Physics 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 Engine tested · Source checked · 6/6 tests · Production surface contract 6/6 · v1.0.0

Input interpretation: Enter values to calculate.
Result: —
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 · 6/6 tests · Production surface contract 6/6 · Source checked · v1.0.0
Named expert review: Optional · Not performed
Sources: Hibbeler, Engineering Mechanics: Statics — Force systems and rigid-body equilibrium
Sources
Evidence: 2 golden · 4 boundary · Production surface contract 6/6 · Artifact integrity PASS
Production: Embedded snapshot: unpublished · Build schema 1.0.0 ready · Semantic contract ✓ · Attestation report not published on origin · Live production status STALE (3 capabilities; 189 remain CURRENT) @ 2026-09-23T10:40:38.140Z
Semantic contract: PASS

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

Enter forces as fx, fy, x, y

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

Enter concentrated moments

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

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.

3-4-5 resultant

Forces 3 N and 4 N at the origin

R = (3, 0) + (0, 4)

|R| = 5 N

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

Engine tested 6/6 tests · Production surface contract 6/6
Named expert review Not performed
Calculation version 1.0.0

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
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.

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

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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.