Work by Variable Force
When a constant force acts on an object, calculating work is straightforward: multiply force by displacement. But in the real world, forces rarely stay constant. A spring pushes harder the more you compress it; a rocket engine produces different thrust as fuel burns; air resistance changes with speed. In all these situations, the force varies with position, and we need a more powerful method to calculate the work done. The solution lies in mathematics — specifically, in the idea of integration — and in a beautifully intuitive graphical technique. The key insight is that work done by a variable force equals the area under a force–displacement graph. Plot force F on the vertical axis and displacement x on the horizontal axis. The area of the region between the curve and the x-axis, from your starting position to your finishing position, gives the total work done. This works because you can imagine slicing that region into many thin vertical strips. Each strip has width Δx and height F(x), so its area F(x)Δx approximates the tiny bit of work done over that small displacement — exactly matching the definition W = F·d for a nearly-constant force over a tiny step. As the strips become infinitely thin, the approximation becomes exact. Mathematically, this is the definite integral: W = ∫F(x)dx evaluated between the initial and final positions. This connects the graphical method to calculus, confirming that area under a force–displacement curve is not just a visual trick but a precise physical result. Understanding this principle unlocks the analysis of springs, molecular bonds, and almost every realistic mechanical system.
- Why constant-force methods fail
- Slicing into small steps
- Summing all the rectangles
- Taking the limit — the definite integral
- Reading graphs in practice
- Sign convention
Why constant-force methods fail
The simple formula W = Fd assumes F is constant throughout the displacement. If F changes — even smoothly — using a single average force carelessly introduces errors. We need to account for exactly how F varies with position x.
Slicing into small steps
Divide the total displacement from x₁ to x₂ into N tiny intervals, each of width Δx. Over one tiny interval, the force is approximately constant at value F(xᵢ). The work done in that step is ΔW ≈ F(xᵢ)Δx. This is just the area of a thin rectangle on the F–x graph.
Summing all the rectangles
Total work W ≈ Σ F(xᵢ)Δx. Visually, this is the sum of all the rectangle areas stacking up under the curve. The approximation improves as Δx shrinks, because each rectangle hugs the curve more tightly. This sum is called a Riemann sum — the foundation of integral calculus.
Taking the limit — the definite integral
As Δx → 0 and N → ∞, the Riemann sum converges to the exact area under the curve: W = ∫[x₁ to x₂] F(x) dx. This definite integral gives the precise work done by a variable force. On a graph, it is simply the shaded area between the F(x) curve, the x-axis, and the two vertical lines at x₁ and x₂.
Reading graphs in practice
You don't always need calculus explicitly. For straight-line F–x graphs (like an ideal spring, F = kx), the area is a triangle: W = ½ × base × height. For more complex shapes, you can count grid squares on graph paper, use the trapezoid rule, or apply known geometric formulas. The rule is universal: area under F–x graph = work done, regardless of the shape of the curve.
Sign convention
Area above the x-axis (force in the direction of displacement) counts as positive work — the force adds energy to the object. Area below the x-axis (force opposing displacement) counts as negative work — energy is removed. Net work is the algebraic sum of positive and negative areas.
The Key Equation
- W — Work (Joule (J)): The total energy transferred to or from the object by the force over the displacement.
- F(x) — Force as a function of displacement (Newton (N)): The magnitude of the force at each position x; this may follow any mathematical rule (e.g. F = kx for a spring).
- x — Displacement (Metre (m)): The position variable along the direction of motion; integration is performed along this axis.
- x_1,\, x_2 — Initial and final positions (Metre (m)): The limits of integration marking where the displacement begins and ends.
Worked Example
Problem. A spring has spring constant k = 200 N/m. It is compressed from its natural length by 0.10 m, then compressed a further 0.20 m (so it moves from x = 0.10 m to x = 0.30 m compression). Calculate the work done by the applied force compressing the spring over this additional displacement. Use F = kx.
Answer. The work done compressing the spring over this displacement is 8.0 J.
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
Leibniz independently developed calculus alongside Newton, and crucially, it is his notation — ∫ for the integral and dx for an infinitesimal — that physicists and mathematicians use to this day. The expression W = ∫F(x)dx, used to compute work done by variable forces, is written entirely in Leibniz's notation. He recognised that integration is the process of summing infinitely many infinitesimal products, directly corresponding to the 'area under a curve' interpretation that makes graphical methods work. The elongated 'S' symbol ∫ that Leibniz invented stands for 'summa' — Latin for 'sum' — a direct reference to his vision of integration as an infinite summation of tiny rectangles.
Related Concepts
- Work Done by a Constant Force — The variable-force method generalises the constant-force formula W = Fd; understanding the simpler case first makes the area-under-graph approach much clearer.
- Elastic Potential Energy — The energy stored in a spring is derived by computing work done by a variable force F = kx, giving the classic result E = ½kx² — a direct application of the area-under-graph method.
- Work–Energy Theorem — The work–energy theorem states that net work equals the change in kinetic energy; when forces vary, calculating that net work requires the integral method covered here.
- Power — Power is the rate of doing work; when force and velocity both vary, computing total work via integration is a prerequisite for finding average and instantaneous power.