HomeCalculatorsMechanicalPlanar Equilibrium Calculator
Mechanical calculator

Planar Equilibrium Calculator

Sum 2-D point forces and concentrated moments, or solve two or three unknown reactions. Uses the vector and linear-algebra engines. Not a beam calculator.

Instant result
Result
—

Enter values to calculate.

Inputs—
Mode—
Formula—
Trust summary CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance · 2-D point forces and concentrated moments: ΣFx, ΣFy, ΣM, resultant |R|, and 2–3 unknown reactions; + O3 mpmath tabulated ΣF/ΣM.
Input interpretation
Enter values to calculate.
Result
—
Verified scope
2-D point forces and concentrated moments: ΣFx, ΣFy, ΣM, resultant |R|, and 2–3 unknown reactions; + O3 mpmath tabulated ΣF/ΣM.
Assurance
Engineering
Declared partition coverage
PASS · 5/5 declared partitions (resultant, equilibrium, alias, awkward, invalid-domain) · Matrix
Deferred
Not beam deflection, not distributed load, not shear/moment diagrams, and not 3-D.
Numerical scope
O2: planar ΣF/ΣM vs a separate-module identity (≤2 ULP). Not beam deflection, not distributed load, and not 3-D. ≤2 ULP vs O3 applies only to the published tabulated planar-equilibrium vectors. Not beam deflection.
Known limitations
  • Core CVP does not include live graph, viewport, or pointer interaction.
Model
Planar resultant and equilibrium of point forces and concentrated moments.
Scope
Calculation runs locally in the browser; values are not uploaded.
Verification
Engine tested · Source checked · v1.0.0 · CVP VERIFIED · production STALE · CVP protocol 1.0.0-proposed · Engineering assurance · 2-D point forces and concentrated moments: ΣFx, ΣFy, ΣM, resultant |R|, and 2–3 unknown reactions; + O3 mpmath tabulated ΣF/ΣM.· View Manifest · CVP overview · Specification
Versions
Calculation 1.0.0 · CVP protocol 1.0.0-proposed · Evidence 2026-09-27.o2-o3.v2
Verification revision
2026-09-27.o2-o3.v2 · 1/1 property · digest f05ac954683d
Legacy regression
6/6 tests · Production surface contract 6/6
Trust layers
Verification VERIFIED · Production STALE · overall VERIFIED_STALE
CVP status
STALE · attestation snapshot core 4601e9209773 ≠ live cvp_core_sha256 f05ac954683d (2026-09-27.o2-o3.v2)
Reference
O1 model · O3 expected_values · O3 numerical_behavior · O2 expected_values · O2 numerical_behavior
Interfaces
PASS · UI (SSR) / REST / MCP — ui-ssr is query-result HTML, not a live browser session. Error-path engine·REST·MCP 1/1 (status, code, calculation_version). SSR compared on URL-canonical requested calculations; empty query is idle (not an error) and JSON-typed object/array inputs are REST/MCP-only.
Supplemental domain review
Not performed
Named expert review
Not performed
CVP suite
4/4 golden · 3/3 CVP boundary · 6/6 invalid · 1/1 property · 1/1 metamorphic · 8/8 O3 · 2/2 cross-interface · 6/6 CVP contract · Manifest
Sources
  • Hibbeler, Engineering Mechanics: Statics — Force systems and rigid-body equilibrium
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Force systems and rigid-body equilibrium
Sources
Evidence
2 legacy golden · 4 legacy boundary · legacy regression suite · 4/4 oracle-backed golden · 6/6 invalid · Artifact integrity PASS
STALE · Last attested schema matched 1.0.0 snapshot / local build · Semantic contract ✓ · Last attestation PASS · current evidence changed · re-attestation required · CVP STALE · attestation STALE — Production CURRENT withheld · Public/cache ✓ · Origin ✓
Semantic contract
PASS
Full verification

Manifest identity, reference classes, interfaces, suite, and production records.

Formulas

Core equations used by this calculator.

ResultantR = ΣF
Moment about the originM = Σm + Σ(x Fy − y Fx)
iCCW is positive. Positions are meters from the origin. This page is not beam deflection and not a distributed load.

How to use

1

Enter forces as fx, fy, x, y

One force per row, separated by semicolons. x and y default to 0. Newtons and meters.

2

Enter concentrated moments

One moment per row, in newton-meters, CCW positive.

3

Mark unknowns only in equilibrium mode

A force unknown is fx, fy, or fx+fy. A moment unknown is m. Two or three unknowns are solved together.

Example calculations

Common configurations with formula and result.

ϟ

3-4-5 resultant

Forces 3 N and 4 N at the origin

R = (3, 0) + (0, 4)
|R| = 5 N
ϟ

Bracket reactions

0,−100 N at x = 2 m, plus unknown Rx, Ry, and M at the origin

ΣFx = 0, ΣFy = 0, ΣM = 0
Ry = 100 N, M = 200 N·m

Planar Equilibrium calculator specification

Version 1.0.0 · Engine tested

Calculation status

Review policy · Evidence

Definition
2-D point forces and concentrated moments. Resultant is the vector sum. Equilibrium solves ΣFx = 0, ΣFy = 0, and ΣM = 0 for two or three unknowns.
What it calculates
Planar resultant and equilibrium of point forces and concentrated moments.
Inputs
  • mode
  • forces
  • moments?
Outputs
  • sum_fx
  • sum_fy
  • sum_m
  • resultant_n
  • unknowns?
Formula
ΣF and ΣM = 0; M = x Fy − y Fx
Assumptions
  • Calculation runs locally in the browser; values are not uploaded.
  • Keep units consistent with the labels on each field.
  • CCW is positive. Positions are meters from the origin. This page is not beam deflection and not a distributed load.
Units
  • Force in N, moment in N·m, position in m. CCW positive.
Boundary conditions
  • no forces and no moments → MISSING_REQUIRED_INPUT
  • unknowns in resultant mode → INVALID_INPUT
  • 1 or more than 3 unknowns → INVALID_INPUT
  • 2 unknowns that cannot satisfy all three equations → INVALID_INPUT
Example
3,0; 0,4 → |R| = 5 N
Validation cases

3 published on this page · 6/6 tests · Production surface contract 6/6 · View evidence

  • mode=resultant forces=3,0; 0,4 → resultant_n=5
  • mode=equilibrium forces=0,-100,2,0; 0,0,0,0,fx+fy moments=0,m → F2.fy=100, M1=200
  • mode=equilibrium forces=0,-10,2,0; 0,0,0,0,fx+fy → INVALID_INPUT
Sources
  • Hibbeler, Engineering Mechanics: Statics — Force systems and rigid-body equilibrium
    Supports: ΣFx = 0, ΣFy = 0, ΣM = 0 for a planar rigid body
  • Beer, Johnston, and Mazurek, Vector Mechanics for Engineers: Statics — Force systems and rigid-body equilibrium
    Supports: Supports the page formula: R = ΣF
Calculation version
1.0.0

Background

Interpretation and common distinctions.

Balance point forces and concentrated moments in the plane.

Supported and not supported

Supported: a resultant of 2-D forces, the moment of those forces about the origin, and a unique solution for two or three unknown components.

Not supported: beam deflection, distributed loads, frames, finite elements, and 3-D statics. Impending slip on an incline is its own page. /calc/mechanical is not open.

Agent / API notes

Capability id: mechanical.statics.planar_equilibrium · tool id: planar-equilibrium · pin 1.0.0.

{ "mode": "resultant", "forces": "3,0; 0,4" }

A force row is Fx,Fy or Fx,Fy,x,y. Share the link with mode, forces, and moments when used. The result is not written into the link.

Other calculators in this family: Incline friction, Vector, Linear system .

μ Incline friction › u Vector › Ax=b Linear system ›

Frequently asked questions

Key distinctions behind the calculation.

Is this a beam calculator?

No. It balances point forces and concentrated moments in the plane. Distributed loads, shear and moment diagrams, and deflection are out of scope.

Where does the solve run?

Locally. The force sum calls the vector engine. Two or three unknowns call the linear-algebra engine. This page does not contain a second solver.