Once you know a system is undergoing SHM, three related quantities describe how fast it's oscillating. The period T is the time for one full cycle. The frequency f is how many cycles happen per second. And the angular frequency ω — the same ω from x(t) = A cos(ωt + φ) — ties them both together:
Adjust the angular frequency ω and watch the period and frequency respond — they're three different ways of describing the exact same repeating motion.
For an object attached to an ideal spring, the period has its own dedicated formula — no need to derive ω from scratch every time:
Adjust the mass and spring constant of an ideal mass–spring oscillator and watch its period respond — heavier masses swing more slowly, stiffer springs swing faster.
In orbit, an astronaut can't just step on a scale — there's no gravity to pull down on it. Instead, mission engineers strap the astronaut into a chair attached to a spring with known spring constant k = 600 N/m, and measure the period of the resulting oscillation: T = 2.1 s. Find the combined mass of the astronaut and chair.
A simple pendulum displaced by a small angle has its own period formula too — and it looks strikingly different from the spring oscillator's:
Adjust a simple pendulum's length and try it on different worlds — notice mass never enters the formula at all.
You want to know the length of a long string hanging from a high ceiling, but you can't reach the top to measure it directly. You tie a small weight to the bottom, set it swinging at a small angle, and time 10 full swings: 10 periods take 25.3 s. Find the string's length. (Use g = 9.8 m/s².)