Deriving Hooke's Law for a Spring

Introduction

A spring that is stretched or compressed tends to push or pull back toward its equilibrium position. The ideal model for this response is called Hooke’s law: the restoring force is proportional to displacement, with a direction opposite to that displacement.

The proportionality constant describes the stiffness of the spring. It can be identified from measurements of applied force and extension, provided the measurements are taken in a range where the spring behaves approximately linearly. The sign convention matters: the spring’s force and the external force used to hold the spring stretched point in opposite directions.

Displacement and the Sign Convention

Choose an axis along the spring. Let \(x=0\) represent the spring’s unstretched equilibrium position, and let positive \(x\) point in the direction of stretching. Then \(x>0\) describes an extension, while \(x<0\) describes a compression. The displacement \(x\) is signed: it records both how far and in which direction the spring’s end has moved from equilibrium.

The minus sign expresses direction, not a negative stiffness. When \(x>0\), the force \(F_s\) is negative and points back toward equilibrium. When \(x<0\), the force is positive and again points back toward equilibrium. The magnitude of the restoring force is \(k|x|\).

In an experiment, it is common to report the magnitude of the external force needed to hold the spring at a particular extension. When the spring is held at rest, the external force balances the spring force. Thus, for a positive extension \(x\), the external force points in the positive direction and has magnitude \(kx\), while the spring’s force points in the negative direction.

These are two descriptions of the same balanced situation, but they refer to different forces. Hooke’s law in signed form describes the force exerted by the spring. A measured positive force-extension graph usually records the magnitude of the opposing external force.

Why the Ideal Law Has This Form

Hooke’s law is a model of an ideal spring, not a claim that every real spring follows an exact linear rule at every deformation. It is useful to derive its form from the assumptions of the model.

Suppose the spring force along the chosen axis depends linearly on displacement from its unstretched equilibrium position. The most general linear expression is \(F_s(x)=ax+b\), where \(a\) and \(b\) are constants. At \(x=0\), the spring is unstretched and exerts no force, so \(F_s(0)=0\). Substituting \(x=0\) gives \(b=0\). Therefore the force is \(F_s(x)=ax\).

A restoring force points opposite to a nonzero displacement. In one dimension this means \(F_s(x)<0\) when \(x>0\), and \(F_s(x)>0\) when \(x<0\). Consequently, the coefficient \(a\) must be negative. Write \(a=-k\), where \(k>0\). This gives Hooke’s law.

This derivation explains the mathematical form of the law once the linear-response assumption is made. The assumption is supported by the behavior of many springs for small enough extensions and compressions. For sufficiently large deformation, a real spring may no longer have force proportional to displacement.

Finding the Spring Constant from Measurements

For a spring stretched by a positive amount \(x\) and held at rest, the external force magnitude \(F_{\text{ext}}\) satisfies \(F_{\text{ext}}=kx\). If a force and extension are known, divide the force magnitude by the extension to find \(k\). Use metres for extension when reporting \(k\) in \(\mathrm{N/m}\).

With several measurements, an ideal spring should give the same value of \(F_{\text{ext}}/x\) for every data pair. Equivalently, a graph of external force magnitude against extension is a straight line through the origin, with slope \(k\). The line passes through the origin because zero extension requires zero external force in this ideal model.

Common Sign and Unit Errors

A frequent error is to give the spring force the same sign as the displacement. For example, at \(x=+0.040\ \mathrm{m}\), a student might calculate \(F_s=(300\ \mathrm{N/m})(0.040\ \mathrm{m})=+12\ \mathrm{N}\). The multiplication gives the magnitude, but the sign is wrong for the force exerted by the spring. The correction is:

The spring force points back toward equilibrium. The \(+12\ \mathrm{N}\) force would instead be the external force required to hold the spring at rest at that extension.

A second common error is to use centimetres as though they were metres. If an extension is \(4.0\ \mathrm{cm}\), convert it first: \(4.0\ \mathrm{cm}=0.040\ \mathrm{m}\). For an external force magnitude of \(12\ \mathrm{N}\), the spring constant is \(12\ \mathrm{N}/0.040\ \mathrm{m}=300\ \mathrm{N/m}\), not \(3\ \mathrm{N/m}\). The ratio must use consistent length units.

When the Reference Position Is Shifted

The simple formula \(F_s=-kx\) uses \(x=0\) at the spring’s unstretched position, where its force is zero. Sometimes a spring is already under a constant load, so the position of the system’s equilibrium is shifted. The spring force at that loaded equilibrium need not be zero: it may balance the load.

For changes measured from that loaded equilibrium, the change in spring force still has the restoring form \(\Delta F_s=-k\Delta x\), if the spring remains in its linear range. This statement concerns the change in force, not necessarily the total spring force. Keeping the reference position clear prevents a nonzero balancing force from being confused with the spring’s incremental restoring response.

Why Hooke’s Law Matters

Hooke’s law turns a measured extension into a force prediction, or a force measurement into a stiffness value. Its compact form also keeps direction explicit: a positive displacement produces a negative restoring force, and a negative displacement produces a positive one. The spring constant \(k\) measures the scale of the response: for the same extension, a larger \(k\) means a larger force magnitude.

The model’s usefulness depends on respecting its assumptions. Check that the displacement is measured from the correct reference position, that lengths are in metres when using \(\mathrm{N/m}\), and that the force being reported is identified as either the spring force or the balancing external force. Finally, compare multiple measurements when available; a constant ratio is evidence that the linear model fits the tested range.