F = kx — Hooke's Law, Spring Constant and Worked Examples

Stretch a spring twice as far and it pulls back with twice the force. That proportionality — F = −kx — is Hooke's Law, and it's deceptively important. It's not just about springs. It describes the restoring force in pendulums (for small angles), the vibration of atoms in a crystal lattice, the behaviour of elastic materials before they yield, and the oscillations in electrical LC circuits. Any system where a restoring force is proportional to displacement is a Hooke's Law system, and those systems all behave the same way mathematically.

The negative sign in F = −kx is doing real work: it tells you the force always opposes the displacement. Pull the spring right and the force points left. That's what makes it a restoring force, and that's what makes simple harmonic motion inevitable once you pull and release.

In this article

Hooke's Law — statement, formula, and what the spring constant k tells you physically
Elastic potential energy stored in a spring: E = ½kx²
The elastic limit — what happens when you stretch too far
How Hooke's Law connects to simple harmonic motion and why springs oscillate

The Spring Constant k

k = F / x (N/m)

Large k = stiff spring (large force for small extension). Small k = soft spring. k is determined by the spring's material, wire thickness, coil diameter, and number of coils.

Use our Hooke's Law Calculator to solve F = kx for force, spring constant, or extension instantly, and to compute elastic potential energy.

Spring type	Typical k (N/m)
Slinky toy	~1
Mattress spring	~10,000
Car suspension	15,000–30,000
Stiff engineering spring	>100,000

Elastic and Plastic Deformation

A graph of force vs extension is linear in the elastic region (Hooke's Law holds). Beyond the elastic limit, the graph curves. Beyond the yield point, permanent (plastic) deformation occurs — the spring does not return to its natural length.

Elastic vs Plastic Deformation

Elastic: material returns to original shape when force removed. Hooke's Law applies within the elastic limit.

Plastic: permanent shape change — material does not return to original dimensions. Occurs beyond the elastic limit. Stretching a spring past its elastic limit, bending metal, or squashing clay are examples.

Worked Examples

Example 1: Finding extension

k = 400 N/m, F = 60 N applied.

x = F/k = 60/400 = 0.15 m

Example 2: Finding spring constant

2 kg mass extends spring 8 cm (0.08 m).

F = mg = 19.6 N; k = 19.6/0.08 = 245 N/m

Example 3: Car suspension

k = 25,000 N/m. Force to compress 3 cm:

F = kx = 25,000 × 0.03 = 750 N

Elastic Potential Energy

A stretched/compressed spring stores energy equal to the area under the F-x graph (a triangle):

EPE = ½kx²

EPE scales with x² — double the extension, four times the energy. When released from extension A, all EPE converts to kinetic energy:

½kA² = ½mv² → v_max = A√(k/m)

Worked Example: Elastic Potential Energy and Maximum Speed

A 0.3 kg mass on a spring (k = 150 N/m) is compressed 0.1 m and released. Find the elastic potential energy stored and the mass's maximum speed.

EPE = ½kA² = 0.5 × 150 × 0.1² = 0.75 J

v_max = A√(k/m) = 0.1 × √(150/0.3) = 0.1 × 22.36 = 2.24 m/s

All 0.75 J of stored elastic energy converts to kinetic energy at the instant the mass passes through the spring's natural length — the point of maximum speed, since all the restoring force's work has already been done.

Springs in Series and Parallel

Series (same force, extensions add): 1/k_eff = 1/k₁ + 1/k₂ → k_eff less than smallest.

Parallel (same extension, forces add): k_eff = k₁ + k₂ → k_eff greater than largest.

Worked Example: Combining Two Springs

Two springs, k₁ = 200 N/m and k₂ = 300 N/m, are combined. Find the effective spring constant (a) in series and (b) in parallel.

(a) Series: 1/k_eff = 1/200 + 1/300 = 0.00833 → k_eff = 120 N/m

(b) Parallel: k_eff = 200 + 300 = 500 N/m

In series, the combination is softer than either spring alone (120 N/m, less than k₁ = 200) — each spring stretches under the same force, so extensions add up, requiring less total force for the same stretch. In parallel, the combination is stiffer than either spring alone (500 N/m, more than k₂ = 300) — both springs resist the same extension together, so their forces add.

Connection to Simple Harmonic Motion

Hooke's Law (F = −kx) is the condition that produces SHM. Newton's second law: ma = −kx → a = −(k/m)x. This is the SHM equation with ω = √(k/m) and period T = 2π√(m/k). Any system with a Hooke's Law-type restoring force (pendulum for small angles, molecular bonds, LC circuits) exhibits approximately SHM.

Real-World Applications

Suspension systems: car springs and shock absorbers use Hooke's Law to absorb road impacts. Spring k is chosen to match the car's mass for optimal ride frequency. Spring scales: the extension of a calibrated spring directly measures force/weight. Seismometers: use mass-spring systems to detect ground motion — the spring force resists acceleration, allowing tiny movements to be amplified and recorded. Atomic force microscopes (AFM): a cantilever tip deflects by Hooke's Law amounts (nanometres) as it scans atom-by-atom across a surface, creating atomic-resolution images.

Hooke's Law in Materials Science

Hooke's Law in one dimension (F = kx for springs) extends to three dimensions as the theory of elasticity. For a uniform material under uniaxial stress: stress σ = E × strain ε, where E is Young's modulus (Pa) — the materials-science equivalent of the spring constant. Young's modulus for steel ≈ 200 GPa, aluminium ≈ 70 GPa, rubber ≈ 0.01–0.1 GPa. The relationship σ = Eε is linear (Hookean) up to the proportional limit; above this, more complex plastic behaviour dominates. Engineers use finite element analysis (FEA) to solve the three-dimensional generalisation of Hooke's Law for complex component geometries and loading conditions.

Common Mistakes and Exam Tips

Always convert to SI units first. Forces in N, distances in m, masses in kg — before plugging into any formula. Spring extensions given in cm must be converted to metres. Masses given in grams need converting to kg.

Remember F = kx gives the spring force, not the net force. When a spring is attached to a hanging mass at equilibrium, the spring force equals the weight. In dynamics (e.g. the mass is accelerating), the spring force and weight don't cancel — write Newton's second law carefully.

The restoring force is always toward equilibrium. If you extend the spring downward, the spring force acts upward (toward the natural length). If you compress it, the force acts downward. The minus sign in F = −kx reflects this: force opposes displacement.

Frequently Asked Questions

What is Hooke's Law?

Hooke's Law states that the force exerted by a spring is proportional to its extension or compression: F = kx, where F is the restoring force (N), k is the spring constant (N/m), and x is the displacement from the natural length (m). It applies within the elastic limit — if the spring is overstretched, it deforms permanently and Hooke's Law no longer holds. The law applies to any elastic material, not just springs: rubber bands, metal wires, and biological tissues all follow it within their elastic range.

What is the spring constant k?

The spring constant k (also called stiffness) measures how much force is needed per unit extension: k = F/x in N/m. A high k means a stiff spring requiring large forces for small deflections. A low k means a soft spring that deflects easily. k is determined by the spring material, wire thickness, coil diameter, and number of coils. It can be measured experimentally by plotting F vs x — the gradient is k. Springs in series give 1/k_eff = 1/k₁ + 1/k₂ (softer); springs in parallel give k_eff = k₁ + k₂ (stiffer).

What is elastic potential energy?

Elastic potential energy is the energy stored in a deformed elastic object. For a spring displaced by x from its natural length: E = ½kx². The energy is proportional to the square of displacement — double the compression stores four times the energy. This stored energy is released as kinetic energy when the spring returns to its natural length, following conservation of energy. The formula E = ½kx² is derived by integrating the Hooke's Law force F = kx over the displacement.

What is the elastic limit?

The elastic limit is the maximum force (or extension) a material can experience and still return to its original shape when the load is removed. Below the elastic limit, deformation is elastic — temporary and fully reversible. Above it, deformation becomes plastic — permanent. For steel, the elastic limit corresponds to a stress of roughly 250 MPa. Engineering components are always designed to operate well below their elastic limit (typically using safety factors of 2–4) to ensure no permanent deformation occurs under normal loads.

How does Hooke's Law relate to simple harmonic motion?

A mass on a spring undergoes simple harmonic motion (SHM) because Hooke's Law provides a restoring force proportional to displacement: F = −kx (negative because force opposes displacement). By Newton's second law: ma = −kx, giving a = −(k/m)x. This is the defining equation of SHM with angular frequency ω = √(k/m) and period T = 2π√(m/k). Larger k → stiffer spring → faster oscillation. Larger m → more inertia → slower oscillation. The period is independent of amplitude — a key property of SHM.