Spring-Mass System

Introduction

A mass attached to a spring is one of the simplest and most important systems in physics. When the mass is displaced from its rest position and released, the spring pulls it back with a force proportional to how far it was stretched or compressed. This produces a smooth, repeating oscillation known as simple harmonic motion. The spring-mass system is the foundation for understanding vibration in everything from car suspensions to musical instruments to the atoms in a crystal lattice.

The Physics Explained

The behaviour of a spring-mass system is governed by Hooke's law: the restoring force exerted by a spring is proportional to the displacement from equilibrium and acts in the opposite direction. If you stretch the spring downward by a distance x, it pulls the mass back upward with a force F = -kx, where k is the spring constant — a measure of how stiff the spring is. A larger k means a stiffer spring and a stronger restoring force.

When the mass is released from an initial displacement, this restoring force accelerates it back toward equilibrium. But by the time it reaches the rest position it has built up speed, so it overshoots and compresses the spring on the other side. The spring then pushes it back, and the cycle repeats. In an ideal system with no friction, this oscillation continues forever at a constant amplitude and frequency.

The natural frequency of oscillation depends only on the spring constant and the mass: stiffer springs oscillate faster, heavier masses oscillate slower. The period — the time for one complete cycle — is T = 2pi * sqrt(m/k). This is independent of the amplitude, which is why springs make reliable timing mechanisms.

In practice, friction and air resistance gradually drain energy from the system. This is modelled by a damping force proportional to the velocity: F_damp = -bv, where b is the damping coefficient. Light damping causes the amplitude to decay slowly over many cycles. Heavy damping can prevent the mass from completing even a single full oscillation — it simply creeps back to equilibrium without overshooting. The boundary between these behaviours is called critical damping, which occurs when b = 2 * sqrt(k * m).

Key Equations

F = -k * x (Hooke's law — restoring force)

x''(t) = -(k/m) * x - (b/m) * x' (equation of motion with damping)

T = 2pi * sqrt(m / k) (natural period, undamped)

f = 1 / T = (1/2pi) * sqrt(k / m) (natural frequency)

omega = sqrt(k / m) (angular frequency, rad/s)

E = (1/2) * k * x^2 + (1/2) * m * v^2 (total mechanical energy)

Key Variables

Symbol	Name	Unit	Meaning
x	Displacement	m	Distance from the equilibrium position
v	Velocity	m/s	Rate of change of displacement
k	Spring constant	N/m	Stiffness of the spring — force per unit extension
m	Mass	kg	Mass of the object attached to the spring
b	Damping coefficient	N·s/m	Controls how quickly friction removes energy
T	Period	s	Time to complete one full oscillation
f	Frequency	Hz	Number of complete oscillations per second
omega	Angular frequency	rad/s	Rate of oscillation in radians per second
A	Amplitude	m	Maximum displacement from equilibrium

Real-World Examples

Vehicle suspension: Car shock absorbers are damped spring-mass systems. The spring absorbs bumps in the road, and the damper prevents the car from bouncing indefinitely — a carefully tuned balance between comfort and control.
Seismometers: A spring-mass system inside a seismometer stays still while the ground shakes beneath it. The relative motion between the mass and the casing is recorded as seismic data.
Musical instruments: Guitar strings, drumheads, and piano hammers all involve spring-like restoring forces. The frequency of vibration determines the pitch of the sound produced.

How the Simulation Works

Four sliders let you set the spring constant (N/m), the mass (kg), the damping coefficient (N·s/m), and the initial amplitude (m). Pressing Start releases the mass from the initial displacement. The simulation integrates the full damped equation of motion using sub-step integration at 1/240 s, so the motion is smooth and accurate even at high stiffness. The trail shows the recent path of the mass, making it easy to see the amplitude decay when damping is active. The readouts display time, instantaneous displacement, and velocity. The period display updates as you change the spring constant or mass, letting you verify T = 2pi * sqrt(m/k) in real time.