Equal Angle Weight Calculator

Example Data Table

Sample case: n = 3, W = 100 N, θ = 30°, α = 0°, counterclockwise.

Formula Used

This tool models n concurrent vectors of equal magnitude W, separated by a constant angle step θ.

• Vector directions: α, α+θ, α+2θ, …, α+(n−1)θ
• Components of each vector: Fx = W cos(αi), Fy = W sin(αi)
• Resultant components: Rx = ΣFx, Ry = ΣFy
• Resultant magnitude: R = √(Rx² + Ry²)
• Resultant angle: φ = atan2(Ry, Rx)

For equal magnitudes at equal steps, an analytic magnitude check is:

R = W · | sin(nθ/2) / sin(θ/2) |

If the sine ratio becomes zero, the vectors form a closed polygon and the resultant can be zero.

How to Use

1. Select Forward to compute resultant from a known weight W.
2. Select Inverse to estimate W from a target resultant R.
3. Enter n, the number of equal vectors in the system.
4. Set θ (angle step) and α (start angle).
5. Choose rotation if you want θ applied clockwise.
6. Press Calculate to view results above the form.
7. Use Download CSV or Download PDF to export tables.

Technical Article

1) Meaning of equal-angle loading

Equal-angle loading describes n concurrent forces (or weights) that share a common point and are separated by a constant angular step θ. When each force has the same magnitude W, the system becomes a clean testbed for symmetry, balance checks, and quick validation of multi-direction supports.

2) Component resolution for design checks

Engineering decisions typically rely on components. Each vector resolves into Fx = W cos(αi) and Fy = W sin(αi), where αi = α + iθ. The table helps verify that peak components remain within allowable limits, especially when α shifts the entire pattern.

3) Resultant magnitude trends with θ and n

For small θ, the forces cluster and the resultant approaches R ≈ nW. As θ increases, directions spread and partial cancellation grows. For example, with n = 6 and θ = 60°, the vectors form a closed hexagon and the resultant can approach zero.

4) Analytic sine-ratio cross-check

A fast validation uses R = W · | sin(nθ/2) / sin(θ/2) |. This factor captures geometric reinforcement and cancellation without summing every vector. If sin(nθ/2) = 0, the polygon closes and ideal balance is possible, which is a useful sanity check before exporting.

5) Resultant direction and the midline

When the set does not span a full closed polygon, the resultant tends to align near the angular midline, approximately α + (n−1)θ/2. This is practical for estimating where the net load “points” before detailed analysis. Compare the midline angle to the computed direction to spot unusual configurations.

6) Forward vs inverse workflows

Forward mode answers: “Given W, what is the net resultant?” Inverse mode answers: “Given a target R, what equal W is needed?” Inverse mode uses the analytic factor, so it is most reliable when the factor is non-zero and not near a numerical singularity.

7) Practical parameter ranges

Typical use cases include n from 3 to 12 and θ from 10° to 120°, but the calculator supports broader exploration. Small θ highlights reinforcement; larger θ reveals cancellation and balance points. Always keep consistent units (N, kN, lbf, kgf) for interpretation.

8) Reporting and traceability

The exported CSV/PDF captures inputs, vector components, and the resultant summary so calculations remain reproducible. For audits, include n, α, θ, rotation direction, and precision. When reviewing changes, compare both the vector-sum resultant and the analytic cross-check to confirm stability. It also supports quick peer review during design iterations.

FAQs

1) What does “equal-angle” mean here?

It means each consecutive force direction differs by the same angle step θ. The first direction starts at α, then continues at α+θ, α+2θ, and so on.

2) Why can the resultant become zero?

If the vectors form a closed polygon, their sum cancels. This happens when the geometry makes sin(nθ/2) equal to zero, such as n=6 with θ=60°.

3) When should I use inverse mode?

Use it when you know the desired resultant magnitude R and want the equal per-vector weight W. It works best when the analytic factor is not zero or extremely small.

4) Does rotation direction change the magnitude?

Magnitude is usually unchanged because the set of angles is mirrored. However, the sign convention affects the resultant direction (clockwise vs counterclockwise progression), so the angle output can differ.

5) What is the bisector (midline) angle used for?

It is a quick estimate of where the net resultant points when the forces do not close perfectly. Comparing midline and computed direction helps detect input mistakes or unexpected symmetry.

6) How do I interpret Fx and Fy values?

Fx and Fy are the horizontal and vertical components of each weight. Their sums Rx and Ry give the resultant components, which then produce the resultant magnitude and direction.

7) What precision should I choose for reporting?

Use 3–6 decimals for most engineering reports. Increase precision when θ is small or when cancellation makes R close to zero, because small rounding differences can matter.