Rotational work calculator
A constant 10 N·m torque turning a shaft through 2 radians does 20 J of work. Enter the torque and the angle in radians to get the work in joules, negative when the torque opposes the rotation, as in braking. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Torque
- 10
- Angular displacement
- 2
- Work
- 20
- Work sign
- positive
Rotational work measures how much energy a torque transfers when a body turns through an angular displacement. For a constant torque about a fixed axis the closed-form relation is W equals tau times theta, where theta is the angular displacement in radians. When torque is in newton-metres and the angle is in radians, work is reported in joules, the same SI unit used for linear mechanical work. Students use this one-line formula for flywheels, shafts, and winches on homework; teachers and auto-graders need a deterministic oracle; lab engineers verify motor duty cycles without re-deriving the product each session. This calculator accepts torque and angular displacement in radians, returns signed work in joules with unit hints and a formula string, and rejects non-numeric or non-finite torque so bad data never becomes a silent NaN.
How rotational work is defined and computed
Begin with a constant torque tau applied about a fixed rotation axis and an angular displacement theta of the body about that same axis. In linear mechanics, work is force times displacement along the force; the rotational analogue replaces force with torque and linear displacement with angle, giving W equals tau times theta. The angle must be expressed in radians for the product to have the dimensions of energy: one newton-metre of torque through one radian equals one joule.
Positive torque and positive angle yield positive work, meaning energy is added to the rotating system. Opposite signs yield negative work, meaning the torque removes energy—useful when modelling braking, opposing friction, or a motor being back-driven. Zero torque or zero angular displacement both produce zero work and are accepted as valid physical edge cases.
Where rotational work calculators show up in real work
Secondary-school and introductory university physics courses assign shaft, winch, and flywheel work drills, compare linear and rotational energy transfer, and require signed joules before energy-conservation problems; a single trusted oracle speeds grading and demo setup. Mechanical design and manufacturing staff convert applied motor torque and commanded angular stroke into work estimates for indexers, clamps, and rotary actuators without re-deriving the product each session. Robotics and control labs report work done at a joint from measured torque and encoder angle change as a first filter before full multi-body dynamics models.
Sports science and biomechanics staff estimate rotational work of limbs and training machines from approximate torque and range of motion in radians. If you later need power from torque and angular velocity, rotational kinetic energy from inertia and spin, linear work from force and displacement, or variable-torque numerical integration, treat those as separate search intents and landings rather than overloading this rotational work calculator page.
What you can do with it
Homework and lab answer keys
Compute signed work W=τθ from torque and angular displacement in radians so students and auto-graders share one closed-form ground truth in joules.
Motor stroke and actuator duty
Convert applied torque and commanded angular stroke into joules for rotary actuators, indexers, and winches without manual unit mistakes.
Joint work in robotics labs
Estimate rotational work at a joint from measured torque and encoder angle change before full multi-body energy models.
FAQ
What formula does this rotational work calculator use?
Rotational work W = τ × θ, where τ is torque and θ is angular displacement in radians. The result is reported in joules (J) when torque is in newton-metres and the angle is in radians.
Must angular displacement be in radians?
Yes. The SI product τ·θ is joules only when θ is in radians.
What happens if torque is non-numeric?
Torque must be a finite number; strings that are not numeric, NaN, Infinity, and missing torque are rejected and not billed.
Can rotational work be negative?
Yes. When torque and angular displacement have opposite signs, work is negative, meaning the torque removes energy from the rotating body (for example during braking).
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-work \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"torque":10,"angular_displacement":2}'const res = await fetch("https://api.kit.forhosting.com/mech/rotational-work", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"torque": 10,
"angular_displacement": 2
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/rotational-work",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"torque": 10,
"angular_displacement": 2
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/rotational-work", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"torque":10,"angular_displacement":2}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"torque":10,"angular_displacement":2}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/rotational-work", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"torque": 10,
"angular_displacement": 2
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.rotational_work",
"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 work is defined and computed
Degrees are not SI for this product; if your instrument reports degrees you must convert with theta_rad equals theta_deg times pi over one hundred eighty before calling the calculator. 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. Both fields must be finite numbers; non-numeric strings, NaN, and Infinity on torque or angle are rejected with invalid_input so clients can fix types and units rather than guessing. This page is intentionally limited to the constant-torque fixed-axis product; variable-torque path integrals, power as torque times angular velocity, rotational kinetic energy one half I omega squared, and linear work F d cos theta are separate search intents and separate landings.
What you send and what you get back
Keep units consistent across a multi-step lab: if you measure rotation in revolutions or degrees you must convert to radians before calling, so the numeric product still matches the SI interpretation your report claims. Do not pass angular velocity in place of angular displacement; velocity would give power-like dimensions, not work. Send two required finite numbers under the fields torque and angular_displacement. Torque is typically in newton-metres; angular_displacement is the angle of rotation in radians about the same axis. Missing fields, empty strings, non-numeric torque, non-numeric angle, NaN, and Infinity fail with invalid_input and are not charged. On success the response echoes cleaned torque and angular_displacement, provides work as the primary SI product tau times theta in joules, work_sign as positive, negative, or zero, a units hint object naming N·m, rad, and J, method set to rotational_work_torque_times_angle, and the short formula string W equals tau times theta for display. Classic textbook checks include ten newton-metres through two radians for twenty joules, five newton-metres through pi radians for about fifteen point seven joules, and equal-magnitude opposite signs that reverse work_sign while preserving magnitude. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where rotational work 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.