SHM Displacement at Time t

Any system whose restoring force grows in proportion to displacement — a mass on a spring, a small-swing pendulum, a quartz tuning fork — traces a cosine in time: x = A cos(ωt), where ω = 2πf is the angular frequency in rad/s. Released from full stretch, a 5 cm amplitude oscillator at ω = 4 rad/s sits at 0.05 × cos(2 rad) ≈ −0.0208 m after half a second, already past centre and heading for the far side.

The cosine form assumes the clock starts at maximum displacement; start from the equilibrium point instead and you need a sine, or equivalently a phase shift. Christiaan Huygens turned this regularity into the first pendulum clock in 1656, cutting timekeeping error from about 15 minutes a day to roughly 15 seconds; today the same equation runs the 32,768 Hz quartz crystal in every wristwatch. Note that solving for t or ω returns only the first solution from arccos — the motion repeats forever, so infinitely many later times give the same displacement.