Hooke's Law

The law

When a spring is stretched or compressed by a displacement $x$ from its natural (equilibrium) length, it exerts a restoring force:

$F = -kx$

• $k$ is the spring constant (also called stiffness) in N/m
• $x$ is the displacement from natural length (positive = stretched, negative = compressed)
• The negative sign means the force opposes the displacement — it always points back toward equilibrium

What the negative sign means

If you stretch the spring ($x > 0$), the spring pulls back: $F < 0$ (toward natural length).

If you compress the spring ($x < 0$), the spring pushes back: $F > 0$ (toward natural length).

The force always acts to restore the spring to its natural length — hence it is called a restoring force.

The spring constant $k$

$k$ measures how stiff the spring is:

• Large $k$: stiff spring — large force for small displacement (car suspension, valve spring)
• Small $k$: soft spring — small force for large displacement (mattress spring, shock absorber)

Units: $[k] = \text{N/m}$

Hooke's Law is an approximation

Hooke's Law holds only within the elastic limit — the range of deformation from which the spring can fully recover.

Beyond the elastic limit:

• The spring is permanently deformed (plastic deformation)
• Hooke's Law no longer holds
• Force-extension relationship becomes nonlinear

For most problems in physics, we assume the spring operates within the elastic limit.

Experimental verification

Plot force $F$ (applied externally to stretch the spring) vs extension $x$:

• Within elastic limit: straight line through origin, slope $= k$
• Beyond elastic limit: line curves, becomes nonlinear
• At breaking point: spring fails

The slope of the linear region gives $k$ directly.

Spring constant from dimensions

A spring of natural length $L$, when cut into two pieces of lengths $l_1$ and $l_2$ ($l_1 + l_2 = L$):

The spring constant of each piece:

$k_1 = \frac{kL}{l_1}, \quad k_2 = \frac{kL}{l_2}$

Shorter pieces are stiffer — cutting a spring makes it stiffer. This is because the same force produces less extension in a shorter spring (less material to deform).