Principle of moments calculator
Hide one of the four values and get it back: 10 N at 1 m on one side balances a force of 20 N placed 0.5 m from the pivot on the other, both moments 10 N·m. Free in your browser.
- Free
- Runs in your browser
- No sign-up
- Your data stays on your device
Run — free
Worked example
- Distance cw
- 0.5
- Force acw
- 10
- Distance acw
- 1
- Calculate
- Force cw
- Force cw
- 20
- Solved for
- force_cw
- Unknown value
- 20
- Moment cw
- 10
The principle of moments is the rotational-equilibrium rule that school statics, workshop checks, and first-year engineering all share: about any chosen pivot, the total clockwise moment equals the total anticlockwise moment when a rigid body is not angularly accelerating. Typical exam and lab items give one force and its perpendicular lever arm on each side of a seesaw, beam, or meter stick, then hide either a force or a distance and ask you to restore the balance. This calculator takes that two-contribution form, treats exactly one of the four quantities as unknown, and returns the missing value together with both side moments so you can show the equality. A zero lever arm is rejected because a force whose line of action passes through the pivot produces no turning effect and would make the rearranged equation divide by zero.
What the principle of moments states and what this calculator returns
A moment about a pivot is the turning effect of a force, equal to the force magnitude times the perpendicular distance from the pivot to the force's line of action. Clockwise contributions try to rotate the body one way; anticlockwise contributions try the other way. The principle of moments, also called the moment rule on many secondary-school syllabuses, says that rotational equilibrium about a convenient point is equivalent to the algebraic sum of those moments being zero, which is the same statement as clockwise total equals anticlockwise total.
Optional precision controls how many decimal places are exposed so automated tests stay stable. Use a dedicated lever-effort, net-torque, or single-moment tool when the problem is a different search, not this named principle.
Where a principle-of-moments solver is used, what it costs, and what it will not invent
Students meet this identity when a textbook introduces pivots and hanging masses, when a lab balances a meter stick, or when an exam prints three numbers and asks for the fourth force or distance without calculus. Teachers can pre-build answer keys for modest arm and load values instead of trusting opaque calculator apps that silently switch units or confuse mass with force.
First-order design checks use force times distance balance when only three of the four design loads and arms are known, before safety factors or distributed beam weight enter the model. On this page the work runs entirely in your browser. Keep both forces in one force unit and both distances in one length unit: newtons with metres yield newton-metre moments, and any other coherent pair stays coherent.
The calculator does not convert unit systems, does not net more than one contribution per side, and does not replace a full statics package. Pair it with lever-effort or single-moment helpers when the question is those other searches.
What you can do with it
Homework and exam checks
Solve for the missing force or lever arm from three knowns so students can verify principle-of-moments worksheet answers against a deterministic oracle.
Lab and seesaw demos
Predict the balancing force or pivot distance before hanging weights on a meter stick, classroom lever, or workshop bar.
FAQ
What formula does the principle of moments calculator use?
Exactly one of the four quantities is solved from the other three.
What happens if a lever arm is zero?
A zero lever arm produces no moment and would divide by zero when rearranging for a force, so it is not accepted.
Which units should I use?
Any consistent pair: both forces in the same force unit and both distances in the same length unit. With newtons and metres, moments are in newton-metres. The calculator does not convert unit systems.
What does the principle of moments calculator cost?
The browser tool is free on this page.
Which value can I solve for?
Any one of the four: the clockwise force or its distance, or the anticlockwise force or its distance. Leave that one as the unknown, give the other three, and both side moments come back so you can show they are equal.
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/principle-of-moments \
-H "Authorization: Bearer $KIT_KEY" \
-H "Content-Type: application/json" \
-d '{"force_cw":20,"force_clockwise":1}'const res = await fetch("https://api.kit.forhosting.com/mech/principle-of-moments", {
method: "POST",
headers: {
"Authorization": `Bearer ${process.env.KIT_KEY}`,
"Content-Type": "application/json"
},
body: JSON.stringify({
"force_cw": 20,
"force_clockwise": 1
})
});
const { task_id } = await res.json();import os, requests
res = requests.post(
"https://api.kit.forhosting.com/mech/principle-of-moments",
headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
json={
"force_cw": 20,
"force_clockwise": 1
},
)
task_id = res.json()["task_id"]<?php
$res = file_get_contents("https://api.kit.forhosting.com/mech/principle-of-moments", false, stream_context_create([
"http" => [
"method" => "POST",
"header" => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
"content" => '{"force_cw":20,"force_clockwise":1}',
],
]));
$task = json_decode($res, true);body := bytes.NewBufferString(`{"force_cw":20,"force_clockwise":1}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/mech/principle-of-moments", 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_cw": 20,
"force_clockwise": 1
}Example response
{
"task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
"type": "mech.principle_of_moments",
"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.
Limits
max_abs | 1000000000000000 |
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 same closed-form arithmetic runs free in the browser and as a prepaid API item at $0.002 per successful call, with no network, sampling, or clock involved.
What the principle of moments states and what this calculator returns
This page models the standard classroom pair: one clockwise force force_cw acting at lever arm distance_cw, and one anticlockwise force force_acw acting at lever arm distance_acw. Equilibrium then requires force_cw times distance_cw to equal force_acw times distance_acw. You name the unknown with solve_for, or you simply omit that one field, and the engine rearranges the product identity to fill the gap. The response echoes all four quantities with the unknown completed, reports solved_for and unknown_value for display, returns moment_cw and moment_acw so both sides of the ledger are visible, and sets balanced true when the floating-point cross-check agrees. There is no multi-load beam model, no friction couple, and no distributed self-weight: either the closed form evaluates inside the supported range or the request fails with invalid_input.
How the four quantities are read, how the unknown is chosen, and why a zero lever arm fails
Exactly three of those four values must be known. When the unknown is a distance, the force on that same side must also be non-zero or the division is undefined. Numeric strings in ordinary decimal or scientific notation are accepted; empty strings, NaN, and Infinity are not. The arithmetic is one product of the known side divided by the remaining known factor on the unknown side, then stable rounding with negative zero collapsed. Send JSON with the clockwise force as force_cw, the clockwise perpendicular arm as distance_cw, the anticlockwise force as force_acw, and the anticlockwise arm as distance_acw. Aliases are accepted so spreadsheets and typed forms can reuse the same contract: force_clockwise, F_cw, fcw, and clockwise_force for the clockwise load; distance_clockwise, d_cw, arm_cw, lever_arm_cw, and clockwise_distance for its arm; force_anticlockwise, force_ccw, F_acw, facw, anticlockwise_force, and counterclockwise_force for the opposite load; distance_anticlockwise, distance_ccw, d_acw, arm_acw, lever_arm_acw, and anticlockwise_distance for its arm. Either omit the unknown so the engine infers the single blank field, or set solve_for — aliases unknown, find, and for — to force_cw, distance_cw, force_acw, or distance_acw and supply the other three. A known lever arm must be finite and non-zero: zero or numerically zero distance_cw or distance_acw fails with invalid_input and a message that names the field, because a zero lever arm makes F times d useless for solving a force by division. Optional precision, with aliases decimals and dp, chooses decimal places from zero to fifteen and defaults to ten.
Where a principle-of-moments solver is used, what it costs, and what it will not invent
Product and education tools use the endpoint when a learner types three knowns and the interface must show the missing force or lever arm together with both side moments. Through the API each successful lookup is one billable item at $0.002; failed validation is not charged; nothing is stored after the response returns.
Integration use cases
Deterministic test fixtures
Generate reproducible unknown_value and side-moment pairs for unit tests, tutoring software, and continuous-integration assertions.
Technical questions
How do I choose which quantity is unknown?
Omit exactly one of force_cw, distance_cw, force_acw, or distance_acw, or set solve_for to that field name and supply the other three known values.