2.6 Spring Force

Hooke's Law

The spring force is a restoring force given by F = -kx, where k is the spring constant (N/m) and x is displacement from equilibrium. The negative sign shows the force always points opposite the displacement, pulling or pushing the object back toward equilibrium.

Spring Constant & Behavior

k measures spring stiffness — larger k means a stiffer spring requiring more force per unit stretch/compression. Ideal springs obey Hooke's Law only within their elastic limit; beyond that, the linear relationship breaks down. Springs can both stretch (x > 0) and compress (x < 0), with force direction always opposing displacement.

Springs in Dynamics Problems

When analyzing systems with springs, apply Newton's Second Law: F_net = ma, substituting the spring force -kx for one of the forces. This often leads to differential equations describing simple harmonic motion when the spring force is the net restoring force acting on a mass, foundational to later SHM topics.

Key terms

Hooke's Law

F = -kx; the spring force is proportional to displacement and directed opposite to it.

Spring constant (k)

A measure of spring stiffness, in N/m; larger k means a stiffer spring.

Restoring force

A force that always acts to push/pull a system back toward equilibrium, like the spring force.

Elastic limit

The maximum displacement/force within which a spring obeys Hooke's Law linearly.

Spring force + Newton's 2nd Law

Substituting F = -kx into F_net = ma leads to equations describing simple harmonic motion.

Try an AP-style question

A 50 N force stretches a spring 0.25 m. What is its spring constant?

1. 50 N/m
2. 12.5 N/m
3. 0.005 N/m
4. 200 N/m