3.4MechanicsCore

Hooke’s Law and Elastic Strain Energy

Stretch a spring gently and it pulls back proportionally — stretch it too far, and that neat proportionality breaks down completely, sometimes permanently. Knowing exactly where that line is, and how much energy gets stored along the way, is the foundation of this whole materials topic.

40 min Video by Kit Betts-Masters Module 3.4 Materials

Hooke’s Law and Elastic Strain Energy — A Level Physics RevisionWatch the full walkthrough before the notes below.

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What you'll be able to do

State and use Hooke’s law, $F = k Δ x$
Define the elastic limit and explain what happens beyond it
Calculate the elastic strain energy stored in a stretched spring or wire
Interpret a force–extension graph, including for springs in series and parallel

Hooke’s law

$Hooke’s law$ states that the extension of a spring (or wire) is directly proportional to the force applied to stretch it, provided the $elastic limit$ is not exceeded. The constant of proportionality, $k$ , is the $spring constant$ (or stiffness), measured in N m⁻¹ — a stiffer spring has a larger spring constant.

F = k Δ x

Hooke’s law; F = applied force, k = spring constant, Δx = extension.

1Use

k = \frac{F}{Δ x}

k = \frac{8.0}{0.040}

Answer

k = 200

N m⁻¹

Tip — Beyond the elastic limit, a material no longer obeys Hooke’s law and — crucially — no longer returns fully to its original length once the force is removed; it has been permanently (plastically) deformed.

Force–extension graphs

A force–extension graph for a material obeying Hooke’s law is a straight line through the origin, with gradient equal to the spring constant $k$ . Beyond the elastic limit, the graph curves, showing the material becoming easier (or harder) to stretch further, and no longer returning along the same line if unloaded.

Springs can also be connected in series or parallel, changing the overall (combined) stiffness of the arrangement — springs in series stretch more easily (a smaller combined spring constant) than a single spring, while springs in parallel are stiffer (a larger combined spring constant).

Elastic strain energy

The energy stored in a stretched spring or wire — as long as it remains within its elastic limit — equals the work done stretching it, which is the area under its force–extension graph. Because force builds up proportionally with extension (starting from zero), this area is a triangle, giving exactly half of the final force times the final extension, not the full product.

E = \frac{1}{2} F Δ x = \frac{1}{2} k (Δ x)^{2}

Elastic strain energy stored (two equivalent forms, valid within the elastic limit).

1Use

E = \frac{1}{2} k (Δ x)^{2}

E = 0.5 \times 150 \times 0.06 0^{2} = 0.5 \times 150 \times 0.0036

Answer

E = 0.27

Tip — The formula E=½FΔx (not simply FΔx) exists precisely because the force isn’t constant while stretching — it grows from zero up to its final value, so only "half" the final force, on average, acted over the whole extension.

Equation recap

F = k Δ x

Hooke’s law.

E = \frac{1}{2} F Δ x = \frac{1}{2} k (Δ x)^{2}

Elastic strain energy stored.

Common mistakes to avoid

Using E=FΔx for elastic strain energy.

The correct formula is E=½FΔx — the factor of ½ accounts for the force building up from zero as the spring extends, exactly like the area of a triangle rather than a rectangle.

Assuming a material always returns to its original length once unloaded.

This is only true within the elastic limit — beyond it, the material undergoes permanent (plastic) deformation and does not fully recover.

Believing springs in series are stiffer than a single spring.

Springs in series are actually LESS stiff (extend more for the same force) than a single spring — it is springs in PARALLEL that increase overall stiffness.

Key takeaways

Hooke’s law: F = kΔx, valid up to the elastic limit.
Beyond the elastic limit, extension is no longer proportional to force, and deformation becomes permanent.
Elastic strain energy stored: E = ½FΔx = ½k(Δx)², found from the area under a force–extension graph.
Springs in series reduce overall stiffness; springs in parallel increase it.

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Hooke’s Law and Elastic Strain Energy

40 min Video by Kit Betts-Masters Module 3.4 Materials