Force-vector triangle and allowable envelopeLive current inputs
Unit Converters
Force Comparison Calculator
Compare two forces as vectors rather than unrelated magnitudes. Resolve signed components, reconstruct the resultant and direction, estimate moments from entered perpendicular radii, and check both translational and rotational allowables on one free-body sketch.
Input evidence: define one coordinate system, positive rotation, point of application, free-body boundary, load case, and allowable basis before comparing forces.
Resultant force-
Resultant direction-
Force difference-
Vector angle separation-
Net moment proxy-
Envelope status-
Vector comparison plane
Compare magnitude, direction, resultant, and moment in one load case
The free-body sketch preserves direction; the ledger reconciles rectangular components and limits.
| Vector | Magnitude | Direction | x component | y component | Moment proxy | Decision use |
|---|
How to use
Compare two forces only after defining one free body and coordinate system
- Draw the body or joint being assessed and choose the positive x-axis and angle direction.
- Enter both force magnitudes and directions using that same convention.
- Enter application radii only when they represent signed perpendicular moment arms under one declared reference point.
- Compare the reconstructed resultant and net moment with compatible allowable values.
- Review the component ledger and force triangle, then repeat for each independent load case rather than mixing mutually exclusive events.
Vector fundamentals
Magnitude, direction, components, resultant, and moment are different quantities
Two equal force magnitudes can reinforce, cancel, or redirect one another. A valid comparison must keep vector direction and moment geometry attached to each load.
Free-body rule: forces from different bodies, reference frames, or load cases cannot be added merely because they share units.
Calculation method
Resolve components, sum them, then reconstruct the resultant
Each force is projected onto the common axes. Signed components are added, and the resultant magnitude and direction are reconstructed from those sums. Moment uses the entered signed arm convention and is evaluated separately.
Rₓ = A cos α + B cos β; Rᵧ = A sin α + B sin β; R = √(Rₓ² + Rᵧ²)The resultant direction is atan2(Rᵧ, Rₓ); the moment proxy follows the entered signed radius convention.
- Use atan2 rather than a simple arctangent so the quadrant is retained.
- Do not add magnitudes unless the forces are collinear and act in the same direction.
- Angle separation describes orientation, not the size of the resultant.
- Check component sums as the reconciliation basis for the force triangle.
Coordinate discipline
An angle has no meaning without an axis and sign convention
Record whether angles are global or local, clockwise or counterclockwise, and whether the coordinate frame rotates with the body.
- Convert all directions to one convention before entry.
- Retain signed components for audit.
- Check near-180° cancellation with adequate precision.
- Transform local loads before combining them globally.
Moment interpretation
Radius is not automatically a perpendicular lever arm
True moment magnitude depends on the cross product of position and force. The entered radius is valid only when it already represents the signed perpendicular distance.
- Choose and document the moment reference point.
- Use consistent clockwise and counterclockwise signs.
- Do not combine forces and moments with incompatible origins.
- Use a full vector moment model when line-of-action geometry matters.
Decision interpretation
Resultant force and net moment can govern different failure modes
A low resultant can coexist with a large force difference or moment. Conversely, forces may create a large resultant while their signed moments partially cancel.
How to read the force triangle
The plotted arrows show Force A, Force B placed head-to-tail, and the closing resultant; the allowable envelope provides scale context. Editing magnitude changes arrow length and editing direction rotates it. The diagram can mislead when the forces act at different points, in different planes, or under incompatible load cases.
Load combinations and allowable checks
Compare like load cases with like allowables
Allowable force and moment must correspond to the same component, direction, duration, safety basis, and environmental condition as the modeled loads.
- Keep service, proof, ultimate, fatigue, impact, and accidental cases separate.
- Apply prescribed load factors before or after combination according to the governing method.
- Do not treat a scalar envelope as an interaction equation unless it was defined that way.
- Escalate to structural analysis when stiffness, deformation, contact, or stress distribution governs.
Detailed calculation process
Resolve every force and reconcile the resultant
1. Force A components: Aₓ = A cos α; Aᵧ = A sin α.
2. Force B components: Bₓ = B cos β; Bᵧ = B sin β.
3. Resultant components: Rₓ = Aₓ + Bₓ; Rᵧ = Aᵧ + Bᵧ.
4. Resultant: R = √(Rₓ² + Rᵧ²); θ = atan2(Rᵧ, Rₓ).
5. Moment proxy: M = Arₐ + Brᵦ under the entered signed perpendicular-arm convention.
- A, B
- force magnitudes; N
- α, β
- force directions from positive x; degrees
- Aₓ, Aᵧ
- Force A signed components; N
- Bₓ, Bᵧ
- Force B signed components; N
- Rₓ, Rᵧ
- resultant signed components; N
- R, θ
- resultant magnitude in N and direction in degrees
- rₐ, rᵦ
- signed perpendicular moment arms; m
Default substitution and reconciliation
Aₓ = 850 cos 20° = 798.739 N and Aᵧ = 290.717 N. Bₓ = 620 cos 135° = -438.406 N and Bᵧ = 438.406 N. Thus Rₓ = 360.333 N, Rᵧ = 729.123 N, and R = √(360.333² + 729.123²) = 813.272 N at 63.71°. The closing vector equals the component sum, reconciling the force triangle.
Evidence requirements
Retain the load case and geometry
- Free-body diagram, coordinate axes, angle origin, and sign convention
- Force magnitudes, directions, uncertainty, duration, and source
- Application points, lines of action, moment reference, and lever-arm derivation
- Allowable source, load factors, environmental condition, and revision
Scope and limitations
What this planar static comparison excludes
- Three-dimensional vectors, distributed loads, couples, and changing geometry
- Acceleration, impact, vibration, fatigue, buckling, and time history
- Stiffness, displacement, contact, stress concentration, and material nonlinearities
- A full cross-product moment unless the entered arms are perpendicular and signed
Key terminology
Force-comparison glossary
- Free body
- The isolated body on which all external loads are shown.
- Component
- Signed projection of a vector onto an axis.
- Resultant
- Single vector equal to the vector sum of applied forces.
- Line of action
- Infinite line along which a force acts.
- Moment arm
- Perpendicular distance from a reference point to a line of action.
- Load case
- A defined set of loads assumed to act together.
Practical examples
Two different comparison uses
Bracket load case
An engineer combines two simultaneous planar cable forces at a bracket, checks the resultant against the joint rating, and separately evaluates the signed moment about the mounting face.
Mutually exclusive actuator states
A machine has extend and retract loads that never occur together. The analyst runs two separate cases instead of adding them, preventing a fictitious resultant.
Important note
Before relying on this result
The planar static model excludes supports, distributed loads, 3D geometry, dynamics, deformation, fatigue, and code combinations.
Additional Force Comparison Calculator questions
Why can the resultant be smaller than either force?
Vectors may oppose in one or both components.
Which angle convention is used?
Degrees counterclockwise from positive x.
Is the moment calculation complete equilibrium?
No. It uses entered perpendicular radii as a planning proxy.
Can three-dimensional forces be entered?
No. This model is planar.