Tangential acceleration calculator
An angular acceleration of 2 rad/s² at a radius of 3 m gives a tangential acceleration of 6 m/s². Enter both values to get the signed result and its magnitude for any point on a rotating body. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angular acceleration
- 2
- Radius
- 3
- Tangential acceleration
- 6
- Magnitude
- 6
Tangential acceleration is the linear acceleration of a point along the tangent of its circular path when a rigid body or particle changes angular speed. The defining closed form is a sub t equals alpha times r: angular acceleration in radians per second squared multiplied by the radial distance from the rotation axis. When alpha is in rad per s squared and radius is in metres, the result is metres per second squared, the same SI unit as ordinary linear acceleration. Students, machine designers, and vehicle-dynamics learners use this identity constantly for homework, flywheel spin-up checks, and first-pass load estimates on rotating arms. This calculator accepts angular acceleration and path radius, returns signed tangential acceleration with magnitude, and rejects a negative radius with a clear validation error instead of inventing a geometric distance.
How tangential acceleration is defined and computed
Begin with a point at fixed distance r from a rotation axis. Its angular position theta may change with time; angular velocity omega is the time derivative of theta, and angular acceleration alpha is the time derivative of omega. The linear speed along the arc is v equals omega times r when r is constant, so differentiating with respect to time yields a tangential component a sub t equals alpha times r.
That component is parallel to the velocity vector and changes how fast the point moves along the path; it is distinct from the centripetal (radial) acceleration v squared over r, which always points toward the centre and changes the direction of velocity even when speed is constant. In uniform circular motion alpha is zero, so tangential acceleration vanishes and only the centripetal term remains.
When a motor spins a shaft up or a brake slows a wheel down, alpha is nonzero and a sub t is the linear acceleration felt along the rim. Because the product preserves the sign of alpha, a negative angular acceleration correctly yields a negative tangential acceleration for spin-down problems. Radius must be non-negative: a geometric path distance cannot be negative, and this tool rejects that case with invalid input rather than silently flipping a sign.
Both fields must be finite; NaN and Infinity are rejected. Zero radius is allowed and yields zero tangential acceleration, which is the correct limit when the point sits on the axis.
Where tangential acceleration calculators show up in real work
Secondary-school and introductory university physics courses assign circular-motion drills, compare a sub t equals alpha r with centripetal a equals v squared over r, and require SI consistency on free-body diagrams of points on rotating rods; a single trusted oracle speeds grading and demo setup. Machine and drivetrain engineers estimate rim linear acceleration from motor angular acceleration and pulley or gear radius before full multi-body simulation. Vehicle dynamics and motorsport teams convert wheel angular acceleration into tangential acceleration at the contact patch for traction and braking sketches.
Amusement-ride and robotics designers check linear accelerations at passenger or end-effector positions on circular arcs when angular rates change. If you later need centripetal acceleration from speed and radius, angular-acceleration unit conversion, total acceleration magnitude from radial and tangential components, or period of uniform circular motion from radius and speed, treat those as separate search intents and landings rather than overloading this tangential acceleration calculator page.
What you can do with it
Drivetrain and flywheel spin-up checks
Turn motor angular acceleration and pulley or rim radius into tangential acceleration for early load and traction estimates.
Ride and robotics path design
Estimate linear acceleration along circular arcs when angular rate changes, before full multi-body simulation.
FAQ
What happens if radius is negative?
A geometric path radius cannot be negative; fix the distance and resubmit.
Can angular acceleration be negative or zero?
Yes. Negative alpha models spin-down and yields negative tangential acceleration. Zero alpha yields zero tangential acceleration (purely uniform angular rate).
Is this the same as centripetal acceleration a = v²/r?
No. Centripetal acceleration is radial and depends on speed; tangential acceleration is along the path and depends on angular acceleration. They are orthogonal components of the full acceleration vector in polar coordinates.
What does it cost?
The browser tool is free and runs locally.
How do I calculate tangential acceleration?
Multiply the angular acceleration by the radius. An angular acceleration of 2 rad/s² at 3 m from the axis gives 6 m/s² along the path, in the same unit as ordinary linear acceleration.
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/tangential-acceleration \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angular_acceleration":2,"radius":3}'const res = await fetch("https://api.kit.forhosting.com/mech/tangential-acceleration", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angular_acceleration": 2,
"radius": 3
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/tangential-acceleration",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angular_acceleration": 2,
"radius": 3
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/tangential-acceleration", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angular_acceleration":2,"radius":3}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angular_acceleration":2,"radius":3}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/tangential-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
{
"angular_acceleration": 2,
"radius": 3
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.tangential_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 tangential 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
Angular acceleration is alpha in radians per second squared or any consistent angular-acceleration unit; zero and negative values are allowed so coasting and braking cases work without a separate flag. Radius is the distance r from the axis in metres or any consistent length unit and must be non-negative. Classic textbook checks include two rad per s squared at three metres for six metres per second squared, zero alpha for zero tangential acceleration, and a failed negative-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. Send two required finite numbers under the fields angular_acceleration and radius. Aliases alpha, alpha_rad_s2, and ang_acc are accepted for the first field, and r, R, and path_radius for the second, so spreadsheets and query strings can reuse the same contract. Missing fields, non-numeric strings, NaN, and Infinity fail with invalid_input and are not charged. When radius is negative the request fails with a message that tangential acceleration a sub t equals alpha times r rejects a negative path radius, so clients can fix geometry rather than guessing. On success the response echoes cleaned inputs, tangential_acceleration as the primary result alpha times r, magnitude as the absolute value for tables that drop sign, a units hint object, method set to tangential_acceleration_alpha_times_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 tangential 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.
Integration use cases
Homework and lab answer keys
Compute a_t = α · r from angular acceleration and radius so students and auto-graders share one closed-form ground truth in m/s².
Technical questions
What formula does this tangential acceleration calculator use?
a_t = α · r, where α is angular acceleration and r is the radial distance from the rotation axis. The response also returns the absolute magnitude |a_t|.