Oscillations and waves

The simple pendulum

At small angles, a simple pendulum oscillates approximately harmonically; its period can be used to measure gravitational acceleration.

A simple pendulum consists of a small mass $m$ suspended from a light, inextensible string of length $l$. Released from a small angular displacement, the tangential component of gravity restores it toward equilibrium, producing approximately harmonic motion.

Measuring gravitational acceleration

Time many cycles to reduce stopwatch error. Using the measured period and the length from pivot to the bob's center, calculate $g=4\pi^2l/T^2$. A plot of $T^2$ against $l$ is linear, with slope $4\pi^2/g$.

Quick check

The period grows as the square root of length: quadrupling $l$ doubles $T$. Measure length from the pivot to the bob’s center. Timing several cycles and dividing by their number reduces stopwatch reaction error. Keep the angular displacement small enough for $\sin\theta\approx\theta$ to be valid.

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.