3.1.2.1. Hooke's Law

Table of Contents

Restoring force in a spring

Hooke's law describes how an ideal spring pushes or pulls back when it is stretched or compressed. If a spring is moved a small distance away from its natural length, the spring exerts a force that tries to return it to that natural length. This is called a restoring force.

If the displacement from the natural length is $x$, then Hooke's law is

$$F = -kx$$

Here, $F$ is the force exerted by the spring, $k$ is the spring constant, and $x$ is the displacement from equilibrium.

The negative sign is very important. It shows that the spring force always points in the direction opposite to the displacement. If the spring is stretched to the right, the force points to the left. If the spring is compressed to the left, the force points to the right.

For an ideal spring, the spring force is
$$F = -kx$$
The minus sign means the force is a restoring force, it always points back toward equilibrium.

Meaning of the spring constant

The quantity $k$ tells us how stiff the spring is. A large value of $k$ means the spring is hard to stretch or compress. A small value of $k$ means the spring is easier to deform.

In SI units, since force is measured in newtons and displacement in meters, the unit of $k$ is

$$\frac{\text{N}}{\text{m}}$$

If two springs are stretched by the same amount, the one with the larger $k$ produces the larger restoring force.

Spring constant $k$	Meaning
Small $k$	Soft spring
Large $k$	Stiff spring

Natural length and displacement

A spring has a natural, or unstretched, length. Hooke's law does not use the total length of the spring directly. It uses the change in length from the natural length.

If the natural length is $L_0$ and the current length is $L$, then the displacement is

$$x = L - L_0$$

If $x > 0$, the spring is stretched. If $x < 0$, the spring is compressed. If $x = 0$, the spring is at equilibrium and the spring force is zero.

In Hooke's law, $x$ means change in length from the natural length, not the full length of the spring.

Stretching and compressing

Hooke's law works for both stretching and compressing, as long as the spring behaves elastically. That means the spring returns to its original shape when the force is removed.

Suppose a spring has $k = 200 \, \text{N/m}$.

If it is stretched by $x = 0.05 \, \text{m}$, then

$$F = -kx = -(200)(0.05) = -10 \, \text{N}$$

The force is $10 \, \text{N}$ toward equilibrium.

If it is compressed by $x = -0.03 \, \text{m}$, then

$$F = -kx = -(200)(-0.03) = 6 \, \text{N}$$

The positive result means the force points in the positive direction, again back toward equilibrium.

Force needed to hold a spring

The spring exerts a restoring force, but sometimes we are interested in the external force needed to hold the spring in place. If a spring is held stretched and not moving, the external force must balance the spring force.

So, in magnitude,

$$F_{\text{external}} = kx$$

This external force points away from equilibrium, opposite to the spring's restoring force.

This is why a graph of applied force versus extension is a straight line for an ideal spring.

Force and extension graph

For an ideal spring, the magnitude of the force is proportional to the magnitude of the extension or compression. This gives a straight-line graph.

If we plot applied force $F$ against extension $x$, the graph has slope $k$.

$$F = kx$$

This form is often used when we talk about magnitudes only, without direction.

Rendering scientific sketch…

Force and extension for an ideal spring

For an ideal spring, force is proportional to displacement. A graph of force versus extension is a straight line through the origin.

Elastic limit

Real springs do not follow Hooke's law forever. If a spring is stretched or compressed too much, it may stop behaving linearly. Beyond this range, the force is no longer exactly proportional to displacement. If the spring is pushed too far, it may even become permanently deformed.

So Hooke's law is accurate only within the elastic limit of the spring.

Hooke's law is valid only while the spring remains elastic. Beyond the elastic limit, the relation $F = -kx$ no longer holds exactly.

Physical picture

Hooke's law comes from the fact that the atoms inside the spring resist being moved from their preferred spacing. When the spring is deformed, internal forces appear that oppose the deformation. For small changes in length, this opposition is approximately proportional to the displacement, which leads to the simple linear law.

Simple example

A spring with spring constant $k = 50 \, \text{N/m}$ is stretched by $0.10 \, \text{m}$. The spring force is

$$F = -kx = -(50)(0.10) = -5.0 \, \text{N}$$

This means the spring pulls back with a force of magnitude $5.0 \, \text{N}$.

If we want to hold it at that position without motion, we must apply an external force of magnitude $5.0 \, \text{N}$ in the opposite direction.

Visualizing the restoring force

Rendering scientific sketch…

Stretched spring and restoring force

Key idea

Hooke's law is the basic rule for ideal springs. It says that the farther a spring is displaced from equilibrium, the stronger the restoring force becomes, and that force always points back toward equilibrium. This simple law is the foundation for understanding spring motion and oscillations.

Key formula:
$$F = -kx$$
where $k$ is the spring constant and $x$ is the displacement from equilibrium.

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