Angular momentum calculator
A flywheel with a moment of inertia of 2.5 kg·m² spinning at 4 rad/s carries an angular momentum of 10 kg·m²/s: the calculator multiplies I by ω and keeps the sign of the rotation. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Moment of inertia
- 2.5
- Angular velocity
- 4
- Angular momentum
- 10 kg·m²/s
This angular momentum calculator applies the standard scalar rigid-body relationship L = Iω. Enter a moment of inertia in kilogram metre squared and a signed angular velocity in radians per second, and it returns angular momentum in kilogram metre squared per second. Positive or negative angular velocity preserves the direction chosen for the rotation axis, while moment of inertia must remain non-negative. Missing text, non-numeric values, NaN, Infinity, and an overflowing product are rejected clearly instead of producing a misleading result. No network service, random value, or clock affects the answer.
Understanding angular momentum for a rotating body
Angular momentum describes rotational motion relative to a chosen axis. Moment of inertia measures how strongly the mass distribution resists a change in rotational motion. Mass located farther from the axis contributes more strongly, so two bodies with equal mass and equal angular speed can have different angular momentum.
Angular velocity gives the signed rate of rotation in radians per second. With I expressed in kilogram metre squared and ω expressed in radians per second, the radians are dimensionless and the resulting SI unit is kilogram metre squared per second, written kg·m²/s. The sign of L follows the sign of ω because scalar I is non-negative.
A positive result points along the positive rotation axis selected by your convention; a negative result points in the opposite direction. This calculator intentionally handles that fixed-axis scalar case. General three-dimensional rigid-body motion may require the inertia tensor and vector multiplication, especially when the rotation axis is not a principal axis, so a single scalar I is not enough for every dynamics problem.
Assumptions, checks, and practical applications
Use this result when the body can be represented by one moment of inertia about the same axis used to define angular velocity. Confirm that your I value belongs to that axis: the moment of inertia of a disk about its central symmetry axis differs from its moment about an edge or a diameter. Confirm units before multiplying, because a numerically plausible answer can still be wrong by orders of magnitude when RPM is mistaken for rad/s or centimetres are mistaken for metres.
A quick dimensional check should read kg·m² multiplied by s⁻¹, producing kg·m²/s. A sign check should verify that reversing the angular velocity reverses angular momentum without changing its magnitude. Torque relates to the rate of change of angular momentum; rotational kinetic energy uses one half Iω² and therefore loses the directional sign; neither should be substituted for L.
For off-axis or freely tumbling objects, use a vector and inertia-tensor treatment instead.
What you can do with it
Check a flywheel calculation
Multiply a known flywheel moment of inertia by its signed shaft speed in rad/s and verify the resulting angular momentum before a wider energy or torque analysis.
Validate physics coursework
Compare a hand calculation of L = Iω with a deterministic response that repeats the inputs, formula, SI unit, and signed result.
FAQ
What formula does the angular momentum calculator use?
It uses L = I × ω for a rigid body rotating about a fixed axis, with I in kg·m² and ω in rad/s.
What unit does the result use?
The returned angular momentum uses kilogram metre squared per second, written kg·m²/s.
Can angular velocity be negative?
Yes. Its sign represents rotation direction relative to your chosen axis, and angular momentum receives the same sign.
What should I do if my speed is in RPM?
Convert it to radians per second first, and turn a moment of inertia given in g·cm² into kg·m². The calculator multiplies the numbers as entered, so RPM taken as rad/s gives an answer that looks plausible but is off by orders of magnitude.
Does this work for any rotating object?
It covers rotation about one fixed axis, with a single moment of inertia measured about that same axis. Freely tumbling bodies, or rotation off a principal axis, need the full inertia tensor and vector maths, which this tool does not do.
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/angular-momentum \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"moment_of_inertia":2.5,"angular_velocity":4}'const res = await fetch("https://api.kit.forhosting.com/mech/angular-momentum", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"moment_of_inertia": 2.5,
"angular_velocity": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/angular-momentum",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"moment_of_inertia": 2.5,
"angular_velocity": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/angular-momentum", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"moment_of_inertia":2.5,"angular_velocity":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"moment_of_inertia":2.5,"angular_velocity":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/angular-momentum", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"moment_of_inertia": 2.5,
"angular_velocity": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.angular_momentum",
"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
The browser calculation is free, while an API request costs $0.002.
Understanding angular momentum for a rotating body
For a rigid body rotating about a fixed principal axis, its scalar angular momentum L equals its moment of inertia I multiplied by its angular velocity omega: L = Iω.
Entering values and interpreting the response
The first value is I in kg·m² and cannot be negative. The second is ω in rad/s and may be positive, zero, or negative. If that same flywheel rotates at -4 rad/s relative to the chosen axis, the magnitude remains 10 while the returned angular momentum is -10 kg·m²/s. Convert source units before calling: RPM must first become rad/s, and moment of inertia supplied in g·cm² must first become kg·m². Provide moment_of_inertia and angular_velocity as finite numbers. The aliases inertia and omega are accepted for convenient programmatic calls, but using the canonical field names makes payloads easiest to understand. The response repeats both normalized inputs, provides angular_momentum, identifies the unit as kg·m²/s, and includes the formula L = I × ω. For example, a flywheel with I = 2.5 kg·m² rotating at 4 rad/s has L = 10 kg·m²/s. A stationary body has ω = 0 and therefore L = 0 even if its moment of inertia is large. Successful API requests cost $0.002; validation failures return an input error rather than an invented physical answer.
Assumptions, checks, and practical applications
Within the stated scalar model, the implementation performs constant-time validation and multiplication, uses no external data, and produces the same result in the browser and API.
Integration use cases
Create simulation fixtures
Generate stable expected angular-momentum values for rotating-body tests, including zero speed and rotation opposite the selected positive axis.
Technical questions
Can moment of inertia be negative?
No. A scalar physical moment of inertia is non-negative, so negative values return an invalid_input error.
How much does the API calculation cost?
Each successful API request costs $0.002. The browser runner is free, and invalid input is rejected clearly.