Rotational kinetic energy calculator
A flywheel with a moment of inertia of 2 kg·m² spinning at 4 rad/s stores 16 J. Enter the moment of inertia and the angular velocity to get the energy held in the spin, in joules, whichever way it turns. Free in your browser.
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- Your data stays on your device
Run — free
Worked example
- Moment of inertia
- 2
- Angular velocity
- 4
- Omega squared
- 16
- Kinetic energy
- 16
- Energy class
- positive
Rotational kinetic energy measures how much energy is stored in a rigid body simply because it is spinning about an axis. For a fixed moment of inertia I and angular velocity omega the closed-form relation is one half times I times omega squared, reported in joules when I is in kilogram metres squared and omega is in radians per second. Students use this formula for flywheels, disks, rods, and planets on homework; teachers and auto-graders need a deterministic oracle; lab engineers check spinning-shaft energy budgets without re-deriving the square each time. This calculator accepts moment of inertia and angular velocity, returns kinetic energy with the omega-squared intermediate and unit hints, and rejects negative inertia plus any non-finite input so bad data never becomes a silent NaN.
How rotational kinetic energy is defined and computed
Begin with a rigid body that rotates about a fixed axis with moment of inertia I and instantaneous angular velocity omega. Translational kinetic energy is one half mass times speed squared; the rotational analogue replaces mass with I and linear speed with angular speed, giving KE equals one half times I times omega squared.
Moment of inertia already folds in how mass is distributed relative to the axis: a thin hoop with mass far from the center has a larger I than a solid disk of the same mass and radius, so it stores more energy at the same spin rate. Angular velocity may be positive or negative depending on rotation sense; because energy depends on omega squared, both signs yield the same non-negative kinetic energy.
When I is in kilogram metres squared and omega is in radians per second, the product is in joules, the SI unit of energy. Both fields must be finite numbers; NaN and Infinity are rejected with invalid input so clients can fix units and types rather than guessing. Negative moment of inertia is rejected because inertia about a fixed axis cannot be negative for a physical rigid body; zero inertia is allowed and produces zero energy.
This page is intentionally limited to the scalar fixed-axis formula; computing I from shape tables, angular momentum I omega, torque and angular acceleration, or full 3D inertia tensors are separate search intents and separate landings.
Where rotational kinetic energy calculators show up in real work
Secondary-school and introductory university physics courses assign flywheel, disk, and rod energy drills, compare translational and rotational kinetic energy in rolling without slipping, and require joules before energy-conservation problems; a single trusted oracle speeds grading and demo setup. Mechanical design and manufacturing staff convert shaft inertia and operating angular velocity into stored energy estimates for spindles, centrifuges, and machine-tool flywheels without re-deriving the half-I-omega-squared product each session.
Robotics and control labs report rotational energy of joints and reaction wheels from known inertias and encoder rates as a first filter before full multi-body dynamics models. Sports science and biomechanics staff estimate rotational energy of limbs, rackets, and wheels from approximate moments of inertia and measured angular speeds.
If you later need translational kinetic energy from mass and speed, moment of inertia from geometry, angular momentum, torque, or power from torque and angular velocity, treat those as separate search intents and landings rather than overloading this rotational kinetic energy calculator page.
What you can do with it
Homework and lab answer keys
Compute KE=½Iω² from moment of inertia and angular velocity so students and auto-graders share one closed-form ground truth in joules.
Flywheel and shaft energy budgets
Convert known rotor inertia and operating spin rate into stored rotational energy for spindles, centrifuges, and machine-tool flywheels.
Robot joint and reaction-wheel checks
Estimate rotational kinetic energy of joints or reaction wheels from inertia and encoder angular velocity before full dynamics models.
FAQ
What formula does this rotational kinetic energy calculator use?
KE = ½ I ω², where I is moment of inertia about the rotation axis and ω is angular velocity about the same axis. With SI units the result is in joules.
What happens if moment of inertia is negative?
Moment of inertia must be ≥ 0 for a rigid body about a fixed axis; negative I is almost always a unit or sign error.
Can angular velocity be negative?
Yes. Sign indicates rotation sense. Because energy depends on ω², positive and negative ω of equal magnitude yield the same non-negative kinetic energy.
What units should I use?
Prefer SI: I in kg·m² and ω in rad/s so KE is in joules. Any consistent pair of units works if you interpret the product accordingly.
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/rotational-kinetic-energy \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"moment_of_inertia":2,"angular_velocity":4}'const res = await fetch("https://api.kit.forhosting.com/mech/rotational-kinetic-energy", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"moment_of_inertia": 2,
"angular_velocity": 4
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/rotational-kinetic-energy",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"moment_of_inertia": 2,
"angular_velocity": 4
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/rotational-kinetic-energy", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"moment_of_inertia":2,"angular_velocity":4}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"moment_of_inertia":2,"angular_velocity":4}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/rotational-kinetic-energy", 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,
"angular_velocity": 4
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.rotational_kinetic_energy",
"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 rotational kinetic energy is defined and computed
The entire pipeline is deterministic: 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
Moment of inertia is typically in kilogram metres squared; angular velocity is in radians per second about the same axis. Classic textbook checks include I equals two and omega equals four for sixteen joules, I equals one and omega equals zero for zero energy, and equal-magnitude opposite omega signs that share the same kinetic energy because of the square. Keep units consistent across a multi-step lab: if you measure spin in revolutions per minute you must convert to radians per second before calling, or scale carefully after, so the numeric product still matches the SI interpretation your report claims. Send two required finite numbers under the fields moment_of_inertia and angular_velocity. Missing fields, non-numeric strings, NaN, Infinity, and negative moment of inertia fail with invalid_input and are not charged. On success the response echoes cleaned moment_of_inertia and angular_velocity, provides omega_squared as the squared intermediate used in the product, kinetic_energy as the primary SI value one half I omega squared in joules, energy_class as positive or zero, a units hint object naming kg·m², rad/s, and J, method set to rotational_kinetic_energy_half_I_omega_squared, and the short formula string KE equals one half I omega squared for display. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where rotational kinetic energy 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.