Force Composition
Compute resultant magnitude and direction of two concurrent forces P, Q at angle α using the parallelogram law (law of cosines).
Inputs
Formula
R = √(P² + Q² + 2PQ cosα); β = atan(Q·sinα/(P + Q cosα))
Fundamentals
Force composition replaces several concurrent forces by one resultant — the most basic force operation.
Parallelogram (or polygon) law: $R=\sqrt{F_1^2+F_2^2+2F_1F_2\cos\theta}$ for two forces at angle $\theta$. Orthogonal resolution then summing is often easier.
History
The parallelogram law of forces was set by Stevin (1586) and Newtonian mechanics.
Engineering applications
Used in force analysis of trusses, cables, inclined planes and mechanisms.
Glossary
| Resultant $R$ | Single force equivalent to several. |
| Parallelogram law | Diagonal of the parallelogram on the two forces. |
| Resolution | Project forces on axes, then sum. |
How to use
- Fill in Force P, Force Q, Angle α in the Inputs section (watch the unit on each field).
- Click Calculate; the tool evaluates the formula shown above.
- Read Resultant R, Direction β in the results area.
Formula notesForce composition (angle θ): R = √(F₁²+F₂²+2·F₁·F₂·cosθ); angle tanφ = F₂·sinθ/(F₁+F₂·cosθ).
Formula · Worked Example · Knowledge
Formula
Resultant of two concurrent forces $R=\sqrt{P^2+Q^2+2PQ\cos\alpha}$, direction $\beta=\arctan(Q\sin\alpha/(P+Q\cos\alpha))$.
Worked Example
P=100, Q=80, α=60° → R≈148.4 N, β≈26.3°.
Key Points
- The resultant is the vector sum of components.
- Larger angle → resultant near the difference.
- Use polygon/analytic method for many forces.
Parameters
Inputs
| Parameter | Symbol | Unit | Default |
|---|---|---|---|
| Force P (N) | P | N | 100 |
| Force Q (N) | Q | N | 80 |
| Angle α (°) | alpha | ° | 60 |
Outputs
| Result | Symbol | Unit |
|---|---|---|
| Resultant R | R | N |
| Direction β | beta | ° |
Applications
- Simply-supported/cantilever reactions
- Structural equilibrium analysis
- Shear & moment distribution
FAQ
What formula does this tool use?
This tool computes per ISO / AGMA / ASME standard formulas: R = √(P² + Q² + 2PQ cosα); β = atan(Q·sinα/(P + Q cosα))
How accurate are the results?
Results match input precision, based on SI units and common engineering approximations; for critical duty re-check with a safety factor.
Where is it used?
Simply-supported/cantilever reactions