What does Hooke's law state, where does it stop holding good, and how do you verify it in the lab? A complete guide with the F = −kx formula, stress-strain curve, Young's modulus, and a step-by-step spring experiment.
In physics, Hooke's law describes how elastic objects respond to being stretched or compressed. It is named after the English scientist Robert Hooke, who announced the principle in 1676 and published it in 1678 as the Latin phrase "ut tensio, sic vis" — "as the extension, so the force."
Put simply: if you double the stretch of a spring, you double the force it pulls back with. This straight-line relationship is the reason springs make reliable scales, shock absorbers and clocks, and it is the starting point for the entire study of elasticity in Class 11 physics.
Within the limit of proportionality, the stress applied to a body is directly proportional to the strain produced in it. For a spring, the restoring force is directly proportional to the extension and acts in the direction opposite to it.
For a spring, Hooke's law equation is written as:
F = restoring force (N) | k = spring constant (N/m) | x = extension or compression from natural length (m)
The negative sign means the restoring force always points opposite to the displacement: stretch the spring and it pulls back; compress it and it pushes out. When we measure only the size of the force, as in a lab, we simply write F = kx.
Figure 1: A helical spring at natural length, then stretched by a load m and a load 2m. The extension doubles when the force doubles, which is Hooke's law.
The spring constant k measures stiffness: it is the force needed to stretch the spring by one metre. A large k (say 500 N/m) means a stiff spring; a small k (say 10 N/m) means a soft one. For a given spring, k stays constant as long as Hooke's law holds.
| Quantity | Symbol | SI Unit | Formula |
|---|---|---|---|
| Restoring force | F | newton (N) | F = −kx |
| Spring constant | k | N/m | k = F / x |
| Extension | x | metre (m) | x = F / k |
| Elastic potential energy | U | joule (J) | U = ½ k x² |
| Springs in series | keq | N/m | 1/keq = 1/k₁ + 1/k₂ |
| Springs in parallel | keq | N/m | keq = k₁ + k₂ |
If you plot force against extension, Hooke's law gives a straight line through the origin. The slope of that line is the spring constant. But the law does not hold forever. Hooke's law holds good up to the limit of proportionality; beyond that point the graph bends and extension grows faster than force.
Figure 2: Up to point P the force-extension graph is a straight line whose slope equals the spring constant k. Beyond P the curve bends away from the dashed straight-line extension.
These two are not the same. The limit of proportionality is where the straight line ends (Hooke's law stops). The elastic limit is slightly further along: up to it the material still returns to its original shape when the load is removed. Beyond the elastic limit, deformation is permanent.
The spring form F = −kx depends on the size and shape of the spring. To describe the material itself, physicists use stress and strain, which do not depend on dimensions.
Restoring force per unit cross-sectional area. σ = F / A, measured in pascal (N/m²).
Fractional change in dimension. Longitudinal strain = ΔL / L. It is a ratio, so it has no unit.
E = Young's modulus (Pa) | F = applied force | A = cross-sectional area | L = original length | ΔL = change in length
Young's modulus (E) measures stiffness of a solid. A material with a high Young's modulus, like steel (about 200 GPa), barely stretches under load; rubber, with a very low modulus, stretches a lot. The type of strain decides which modulus applies:
| Type of Strain | Definition | Modulus |
|---|---|---|
| Longitudinal | ΔL / L — change in length per unit length | Young's modulus, Y = (F/A)/(ΔL/L) |
| Volumetric | ΔV / V — change in volume per unit volume | Bulk modulus, B = −p / (ΔV/V) |
| Shear | Δx / L = tan θ — angular deformation | Shear modulus, G = (F/A)/θ |
When a wire is loaded until it breaks, plotting stress against strain gives the stress-strain curve. It shows every stage of elastic and plastic behaviour in a single picture, and the straight initial portion is Hooke's law.
Figure 3: Stress-strain curve. The straight section OP obeys Hooke's law; beyond E the wire deforms permanently, and it finally breaks at F after passing the ultimate tensile strength U.
| Point | Name | What Happens |
|---|---|---|
| O → P | Proportional region | Stress ∝ strain; Hooke's law valid; slope = Young's modulus |
| P → E | Elastic region | Not proportional, but the wire still returns to its original length when unloaded |
| Y | Yield point | Strain rises quickly with little extra stress; permanent deformation begins |
| U | Ultimate tensile strength | Maximum stress the material can bear |
| F | Fracture point | The wire breaks |
When you stretch a rubber band it gets thinner. This sideways contraction is captured by Poisson's ratio (ν), the negative ratio of lateral strain to longitudinal strain.
For most common materials ν lies between about 0.2 and 0.5 | Steel ≈ 0.3 | Rubber ≈ 0.5
The generalized Hooke's law extends F = −kx to three dimensions. Instead of a single force and a single extension, it relates the whole stress tensor to the whole strain tensor. For an isotropic material only two independent constants are needed — typically Young's modulus E and Poisson's ratio ν — and the three elastic moduli are connected by:
G = shear modulus | B = bulk modulus | E = Young's modulus | ν = Poisson's ratio
This is the standard Class 11 / first-year undergraduate experiment to verify Hooke's law and to determine the spring constant of a helical spring.
To verify Hooke's law by measuring the extension of a helical spring for different loads, and to determine its spring constant from the force-extension graph.
A light steel spring of known natural length, with hooks at both ends.
A retort stand with a horizontal clamp to hang the spring vertically.
Set of 50 g slotted masses with a light hanger to apply the load.
Vertical metre scale fixed behind the spring; a pointer or marker at the lower hook reads the position.
In this setup the force is the weight of the hanging mass, F = mg, so a balance (or the stamped mass values on the slotted weights) gives the force. If you want to measure force directly, use a newton meter (spring balance) or a force sensor connected to a data logger.
Figure 4: Experimental arrangement. The spring hangs from a rigid clamp, slotted masses load the hanger, and a pointer on the lower hook moves against a vertical metre scale.
The table below uses sample readings for a spring with k ≈ 20 N/m, taking g = 9.8 m/s². Your own readings will differ; use this table as a template.
| S.No. | Load, m (g) | Force, F = mg (N) | Reading, L (cm) | Extension, x (cm) | Extension, x (m) | k = F / x (N/m) |
|---|---|---|---|---|---|---|
| 0 | 0 | 0.00 | 20.0 | 0.0 | 0.000 | — |
| 1 | 50 | 0.49 | 22.5 | 2.5 | 0.025 | 19.6 |
| 2 | 100 | 0.98 | 24.9 | 4.9 | 0.049 | 20.0 |
| 3 | 150 | 1.47 | 27.4 | 7.4 | 0.074 | 19.9 |
| 4 | 200 | 1.96 | 29.8 | 9.8 | 0.098 | 20.0 |
| 5 | 250 | 2.45 | 32.3 | 12.3 | 0.123 | 19.9 |
| 6 | 300 | 2.94 | 34.7 | 14.7 | 0.147 | 20.0 |
The nearly constant F / x ratio confirms that extension is proportional to force
Plot force F (N) on the y-axis and extension x (m) on the x-axis, and draw the best-fit straight line through the origin. The slope of the line equals the spring constant.
Figure 5: Plot of the sample data. All six points lie on a straight line through the origin, confirming F ∝ x. The slope ΔF/Δx gives the spring constant, here about 20 N/m.
The graph of force against extension is a straight line passing through the origin, so the extension of the spring is directly proportional to the applied force within the range studied. Hooke's law is verified. The spring constant of the given spring is k ≈ 19.9 N/m (from the mean of F/x) and ≈ 20 N/m (from the graph slope).
| Source of Error | Effect | How to Reduce It |
|---|---|---|
| Parallax when reading the scale | Random error in L | View the pointer horizontally; use a mirror strip behind the scale |
| Spring still oscillating | Reading too high or low | Wait a few seconds for the spring to settle |
| Mass of spring itself | Spring's own weight adds to extension | Use a light spring and measure from the loaded position, not the unloaded one |
| Exceeding the elastic limit | Permanent stretch; non-linear graph | Limit the maximum load; check unloading readings return to L₀ |
| Inaccurate masses | Systematic error in F | Check mass labels on a balance; use g = 9.8 m/s² |
A good Hooke's law experiment lab report follows the same structure as the experiment itself. Use this checklist so nothing is missed.
Q. A spring stretches by 8.0 cm when a 400 g mass hangs from it. Find k and the extension for a 600 g mass.
A. F = mg = 0.4 × 9.8 = 3.92 N, x = 0.080 m, so k = 3.92 / 0.080 = 49 N/m. For 600 g, F = 5.88 N, so x = 5.88 / 49 = 0.12 m = 12 cm (assuming the spring is still within its elastic limit).
Weighing scales use a spring whose extension is calibrated to read force or mass directly.
Coil springs in cars absorb road shocks, with stiffness chosen from the spring constant.
The hairspring in a mechanical watch provides a restoring torque proportional to its twist.
Beams, cables and bridges are designed to stay inside the elastic range, where Young's modulus applies.
A mass on a spring obeys F = −kx, so it oscillates with period T = 2π√(m/k).
Elastic wave theory and material testing both rely on stress-strain relations from Hooke's law.
Hooke's law states that, within the elastic limit, the restoring force produced in a spring or elastic material is directly proportional to its extension or deformation. In equation form, F = −kx, where F is the restoring force, x is the displacement from the natural length and k is the spring constant. The negative sign shows that the force opposes the displacement.
For a spring, the Hooke's law formula is F = −kx, where k is the spring constant in newtons per metre and x is the extension or compression in metres. For a solid material, the stress-strain form is stress = E × strain, where E is Young's modulus.
Hooke's law holds good up to the limit of proportionality (proportional limit). Beyond this point stress is no longer proportional to strain, even though the material may still return to its original shape up to the elastic limit. Past the elastic limit the material deforms permanently.
A graph of force against extension is a straight line through the origin up to the limit of proportionality, and the slope of that line equals the spring constant k. Beyond the limit the line curves, showing that extension is no longer proportional to force.
The standard apparatus is a helical spring, a rigid stand with clamp, a hanger with slotted masses, a metre scale and a pointer. The force exerted by the hanging mass is calculated as weight, F = mg. A newton meter can also be used to measure force directly.
Plot force (N) on the y-axis against extension (m) on the x-axis and draw the best-fit straight line through the origin. The slope of this line, change in force divided by change in extension, is the spring constant k in N/m. Alternatively calculate F/x for every reading and take the mean.
Stress is the restoring force per unit cross-sectional area, measured in pascals (N/m²). Strain is the fractional change in dimension, equal to change in length divided by original length, and has no unit. Their ratio, stress divided by strain, is the elastic modulus of the material.
Young's modulus E is the ratio of tensile stress to tensile strain in the linear elastic region and measures stiffness. Poisson's ratio is the negative ratio of lateral strain to axial strain, and is typically between 0.2 and 0.5 for most materials.
The generalized Hooke's law extends the one-dimensional relation to three dimensions, relating the full stress tensor to the strain tensor through a stiffness tensor. For an isotropic material it can be written using just two independent constants, such as Young's modulus and Poisson's ratio.
