Work done calculator
Pushing with 10 N through 5 m in the direction of motion does 50 J of work; at 90° the same push does none. Enter force, displacement and angle to get signed work in joules. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Force
- 10
- Displacement
- 5
- Angle
- 0
- Angle rad
- 0
- Cos theta
- 1
- Work
- 50
- Work sign
- positive
Mechanical work measures how much a force contributes when a body moves through a displacement. For a constant force and a straight path the closed-form relation is W equals F times d times cos of theta, where theta is the angle between the force vector and the displacement vector. When force is in newtons and displacement in metres, work is reported in joules, the SI unit of energy transfer by mechanical means. Students use this one-line formula for homework checks; teachers and auto-graders need a deterministic oracle; lab engineers verify pulley, ramp, and towing setups without re-deriving the cosine projection each time. This calculator accepts force, displacement, and angle in degrees, returns signed work in joules with cosine and intermediate fields, and rejects angles above one hundred eighty degrees plus any non-finite input so bad data never becomes a silent NaN.
How mechanical work is defined and computed
Begin with a constant force vector of magnitude F and a straight displacement of magnitude d. The angle theta between those two directions decides how much of the force actually does work along the path. Only the component of force parallel to the displacement contributes: that component is F times cos of theta, and multiplying by d yields work W equals F d cos theta.
When theta is zero the force and path are aligned and work is maximum and positive for positive F and d. When theta is ninety degrees the force is perpendicular to the motion and work is zero, which is why a centripetal force on a circular path does no work. When theta is one hundred eighty degrees the force opposes the displacement and work is negative, meaning the force removes energy from the body.
The SI unit of work is the joule, equal to one newton metre. All three fields must be finite numbers; NaN and Infinity are rejected with invalid input so clients can fix units and types rather than guessing. This page is intentionally limited to the constant-force straight-path formula; variable-force path integrals, power as work over time, and kinetic energy are separate search intents and separate landings.
What you send and what you get back
Send three required finite numbers under the fields force, displacement, and angle. Force is typically in newtons; displacement is in metres; angle is the angle between force and displacement in degrees and must not exceed one hundred eighty. Classic textbook checks include ten newtons through five metres at zero degrees for fifty joules, the same force and displacement at ninety degrees for zero work, and one hundred eighty degrees for negative fifty joules.
Keep units consistent across a multi-step lab: if you measure force in kilonewtons you must convert to newtons before calling, or scale the result accordingly after, so the numeric product still matches the SI interpretation your report claims.
Where work done calculators show up in real work
Secondary-school and introductory university physics courses assign force-displacement-cosine drills, compare positive and negative work on free-body diagrams, and require signed joules before energy-conservation problems; a single trusted oracle speeds grading and demo setup. Mechanical design and manufacturing staff convert applied force, stroke length, and alignment angle into work estimates for press, clamp, and actuator duty cycles without re-deriving projections each session. Sports science and biomechanics staff estimate work against gravity or resistance on ramps and sleds from force, path length, and inclination angle.
If you later need power from work and time, kinetic energy from mass and speed, potential energy change, or variable-force numerical integration along a path, treat those as separate search intents and landings rather than overloading this work done calculator page.
What you can do with it
Homework and lab answer keys
Compute signed work W=Fd cos θ from force, displacement, and angle so students and auto-graders share one closed-form ground truth in joules.
Ramp and towing work estimates
Convert applied force, path length, and alignment angle into joules for ramps, pulleys, and towing setups without manual cosine mistakes.
Actuator and stroke duty checks
Estimate mechanical work along a straight stroke from force, displacement, and force-path angle before full dynamics models.
FAQ
What formula does this work done calculator use?
Mechanical work W = F × d × cos(θ), where F is force, d is displacement, and θ is the angle between them in degrees. The result is reported in joules (J) when force is in newtons and displacement in metres.
What happens if the angle is greater than 180 degrees?
The angle between force and displacement must be at most 180°. Larger values usually indicate a unit mistake and are rejected rather than wrapped.
Can work be negative?
Yes. When the force has a component opposite the displacement (θ between 90° and 180° for positive F and d), work is negative, meaning the force removes energy from the body.
What if force or displacement is zero?
Work is zero. Both zero force and zero displacement are valid physical edge cases and are accepted when all fields are finite numbers.
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/work-done \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"force":10,"displacement":5,"angle":0}'const res = await fetch("https://api.kit.forhosting.com/mech/work-done", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"force": 10,
"displacement": 5,
"angle": 0
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/work-done",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"force": 10,
"displacement": 5,
"angle": 0
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/work-done", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"force":10,"displacement":5,"angle":0}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"force":10,"displacement":5,"angle":0}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/work-done", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)Example request
{
"force": 10,
"displacement": 5,
"angle": 0
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.work_done",
"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 mechanical work 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
Missing fields, non-numeric strings, NaN, Infinity, and angles greater than one hundred eighty fail with invalid_input and are not charged. On success the response echoes cleaned force, displacement, and angle, provides angle_rad as the radian value used for cosine, cos_theta as the cosine of that angle, work as the primary SI product F times d times cos theta in joules, work_sign as positive, negative, or zero, a units hint object naming N, m, degrees, and J, method set to work_force_times_displacement_times_cos_theta, and the short formula string W equals F times d times cos of theta for display. Pricing for successful API calls is the published base of $0.002 per item; validation failures do not consume prepaid balance.
Where work done calculators show up in real work
Robotics and control labs report work done along a straight rail from payload force and encoder displacement as a first filter before full multi-body dynamics models. 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.