Describing Oscillations

• An object oscillates when it moves repeatedly back and forth through an equilibrium position — the place it rests when left alone. A free oscillation runs at the system's own natural frequency after one push; a forced oscillation is driven at whatever frequency the driver imposes (kept for the resonance lesson).
• Amplitude $x_0$: the maximum displacement from equilibrium. Period $T$: the time for one complete oscillation — out and back. Frequency $f = 1/T$: oscillations per second.
• Phase says where in its cycle an oscillator is. Two oscillators a quarter-cycle out of step have a phase difference of $90^\circ$ or $\pi/2$ rad — a fraction of a cycle scaled onto $360^\circ$ or $2\pi$.

• $\omega$ is the same angular frequency as in circular motion: one full cycle is $2\pi$ radians of phase, so an oscillator at frequency $f$ sweeps phase at $2\pi f$ radians per second. Every SHM equation is written with $\omega$, so converting from $T$ or $f$ is usually step one of any calculation — and examiners test the conversion itself (9702/42/M/J/23 Q4(a)(i)).

• Real exam graphs rarely hand you $x_0$ directly. Three traps, all from recent papers:
• Half the peak-to-peak. If a graph swings between two extremes, the amplitude is half the full range: a spread of $7.2 \times 10^{-15}$ m means $x_0 = 3.6 \times 10^{-15}$ m (9702/42/O/N/23 Q4(a)(i)).
• Quarter of the round trip. “In one oscillation the object travels 14 mm” — one cycle covers $4x_0$ (out, back, out the other side, back), so $x_0 = 3.5$ mm (9702/42/F/M/24 Q3(a)).
• Offset variables. When the plotted quantity is measured from a fixed point (height above the floor, depth of a float), equilibrium is the mid-value: a height varying 3.5–9.5 cm oscillates about 6.5 cm with $x_0 = 3.0$ cm (9702/41/O/N/24 Q4(b)(i)).

Amplitude is measured from equilibrium, and equilibrium is not always zero on the graph. Half the peak-to-peak; a quarter of the distance per cycle.