Centripetal acceleration calculator
An object moving at 10 m/s on a 5 m radius needs 20 m/s² of inward acceleration to stay on its circle. Enter the speed and the radius; the tool squares the speed, divides by the radius and shows both steps. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Speed
- 10
- Radius
- 5
- V squared
- 100
- Centripetal acceleration
- 20
- Magnitude
- 20
Centripetal acceleration is the inward radial acceleration that keeps a body on a circular path when it moves with tangential speed v at distance r from the centre. The defining closed form is a equals v squared over r. When speed is in metres per second and radius in metres, the result is metres per second squared, the same SI unit as linear acceleration, even though the velocity vector is continuously changing direction rather than magnitude in uniform circular motion. Students, vehicle dynamics engineers, amusement-ride designers, and orbital-mechanics learners use this one-line formula constantly for homework checks, banked-curve prep, and first-pass load estimates. This calculator accepts tangential speed and path radius, returns centripetal acceleration with the intermediate v squared term, reports magnitude, and rejects zero radius with a clear validation error instead of producing infinity.
How centripetal acceleration is defined and computed
Begin with motion constrained to a circle of radius r. At every instant the velocity is tangent to the path; to stay on the circle the acceleration must have a radial component directed toward the centre. For speed v the magnitude of that component is v squared divided by r, which follows from differentiating the rotating velocity vector or from the limit of the change of velocity over a short arc.
In uniform circular motion the speed is constant, so the only acceleration is centripetal; if speed also changes, a tangential component appears alongside the same radial term a equals v squared over r evaluated at the current speed. Because speed is squared, reversing travel direction does not flip the magnitude of a: the inward requirement depends on how fast the body is moving, not on the sense of circulation. Radius must be nonzero; a vanishing centre distance is not a circular path and would make the quotient undefined.
Both fields must be finite; NaN and Infinity are rejected. The special case of zero or vanishingly small radius is rejected with invalid input because division by zero is undefined and would not represent a physical circular constraint.
What you send and what you get back
Send two required finite numbers under the fields speed and radius. Speed is the tangential speed v in metres per second or any consistent speed unit; zero and negative values are allowed because they are squared. Radius is the path radius r in metres or any consistent length unit and must be nonzero.
Classic textbook checks include ten metres per second on a five-metre radius for twenty metres per second squared, zero speed for zero acceleration, and a failed zero-radius case. Wire the free browser form for homework and quick sanity checks, then call the identical contract from scripts when you need bulk answer keys or design tables.
Where centripetal acceleration calculators show up in real work
Secondary-school and introductory university physics courses assign circular-motion drills, compare a equals v squared over r with a equals omega squared r, and require SI consistency on free-body diagrams; a single trusted oracle speeds grading and demo setup. Vehicle dynamics and motorsport teams estimate cornering accelerations from speed and turn radius before full tyre models. Amusement-ride and rotorcraft designers check load factors on circular arcs from peak tangential speed and path geometry.
Spacecraft and satellite education problems convert orbital speeds and radii into centripetal requirements as a first filter before full inverse-square gravity balance. If you later need centripetal force m a, angular-speed form a equals omega squared r, banked-curve normal and friction free bodies, or orbital period from Kepler laws, treat those as separate search intents and landings rather than overloading this centripetal acceleration calculator page.
What you can do with it
Homework and lab answer keys
Compute a = v²/r from tangential speed and path radius so students and auto-graders share one closed-form ground truth in m/s².
Vehicle cornering estimates
Turn road-speed and curve-radius pairs into centripetal acceleration for early handling and load-factor checks.
Ride and circular-arc design
Estimate inward acceleration on circular path segments from peak speed and geometry before full multi-body simulation.
FAQ
What formula does this centripetal acceleration calculator use?
a = v² / r, where v is tangential speed and r is the radius of the circular path. The response also returns the intermediate v² value.
What happens if radius is zero?
Centripetal acceleration divides by radius, so zero radius is undefined (division by zero).
Can speed be negative or zero?
Yes. Speed is squared, so the sign does not change the magnitude of a. Zero speed yields zero centripetal acceleration.
Is this the same as a = ω² r?
They are equivalent when v = ω r, so a = v²/r = ω² r. This tool takes linear speed v and radius r directly; it does not accept angular speed as an input field.
What does it cost?
The browser tool is free and runs locally.
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/centripetal-acceleration \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"speed":10,"radius":5}'const res = await fetch("https://api.kit.forhosting.com/mech/centripetal-acceleration", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"speed": 10,
"radius": 5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/centripetal-acceleration",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"speed": 10,
"radius": 5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/centripetal-acceleration", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"speed":10,"radius":5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"speed":10,"radius":5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/centripetal-acceleration", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"speed": 10,
"radius": 5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.centripetal_acceleration",
"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
How centripetal acceleration is defined and computed
The entire pipeline is deterministic floating-point arithmetic: no random seeds, no wall-clock timestamps, no network calls, and no model drift, so identical inputs always produce identical JSON.
What you send and what you get back
Missing fields, non-numeric strings, NaN, and Infinity fail with invalid_input and are not charged. When radius is zero or extremely close to zero the request fails with a message that centripetal acceleration is v squared over r and zero radius is undefined, so clients can fix geometry rather than guessing. On success the response echoes cleaned inputs, v_squared as the intermediate product, centripetal_acceleration as the primary result a equals v squared over r, magnitude as the absolute value for tables that drop sign, a units hint object, method set to centripetal_acceleration_v_squared_over_r, and a short formula string for display. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where centripetal acceleration calculators show up in real work
Because this capability is a pure parse engine, API pricing is the published base of $0.002 per successful item and failed validation does not consume balance.