Rolling down incline calculator
A solid cylinder, with inertia factor 0.5, rolling down a 30° ramp speeds up more slowly than a block sliding without friction, because part of the energy goes into spinning. Enter the angle and the inertia factor to get its acceleration. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Angle
- 30
- Inertia factor
- 0.5
- G
- 9.80665
- Angle rad
- 0.5235987756
- Acceleration
- 3.2688833333
- Assumption
- pure rolling without slipping
This rolling down incline calculator finds the linear acceleration of a rigid body that rolls without slipping. Enter the incline angle and the dimensionless inertia factor, defined as the moment of inertia divided by mass times radius squared. The result shows how rotational inertia reduces acceleration below the sliding value. Standard gravity is supplied automatically, or you can provide another positive gravitational acceleration for a different environment or modeling convention.
Describe the incline and the rolling body
Start with the angle between the incline and the horizontal, measured in degrees. A level surface is zero degrees, while a vertical limit is ninety degrees. The calculator deliberately rejects any angle above ninety degrees because that geometry is no longer the supported down-an-incline model.
Next enter the inertia factor, often written as k = I/(m r²). This dimensionless value captures how the body’s mass is distributed about its rotation axis without requiring separate mass and radius inputs. Use the factor for the actual rotation axis.
You may also override standard gravity when modeling another planet or using a specified laboratory value. Gravity must be positive. The calculation assumes a rigid body, a fixed slope, pure rolling without slipping, and enough static friction to maintain that rolling condition.
Understand the acceleration formula
The calculator evaluates a = g sin(θ)/(1 + k). The component of gravity directed down the slope is g sin(θ), but a rolling body must gain both translational kinetic energy and rotational kinetic energy. The denominator 1 + k represents that rotational demand.
A larger inertia factor means more of the gravitational energy goes into rotation, so the center of mass accelerates more slowly. When k is zero, the expression reduces to the familiar frictionless sliding acceleration. That limiting case is mathematically useful even though an ordinary rolling body has positive rotational inertia.
Mass and radius cancel after inertia is expressed through k, which is why neither appears as a separate input. The output includes the angle in radians, the gravity value used, and acceleration in meters per second squared.
Apply the result within the model limits
Use the acceleration as the constant center-of-mass acceleration for a uniform straight incline when the stated assumptions hold. From rest, it can feed standard kinematics such as speed v = at and distance s = at²/2. It is also useful for comparing shapes on the same ramp: with equal angle and gravity, the object with the smaller inertia factor has the greater linear acceleration.
Do not treat this result as proof that rolling actually occurs. The required static friction depends on the body and surface, and a low-friction contact may slip, producing different translation and rotation. Rolling resistance, deformation, aerodynamic drag, changing slope angle, axle friction, and externally applied torque are also outside this closed-form model.
For experimental work, use measured angle and a body-specific inertia factor, then compare the prediction with observations and uncertainty.
What you can do with it
Compare ideal rolling shapes
Compare a hoop, disk, cylinder, or sphere on the same ramp by changing only the inertia factor.
Check a mechanics assignment
Verify the acceleration obtained from translational and rotational equations for rolling without slipping.
Plan a ramp experiment
Predict center-of-mass acceleration before measuring a rolling object in a laboratory setup.
FAQ
What is the inertia factor?
It is k = I/(m r²), where I is moment of inertia about the rolling axis, m is mass, and r is rolling radius.
Why does mass not appear in the result?
For this ideal rolling model, mass cancels when rotational inertia is written as I = k m r².
Does the calculator include slipping?
No. It assumes pure rolling without slipping and sufficient static friction to enforce that condition.
Why are angles above ninety degrees rejected?
The supported geometry is a body rolling down a conventional incline from horizontal through vertical; larger angles fall outside that model.
What does it cost?
The same deterministic calculation is available in the browser.
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/rolling-down-incline-acceleration \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"angle":30,"inertia_factor":0.5}'const res = await fetch("https://api.kit.forhosting.com/mech/rolling-down-incline-acceleration", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"angle": 30,
"inertia_factor": 0.5
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/rolling-down-incline-acceleration",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"angle": 30,
"inertia_factor": 0.5
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/rolling-down-incline-acceleration", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"angle":30,"inertia_factor":0.5}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"angle":30,"inertia_factor":0.5}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/rolling-down-incline-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
{
"angle": 30,
"inertia_factor": 0.5
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.rolling_down_incline_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
Describe the incline and the rolling body
Common ideal bodies illustrate the meaning: a hoop has k = 1, a solid cylinder or disk has k = 1/2, and a solid sphere has k = 2/5.
Understand the acceleration formula
Results are rounded to stable significant precision so repeated runs with identical inputs produce consistent JSON while retaining more accuracy than most practical measurements justify.
Apply the result within the model limits
The API price is $0.002 per request, while the browser version can run the same deterministic calculation locally for quick checks.