AP® Physics 1: Algebra-Based · Unit 2 · Topic 2.8
Hooke's Law and Spring Force
Hooke's Law says a spring's restoring force is proportional to how far it's stretched or compressed: F = -kx, where k is the spring constant and the minus sign means the force always pulls back toward equilibrium.
A spring resists being stretched, and it resists harder the more you stretch it. Pull a spring twice as far from its natural length and it pulls back twice as hard. That relationship is Hooke's Law: F = -kx. Here x is the displacement from the spring's equilibrium (unstretched) length, and k is the spring constant, a fixed number in newtons per meter that tells you how stiff the spring is. A large k means a stiff spring that fights back hard for even a small stretch; a small k means a loose, easy spring.
The minus sign is not decoration. It says the force always points opposite to the displacement — stretch the spring to the right and it pulls left; compress it and it pushes outward. That is what makes it a restoring force: it always acts to bring the spring back toward equilibrium, which is why springs oscillate back and forth instead of flying off in one direction. On a graph of force magnitude versus stretch, Hooke's Law shows up as a straight line through the origin, and the slope of that line is exactly k. Flip the slope's rise and run and you are just solving F = kx for k.
Hooke's Law is only true within a spring's elastic region, the range of stretch where the spring returns to its original shape once released. Push a real spring past its elastic limit and the clean straight-line relationship breaks down — the spring deforms permanently, and force is no longer simply proportional to stretch. This idea connects directly to springs used as restoring forces in oscillation problems, where F = -kx is the starting point for analyzing simple harmonic motion.
Key terms
- Spring constant (k)
- A measure of spring stiffness, in newtons per meter, equal to the force needed per unit of stretch.
- Restoring force
- A force that always points back toward an equilibrium position, opposing displacement from it.
- Equilibrium length
- The natural, unstretched and uncompressed length of a spring where the spring force is zero.
- Elastic limit
- The maximum stretch or compression a spring can undergo and still return to its original shape.
Watch it explained
- How it worksHow does the minus sign in Hooke's law encode the idea of a restoring force?
- Worked exampleFind the spring constant of a spring that stretches 0.12 m under a 30 N pull.
- How it worksHow do you get the spring constant from the slope of a force versus stretch graph?
- Going furtherWhat happens to a real spring once you stretch it past its elastic limit?
Check yourself
End of topic.