- Polar to components: Fx = F·cos(θ), Fy = F·sin(θ)
- Component sums: ΣFx = Σ(Fx), ΣFy = Σ(Fy)
- Resultant magnitude: R = √((ΣFx)² + (ΣFy)²)
- Resultant direction: θ = atan2(ΣFy, ΣFx)
- Equilibrant: same magnitude, θ + 180°
- Pick units and your angle input type.
- For each row, choose polar or component entry.
- Fill the required fields for that entry mode.
- Add more rows for additional forces as needed.
- Press Solve to view totals and direction.
| Label | Mode | F | θ | Fx | Fy |
|---|---|---|---|---|---|
| F1 | Polar | 120 | 25° | 108.76 | 50.71 |
| F2 | Polar | 80 | 140° | -61.28 | 51.42 |
| F3 | Components | — | — | -30 | 60 |
Vector decomposition and sign convention
Each applied force is treated as a 2D vector acting at one point. Polar inputs are resolved using Fx = F·cosθ and Fy = F·sinθ, where θ is measured counterclockwise from the positive x-axis. Component inputs accept signed values directly, so leftward or downward forces must be negative. Consistent signs ensure the summed components represent the true net action.
Why concurrent systems use component sums
For concurrent forces, the net effect depends only on the vector sum, not the application location. The solver therefore computes ΣFx and ΣFy by adding every row’s components. This approach matches free‑body diagram practice and avoids geometry errors when several forces share one joint. If ΣFx and ΣFy are both near zero, the point is close to translational equilibrium.
Interpreting resultant magnitude
The resultant magnitude R is computed as R = √((ΣFx)² + (ΣFy)²). R represents the single equivalent force that produces the same net translation as the entire set. Engineers often compare R to allowable limits or to an expected load path. A large R with small individual forces usually indicates many vectors are aligned rather than cancelling.
Direction handling and quadrant logic
Direction is obtained with θ = atan2(ΣFy, ΣFx) so the quadrant is correct even when ΣFx is negative or zero. The calculator also reports θ in a 0–360° format for easy sketching. When using radians for input, the conversion is applied only during decomposition; reporting remains in degrees to keep the angle intuitive.
Equilibrant and static balance checks
The equilibrant is the force that would balance the system: it has the same magnitude as the resultant but points 180° opposite. In statics, adding the equilibrant to the original set drives ΣFx and ΣFy toward zero. This is useful for verifying tie‑rod directions, selecting counterweights, or checking whether a support reaction should oppose the computed resultant.
Export-ready reporting considerations
For documentation, the calculator stores the normalized component table along with totals, direction, and equilibrant. CSV export supports quick import into spreadsheets for further checks, while PDF export provides a compact record for reports. Use rounding appropriate to your measurement resolution; excessive precision can hide uncertainty, while too little can mask small imbalances. Add labels to track cables, loads, and test cases.
1) What does “concurrent forces” mean?
It means all forces act through the same point, such as a pin joint. In that case, the net translation is governed by the vector sum of all force components.
2) Which angle convention does this solver use?
Angles are measured counterclockwise from the +X axis. Positive Fy points upward and negative Fy points downward. This convention matches standard engineering free‑body diagrams.
3) Why is atan2 used for direction?
atan2(ΣFy, ΣFx) preserves the correct quadrant when ΣFx is negative or near zero. Using a simple arctan ratio can misreport angles for vectors in quadrants II, III, or IV.
4) What is the equilibrant and when is it useful?
The equilibrant is equal in magnitude to the resultant but opposite in direction. Adding it to the system should produce near‑zero component sums, which helps validate support reactions and balance checks.
5) Can I mix polar and component rows?
Yes. Each row can be entered as (F, θ) or as (Fx, Fy). The solver normalizes all entries to components before summing, so mixed input types are handled consistently.
6) How should I choose rounding for results?
Match rounding to your measurement accuracy and reporting needs. For most lab or field data, 3–4 decimals is enough; extra precision can imply certainty that the inputs do not support.