Spring Force & Elastic Work
When you stretch a rubber band or compress a spring, you are doing work against an elastic restoring force. Unlike gravity or friction acting over a constant distance, the force a spring exerts changes continuously as it is stretched or compressed — it grows larger the further you displace it. This makes springs a classic example of a variable force, and calculating the work done requires a different approach from the simple formula W = Fd used for constant forces. Robert Hooke discovered in the 17th century that, for many elastic materials, the restoring force is directly proportional to the displacement from the natural (equilibrium) length. This relationship, known as Hooke's Law, allows us to write the spring force as F = −kx, where k is the spring constant and x is the displacement. Because the force varies linearly with displacement, we can use the area under a force–displacement graph — a triangle — to calculate the work done, giving W = ½kx². Understanding elastic work is essential across physics and engineering: from the energy stored in a car's suspension springs to the mechanics of a bow and arrow, or the vibrations of atoms in a crystal lattice. The elastic potential energy stored in a deformed spring can be released to do useful work, making springs fundamental energy-storage devices in everyday technology.
- Hooke's Law — the spring's restoring force
- Why we cannot use W = Fd directly
- Graphical method — area under the F–x graph
- Elastic potential energy
- Signs and directions
Hooke's Law — the spring's restoring force
A spring in its natural state exerts no force. When stretched or compressed by a displacement x from equilibrium, it pushes or pulls back toward equilibrium. Hooke's Law states this restoring force is F = −kx. The negative sign tells us the force always opposes the displacement (it is a restoring force). The constant k, called the spring constant or stiffness, has SI units of newtons per metre (N m⁻¹). A stiff spring (large k) requires a large force for a small extension; a soft spring (small k) stretches easily.
Why we cannot use W = Fd directly
The formula W = Fd only works when the force is constant throughout the displacement. For a spring, the force grows from 0 at x = 0 to kx at full extension x. Using the maximum force would overestimate the work; using zero would underestimate it. The correct approach is integration: we sum up the tiny contributions dW = F dx over the entire displacement. Because F = kx varies linearly, this integral has a neat geometric interpretation — it equals the area of a triangle on the F–x graph.
Graphical method — area under the F–x graph
Plot the magnitude of the applied force (equal to kx) against displacement x. The graph is a straight line through the origin with gradient k. The work done in stretching the spring from x = 0 to x = X is the area of the triangle formed under this line: Area = ½ × base × height = ½ × X × kX = ½kX². This graphical method works for any variable force: the work done equals the area under the force–displacement curve. For a non-linear spring or any other variable force, you would need to find the area numerically or by integration.
Elastic potential energy
The work done against the spring force is stored as elastic potential energy (EPE) in the spring. Provided the elastic limit is not exceeded (i.e. the material returns to its original shape), this energy is fully recoverable. When the spring is released, it converts EPE back into kinetic energy of whatever object it is attached to. The formula EPE = ½kx² is therefore both the work done on the spring and the energy stored within it. Exceeding the elastic limit permanently deforms the spring, and the F–x graph is no longer linear — Hooke's Law no longer applies.
Signs and directions
When an external agent stretches or compresses a spring, it applies a force in the same direction as the displacement, so the work done by the external agent is positive: W_ext = +½kx². The spring itself exerts a force opposite to the displacement, so the work done by the spring on the agent is negative: W_spring = −½kx². Energy conservation is maintained: the positive work done by the agent equals the elastic potential energy stored, which equals the magnitude of the negative work done by the spring force.
The Key Equation
- W — Work done / Elastic potential energy (J (joules)): The energy stored in the spring due to its deformation, equal to the work done by the external force stretching or compressing it.
- k — Spring constant (stiffness) (N m⁻¹ (newtons per metre)): A measure of the spring's stiffness. A larger k means the spring is harder to deform; determined experimentally for each spring or elastic material.
- x — Displacement from equilibrium (m (metres)): The extension or compression of the spring measured from its natural (unstretched) length. Must be within the elastic limit for Hooke's Law to apply.
Worked Example
Problem. A spring has a spring constant of 400 N m⁻¹. A student compresses it by 0.15 m from its natural length and holds it stationary. (a) Calculate the force required to hold the spring at this compression. (b) Calculate the elastic potential energy stored in the spring. (c) The student then pushes it a further 0.05 m to a total compression of 0.20 m. Calculate the additional work done in this second stage.
Answer. Force = 60 N; EPE at 0.15 m = 4.5 J; additional work from 0.15 m to 0.20 m = 3.5 J
Origins · 1 minRobert Hooke and how this was discovered
Robert Hooke formulated what we now call Hooke's Law in 1676, originally published as an anagram ('ceiiinosssttuv') and later revealed as 'ut tensio, sic vis' — Latin for 'as the extension, so the force.' He demonstrated that elastic materials produce a restoring force proportional to deformation, laying the mathematical foundation for spring mechanics, elasticity theory, and the analysis of elastic potential energy. Hooke was one of the most versatile scientists of the 17th century — he coined the word 'cell' in biology after observing cork under a microscope, designed buildings in London after the Great Fire of 1666, and was a fierce rival of Isaac Newton. Despite his enormous contributions, no confirmed portrait of Hooke survives; the image commonly associated with him is disputed by historians.
Related Concepts
- Work Done by a Constant Force — Elastic work extends the constant-force formula W = Fd to variable forces; understanding the constant case first makes the graphical area method easier to motivate.
- Kinetic Energy and the Work–Energy Theorem — When a compressed spring is released, elastic potential energy converts to kinetic energy; the work–energy theorem connects the elastic work stored to the resulting motion.
- Gravitational Potential Energy — Like elastic PE, gravitational PE is stored energy resulting from work done against a conservative force; comparing both deepens understanding of potential energy as a concept.
- Conservation of Mechanical Energy — Spring systems are ideal for applying conservation of energy, where elastic PE converts to kinetic energy and back, assuming no energy is lost to friction.
- Variable Forces and Graphical Methods — The area-under-the-graph technique used to find elastic work is the same general method applied to all variable forces; spring force is the clearest worked example of this approach.