Work Done by a Variable Force and Potential Energy Curves

When the Force Keeps Changing

Work done is force times distance, W = Fs, but only if the force stays the same. Stretching a spring, lifting a rocket out of the Earth's gravity, or pushing two charges together all involve forces that change as the object moves. To find the work done by a variable force, you either find the area under a force–displacement graph or integrate. This tutorial covers both methods, the work done on springs and against gravity far from the Earth, the link between force and potential energy, and how to read potential energy curves to find equilibrium points and turning points. It is aimed at A-level-style and first-year university physics (PHY 101).

Work Done by a Variable Force: Area Under the Graph

For a constant force, the graph of F against displacement x is a horizontal line, and the area under it, F × s, is the work done. The same is true for any force: the work done equals the area under the force–displacement graph.

Split the motion into many tiny steps Δx. Over each step the force is almost constant, so the work is about F Δx, a thin strip of area. Adding all the strips gives the total area. In the limit of infinitely thin strips, this becomes an integral:

W = ∫ F dx, from the starting position to the final position.

When the graph is made of straight lines, simple shapes are enough. A force that rises steadily from 0 to 20 N over 3 m does work equal to the triangle's area: ½ × 3 × 20 = 30 J.

Work Done Stretching a Spring

By Hooke's law, the force needed to stretch a spring by x is F = kx, where k is the spring constant. The force grows from 0 as the spring stretches, so the force–extension graph is a straight line through the origin.

The work done stretching it from 0 to x is the area of the triangle, or by integration:

W = ∫ kx dx from 0 to x = ½kx².

This work is stored as elastic potential energy. Since F = kx, it can also be written W = ½Fx.

Example 1. A spring with k = 200 N/m is stretched by 0.15 m. Find the energy stored.

E = ½ × 200 × 0.15² = ½ × 200 × 0.0225 = 2.25 J.

Example 2. The same spring is stretched further, from 0.15 m to 0.25 m. How much extra work is needed?

W = ½k(0.25² − 0.15²) = 100 × (0.0625 − 0.0225) = 4.0 J. Notice that this is more than the first 0.15 m needed, even though the extra distance is shorter, because the force is larger further out.

Example 3: A Force That Depends on Position

A force F = 3x² + 2 newtons acts on an object as it moves along the x-axis from x = 1 m to x = 3 m. Find the work done.

W = ∫(3x² + 2) dx from 1 to 3 = [x³ + 2x] from 1 to 3 = (27 + 6) − (1 + 2) = 30 J.

Work Done Against Gravity Far from the Earth

Near the Earth's surface, the weight mg is constant and the work done lifting a mass through h is mgh. But for satellites and rockets, gravity weakens with distance. Newton's law gives F = GMm/r², where r is the distance from the Earth's centre.

The work done moving a mass from r₁ to r₂ against this force is

W = ∫ GMm/r² dr from r₁ to r₂ = GMm(1/r₁ − 1/r₂).

Example 4. How much work is needed to lift a 1,000 kg satellite from the Earth's surface (r₁ = 6.4 × 10⁶ m) to a height of 600 km (r₂ = 7.0 × 10⁶ m)? Take GM = 4.0 × 10¹⁴ N m² kg⁻¹.

W = 4.0 × 10¹⁴ × 1000 × (1/6.4 × 10⁶ − 1/7.0 × 10⁶) = 4.0 × 10¹⁷ × (1.5625 × 10⁻⁷ − 1.4286 × 10⁻⁷) = 4.0 × 10¹⁷ × 1.339 × 10⁻⁸ ≈ 5.36 × 10⁹ J.

Using mgh with g = 9.8 m/s² would give 1000 × 9.8 × 600,000 = 5.88 × 10⁹ J, an overestimate of about 10% because it ignores the weakening of gravity with height.

Taking r₂ to infinity gives the work needed to escape completely: GMm/r₁. Setting this equal to ½mv² gives the escape velocity, v = √(2GM/r₁) ≈ 11 km/s.

Force and Potential Energy

For a conservative force (such as gravity, a spring force or the electric force), the work done by the force equals the decrease in potential energy U. In one dimension this gives a very useful link:

F = −dU/dx.

The force is minus the gradient of the potential energy graph: it always pushes the object towards lower potential energy, like a ball rolling downhill.

Spring: U = ½kx², so F = −kx, the restoring force pointing back towards x = 0.
Gravity near the surface: U = mgh, so F = −mg, downwards.
Gravity far away: U = −GMm/r, so F = −GMm/r², attractive.

Going the other way, U(x) = −∫ F dx, plus a constant that fixes where U = 0.

Reading Potential Energy Curves

A graph of U against x tells you how an object will move.

Equilibrium occurs where the gradient is zero, because then F = −dU/dx = 0.
At a minimum of U, equilibrium is stable: a small push produces a force back towards the minimum, so the object oscillates about it. A marble at the bottom of a bowl is like this.
At a maximum of U, equilibrium is unstable: a small push produces a force away from it. A marble balanced on top of an upturned bowl is like this.
If the total energy is E, the object can only be where U ≤ E, because kinetic energy E − U cannot be negative. The points where U = E are turning points, where the object stops momentarily and reverses.

Example 5. A particle has potential energy U = x³ − 3x joules. Find its equilibrium positions and their stability.

dU/dx = 3x² − 3 = 0 gives x = ±1 m. The second derivative is 6x: at x = 1 it is positive, so U has a minimum and the equilibrium is stable; at x = −1 it is negative, so U has a maximum and the equilibrium is unstable. The force at x = 2 is F = −(3 × 4 − 3) = −9 N, pointing back towards the stable point at x = 1.

The Interatomic Potential

The potential energy of two atoms against their separation r has a characteristic shape: very steep and positive at small r (strong repulsion when electron clouds overlap), a minimum at the equilibrium separation r₀ (typically about 0.1 to 0.3 nm), and rising towards zero at large r (weak attraction). This single curve explains a lot:

Atoms in a solid vibrate about r₀, the minimum of the curve.
The depth of the minimum is the energy needed to separate the atoms, related to the latent heat and the bond energy.
The curve is not symmetric: it is gentler on the far side. As the atoms vibrate more strongly at higher temperatures, their average separation increases, which explains thermal expansion.
Near the minimum the curve is close to a parabola, so the force is approximately proportional to displacement. This is why solids obey Hooke's law for small stretches.

Summary

Work done by a variable force = area under the force–displacement graph = ∫ F dx.
Energy stored in a spring = ½kx² = ½Fx.
Work against gravity from r₁ to r₂ = GMm(1/r₁ − 1/r₂); mgh is only a near-surface approximation.
F = −dU/dx: the force points down the potential energy slope.
Minima of U are stable equilibria, maxima are unstable, and turning points occur where U equals the total energy.