Oscillations and waves

The spring–mass oscillator

A mass attached to an ideal spring oscillates harmonically, with a period set by its mass and the spring stiffness.

Displace a mass from equilibrium and release it: the spring pulls it back. For a light Hookean spring oscillating about equilibrium, the restoring force is proportional to displacement, so the motion is harmonic.

Energy exchange

Without friction, mechanical energy $E=\frac12mv^2+\frac12kx^2=\frac12kA^2$ is conserved. At an endpoint, speed is zero and spring potential energy is greatest; at equilibrium, speed and kinetic energy are greatest.

Quick check

For a horizontal spring, Newton’s second law gives $m\ddot x=-kx$, hence $\omega=\sqrt{k/m}$. For a vertical spring, the static extension $\Delta l_0=mg/k$ balances gravity. Measuring displacement from the shifted equilibrium cancels gravity from the oscillation equation, so the period remains $2\pi\sqrt{m/k}$.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear.