Force Equilibrium
For two known forces P, Q at an angle, compute the resultant magnitude, direction and the equilibrant (equal & opposite) force.
Inputs
Formula
R = √(P² + Q² + 2PQ cosα); Equilibrant Feq = R (opposite)
Fundamentals
Equilibrium means net force and net moment are both zero — the body is at rest or in uniform motion.
Conditions $\sum F_x=0,\sum F_y=0,\sum M=0$. Three concurrent forces in equilibrium obey Lami's theorem: $F_1/\sin\alpha_1=F_2/\sin\alpha_2=F_3/\sin\alpha_3$.
History
Static-equilibrium conditions were founded by Archimedes and Varignon's theorem (1687).
Engineering applications
Used to solve forces in suspensions, brackets, cableways and static structures.
Glossary
| Equilibrium | Zero net force and zero net moment. |
| Lami's theorem | Each of 3 concurrent forces ∝ sine of opposite angle. |
| Concurrent | Forces whose lines meet at one point. |
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, Resultant dir. θ, Equilibrant Feq in the results area.
Formula notesThree-force equilibrium (Lami): F₁/sinα₁ = F₂/sinα₂ = F₃/sinα₃, α = opposite angles.
Formula · Worked Example · Knowledge
Formula
Resultant $R=\sqrt{P^2+Q^2+2PQ\cos\alpha}$; equilibrant $F_{eq}=R$ opposite in direction.
Worked Example
P=100, Q=80, α=60° → R≈148.4 N; equilibrant 148.4 N opposite.
Key Points
- Equilibrium needs zero resultant (∑F=0).
- Equilibrant equals and opposes the resultant.
- Multi-force equilibrium: pair up or set coordinate sums.
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 |
| Resultant dir. θ | theta | ° |
| Equilibrant Feq | Feq | N |
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); Equilibrant Feq = R (opposite)
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