High SchoolExtendedPhysicsOscillationsSpringsEnergy

How does a spring work?

A spring pushes back in proportion to how far you stretch it (Hooke's law) — which is why a mass on a spring oscillates. The physics, with a simulation.

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PhysicsHub CommunityPublished 21/01/2026 · Updated 06/09/2026
Time to read 5 min

How does a spring work?

A spring resists being stretched or compressed with a force that grows in proportion to how far it is pushed from its natural length. Stretch it twice as far and it pulls back twice as hard. That simple rule — Hooke's law, F=−kxF = -kx — is why a mass hanging from a spring does not just fall to a new position and stop: it overshoots, gets pulled back, overshoots the other way, and keeps oscillating.

Key fact
A spring's force is F=−kxF = -kx: proportional to the displacement xx and always pointing back toward the rest position. A linear restoring force like this always produces simple harmonic motion.

Hooke's law: force proportional to stretch

Take a spring at its natural length and pull the attached mass a distance xx away. The spring pulls back with a force

F=−k xF = -k\,x
Hooke's law · formula card →

where kk is the spring constant (or stiffness), measured in newtons per metre. A large kk means a stiff spring that fights hard against small stretches; a small kk means a floppy one. The minus sign is the important part: the force points opposite to the displacement, so it always tries to return the mass to x=0x = 0.

Hooke's law has limits
F=−kxF = -kx only holds while the spring is not stretched too far. Push past its elastic limit and the metal deforms permanently — the force is no longer proportional to xx, and the spring does not return to its original shape.

Why the mass oscillates: simple harmonic motion

Combine Hooke's law with Newton's second law, F=maF = ma, and you get the equation of motion for the mass:

m x¨=−k x⟹x¨=−km xm\,\ddot{x} = -k\,x \quad\Longrightarrow\quad \ddot{x} = -\frac{k}{m}\,x

This says the acceleration is always proportional to the displacement and points the other way. The solution is a sine wave: the mass moves as x(t)=Acos⁡(ωt+φ)x(t) = A\cos(\omega t + \varphi), oscillating forever between +A+A and −A-A. This pattern — a linear restoring force producing sinusoidal motion — is called simple harmonic motion (SHM), and it turns up everywhere from pendulums to sound waves to the vibrations of atoms.

The angular frequency ω=k/m\omega = \sqrt{k/m} sets how fast the oscillation goes. Converting to a period:

T=2πmkT = 2\pi\sqrt{\frac{m}{k}}
Period of a mass on a spring · formula card →

Notice what is not in that formula: the amplitude. Pull the mass 2 cm or 10 cm from rest and it still takes exactly the same time to complete one bounce. Notice also what changes it — a heavier mass slows the oscillation down, a stiffer spring speeds it up. Try your own numbers in the calculator on the Period of a mass on a spring card.

Energy: kinetic and elastic potential trade off

A stretched or compressed spring stores elastic potential energy U=12kx2U = \tfrac12 k x^2. As the mass moves, energy shuttles between this and kinetic energy K=12mv2K = \tfrac12 m v^2:

  • At the extremes (x=±Ax = \pm A) the mass is momentarily still: all the energy is elastic potential.
  • At the centre (x=0x = 0) the spring is relaxed: all the energy is kinetic and the mass is moving fastest.
  • Everywhere in between, the total E=K+U=12kA2E = K + U = \tfrac12 k A^2 stays constant — as long as nothing removes energy.

Damping: why real springs stop

Friction in the spring and drag from the air take a small bite of energy on every swing, so the amplitude decays and the mass eventually settles at equilibrium. The simulation models this with a damping coefficient bb that adds a force −b v-b\,v opposing the motion:

m x¨=−k x−b x˙m\,\ddot{x} = -k\,x - b\,\dot{x}
  • Light damping — the mass oscillates many times, each swing a little smaller (underdamped).
  • Heavy damping — the mass creeps back to rest without a single full oscillation (overdamped).
  • Critical damping — the fastest possible return to rest with no overshoot; this is what car suspensions and door closers aim for.

Frequently asked questions

What is the spring constant k?

It is the stiffness of the spring in newtons per metre — the force needed to stretch it by one metre. A stiff spring has a large k; a soft one has a small k. It is a property of the spring itself, set by its material, wire thickness and number of coils.

Why doesn't the amplitude affect the period?

A bigger pull stretches the spring further, which makes the restoring force proportionally bigger, which makes the mass accelerate proportionally harder. The extra distance and the extra force cancel out, so every oscillation takes the same time. This is the defining feature of simple harmonic motion.

What makes a mass-spring system oscillate faster?

A stiffer spring (larger k) or a lighter mass (smaller m). The period is T=2πm/kT = 2\pi\sqrt{m/k}, so quadrupling the stiffness halves the period, and quadrupling the mass doubles it.

Where does the energy go when the oscillation dies out?

Into heat. Damping forces — internal friction in the spring and air resistance — do negative work on the mass each cycle, converting its mechanical energy into thermal energy in the spring and the surrounding air.

Keep exploring

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