Calculator Inputs
Example Data Table
| Case | Mode | Input Model | Range | Angle | Work Result |
|---|---|---|---|---|---|
| Example 1 | Constant | F = 25 N | 0 m to 8 m | 0° | 200 J |
| Example 2 | Linear | F(x) = 10 + 4x | 0 m to 8 m | 0° | 208 J |
| Example 3 | Polynomial | F(x) = 1.5x² + 2x + 5 | 0 m to 4 m | 20° | 88.27 J |
| Example 4 | Discrete | (0,5), (2,9), (4,15) | 0 m to 4 m | 0° | 38 J |
Formula Used
W = ∫ F(x) cos(θ) dx
Work equals the area under the projected force curve over displacement. The calculator first converts all inputs to consistent base units, then integrates the selected force model between the chosen limits.
Constant Force
W = F cos(θ) × Δx
Linear Force
F(x) = F₀ + m(x - x₁)
W = cos(θ)[F₀Δx + (mΔx²)/2]
Polynomial Force
F(x) = ax³ + bx² + cx + d
W = cos(θ)[aΔx⁴/4 + bΔx³/3 + cΔx²/2 + dΔx] using actual bounds.
Power Law Force
F(x) = kxⁿ
W = cos(θ) × k(x₂ⁿ⁺¹ - x₁ⁿ⁺¹)/(n+1), when n ≠ -1.
Discrete Dataset
W ≈ cos(θ) × Σ[(Fᵢ + Fᵢ₊₁)/2]Δx
This uses the trapezoidal rule for measured or simulated force values.
How to Use This Calculator
- Select a calculation mode that matches your force model.
- Choose force, distance, angle, and result units.
- Enter start and end displacement values.
- Provide the force parameters for the selected mode.
- Set the angle between force and displacement.
- Click Calculate Work Integral to generate the result.
- Review the summary table, graph, and segment details.
- Use the CSV or PDF buttons to export your output.
Frequently Asked Questions
1. What does this calculator compute?
It computes work done by a force over displacement using integration. It supports constant, linear, polynomial, power law, and discrete force data models.
2. Why does angle matter in work calculations?
Only the component of force along the displacement produces work. The calculator multiplies force by cos(θ) before integrating over the selected interval.
3. When should I use discrete dataset mode?
Use it when you have measured force values at several displacement points. The calculator estimates work with the trapezoidal rule between consecutive points.
4. Can the result be negative?
Yes. Negative work occurs when the projected force opposes motion. This commonly happens when the force angle is greater than ninety degrees.
5. Which units are supported?
Force supports N, kN, and lbf. Distance supports m, cm, mm, and ft. Results can be shown in J, kJ, and ft·lbf.
6. How accurate is the discrete method?
Its accuracy depends on data spacing and curve smoothness. More closely spaced points usually produce better estimates of the true integral.
7. What happens when n equals -1 in power law mode?
The integral becomes logarithmic. Both displacement limits must stay positive, because the formula uses the natural logarithm of x₂ divided by x₁.
8. What does the graph show?
It shows projected force on the vertical axis and displacement on the horizontal axis. The filled area under the curve represents the calculated work.