Simple Harmonic Motion

The question

Push an oscillator harder so it swings wider — does it now tick slower, having more distance to cover, or stay perfectly on beat? Galileo timed a swinging chandelier against his own pulse to find out.

The answer is the surprise that runs every clock: the period doesn't depend on the size of the swing. A wider swing means more distance, but also more energy and more speed — and the two cancel exactly.

The key idea

In simple harmonic motion the restoring force is proportional to displacement and always points back toward equilibrium: F = −kx. The period depends on mass and stiffness — but NOT on amplitude. That amplitude-independence is the signature of SHM.

Hooke's Law F = −kx is the engine: the minus sign keeps the force pointing back toward equilibrium, and its size grows with displacement. Feeding it into Newton's second law gives T = 2π√(m/k). Heavier mass → longer period (harder to accelerate); stiffer spring → shorter period (snaps back faster). Amplitude is absent, so a small swing and a large swing share the same period. Amplitude instead sets the energy: total $E = \tfrac{1}{2}kA^2$, so stretching twice as far stores four times the energy. The mass hits top speed at equilibrium, v_max = Aω, where ω = √(k/m). **Limiting case:** at the turning points $x = \pm A$ the speed is zero and the force is maximal; at the centre it's the reverse — and the period never contains $A$: bigger swings travel faster and arrive on time. **Connect it:** SHM is Hooke's law fed into Newton's second — $a = -(k/m)x$ — and it is also the shadow of uniform circular motion projected onto one axis, which is where the $\sin$ and $\cos$ come from.

The formula

• ·T = period (one full oscillation) in seconds; m = mass in kg; k = spring constant in N/m (stiffness — a bigger k pulls back harder for the same stretch). The 2π comes from the circular-motion geometry underlying SHM.

What to remember

• ·SHM occurs when the restoring force is proportional to displacement (F = −kx).
• ·Mass–spring period T = 2π√(m/k): bigger mass or softer spring → slower.
• ·Amplitude does not change the period (isochronism).
• ·Velocity is a quarter-cycle (π/2) ahead of displacement; acceleration is a half-cycle (π) out of step with it, so a = −ω²x.
• ·One complete oscillation is 2π radians of phase, so n oscillations are 2πn rad.