How a Mass on a Spring Oscillates

Quick Answer

A mass on a spring performs simple harmonic motion: the spring pulls it back with a force proportional to how far it is displaced, F = -k x, so it oscillates smoothly. The period is T = 2 pi times the square root of m / k, set only by the mass m and the spring constant k. For example, a 1 kg mass on a 20 N/m spring oscillates with a period near 1.4 seconds. The amplitude does not change the period. Open this simulator, drag the mass, and watch the period, speed, and energy update live.

How the Spring Oscillator Works

The canvas draws a mass on a spring anchored to a wall. When you pull the mass aside, the spring stores energy and pulls it back with a restoring force F = -k x, always pointing toward the equilibrium line. For example, that is why the motion is simple harmonic motion: the further the mass strays, the harder it is pulled back, which produces a smooth cosine oscillation rather than a jerky one. Release it and it sweeps through the centre, overshoots, and returns.

What Sets the Period

Two things fix the period, and amplitude is not one of them.

• Mass (m), in kilograms: more mass means more inertia, so the oscillation is slower and the period longer.
• Spring constant (k), in newtons per metre: a stiffer spring pulls back harder, so the oscillation is faster and the period shorter.
• Amplitude (A), in metres: how far you pull the mass. It changes the energy and peak speed but not the period.

Calculating Period and Angular Frequency

To calculate the period of a mass on a spring, divide the mass by the spring constant, take the square root, and multiply by 2 pi: that is T = 2 pi times the square root of m / k. To find the angular frequency, instead divide the spring constant by the mass and take the square root, giving omega = the square root of k / m. For example, a stiff 45 N/m spring on a light 0.4 kg mass has a much higher angular frequency than a soft spring under a heavy mass, so it buzzes quickly.

Potential vs Kinetic Energy

A spring oscillator constantly trades spring potential energy for kinetic energy. At the turning points the mass stops, so the energy is all potential, stored in the stretched or compressed spring. Through the equilibrium point the mass moves fastest, so it is all kinetic. For example, watch the PE and KE bars in the simulator swap heights every quarter cycle while their total, half k times amplitude squared, never changes. This is energy conservation in a simple harmonic motion setting.

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