius
- 2
- Speed
- 3.1415926536
- Speed
- 3.1415926536 s
- Period
- 4
- Frequency
- 0.25
- Circumference
- 12.5663706144
This period of circular motion calculator finds how long one full revolution takes when an object moves at constant tangential speed on a circular path of known radius. The closed form is T equals two pi times r over v, which is simply the circumference of the circle divided by the speed along the arc. When radius is given in metres and speed in metres per second, the result is reported in seconds; any consistent length and speed units work the same way. The engine also returns frequency, circumference, and angular velocity 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 the period means and what you get back
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 period T is the time for one complete revolution: how many seconds elapse between successive returns to the same point on the circle with the same orientation of the velocity. Because one full lap covers the circumference two pi r, and the object covers that distance at tangential speed v, the identity T equals two pi r over v follows immediately.
Frequency f is the reciprocal of the period and counts revolutions per second; angular velocity omega equals two pi over T, or equivalently v over r when the radius is positive, so the linear and angular pictures stay linked. This calculator answers the finite, closed-form case used in homework and first-order engineering checks when the radius and tangential speed are known and motion is treated as uniform. Zero radius is valid and yields zero period and zero circumference when the speed is non-zero.
Where circular-motion period fits in study, labs, and software
Students meet the period of circular motion when a textbook introduces uniform circular motion, when a lab measures the radius of a turntable or string path and a measured linear speed, or when a contest problem gives r and v and asks for T without calculus. Teachers can pre-generate answer keys for modest radius and speed 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 speed before a visualization, or to satisfy programming-challenge statements that demand the revolution time rather than a hand-waved sketch. First-order engineering checks use the mean definition when only a path radius and a speedometer or sensor reading of tangential speed are known: period equals circumference over speed for that steady interval, before drag, drive torque, or non-circular path corrections are modeled.
What you can do with it
Homework and lab checks
Compute revolution period from measured path radius and tangential speed before grading worksheets or comparing to stopwatch data.
Tutoring and teaching software
Normalize user-entered radius and speed into period for interactive circular-motion lessons and step helpers.
FAQ
What does the period of circular motion calculator cost?
It is free in your browser on this page.
What happens if the speed is zero?
Period is undefined when the tangential speed equals zero because division by zero has no finite value.
Can the radius be zero?
Yes. A zero radius with a non-zero speed yields zero period 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 tangential speed. Variable speed along an arc or orbital eccentricity needs a different model.
What formula does it use?
It divides the circumference of the circle, two pi times the radius, by the speed along it. A circle of 2 m covered at π m/s gives one lap every 4 seconds, which is a frequency of 0.25 Hz.
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/period-circular-motion \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"radius":2,"speed":3.141592653589793}'const res = await fetch("https://api.kit.forhosting.com/mech/period-circular-motion", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"radius": 2,
"speed": 3.141592653589793
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/period-circular-motion",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"radius": 2,
"speed": 3.141592653589793
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/period-circular-motion", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"radius":2,"speed":3.141592653589793}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"radius":2,"speed":3.141592653589793}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/period-circular-motion", 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,
"speed": 3.141592653589793
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.period_circular_motion",
"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_speed | 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 the period means and what the response contains
The response returns the normalized inputs radius and speed, the computed period, the frequency, the circumference, the angular velocity when radius is positive, the precision used, a method tag circular_motion_period, a short formula reminder, and a unit hint 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 speed, non-finite values, magnitudes above the documented ceilings, a negative radius, and any speed that equals zero, because division by zero is undefined and there is no finite period when the object does not move along the path.
How radius and speed are read and how T 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 tangential speed is speed, with aliases v, velocity, tangential_speed, and linear_speed. Speed must be finite and non-zero: a literal zero fails with invalid_input and a message that names the field and explains that T equals two pi r over v is undefined when the tangential speed 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-v formula yourself.
Where circular-motion period fits in study, labs, and software
Product and education tools use the endpoint when a learner types a radius and a speed and the UI must show seconds along with frequency, 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 the circular-velocity calculator when you already know period and need speed instead, 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 period, frequency, and angular-velocity values for unit tests, game-physics demos, and CI assertions on T = 2πr/v.
Technical questions
What formula is used for the period?
Period is T = 2πr / v, where r is the path radius and v is the tangential speed. With SI inputs the unit is seconds. Equivalently T = C/v with circumference C = 2πr.