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A grandfather clock's heartbeat is one swinging pendulum — yet whether it sweeps a wide arc or barely nudges off vertical, each swing takes the same time. How can the period not care about the size of the swing? And does a heavier bob speed it up or slow it down?
A grandfather clock's heartbeat is one swinging pendulum — yet whether it sweeps a wide arc or barely nudges off vertical, each swing takes the same time. How can the period not care about the size of the swing? And does a heavier bob speed it up or slow it down?
Galileo allegedly spotted this watching a chandelier in Pisa Cathedral, timing it against his pulse. The bob's weight doesn't matter either — this isochronism made pendulums the world's best clocks for three centuries.
A pendulum swings because gravity has a component along the arc that always points back toward the lowest point. For small angles (under about 15°) this restoring force is very nearly proportional to displacement — making the pendulum a simple harmonic oscillator.
The striking feature is what's absent. Mass doesn't appear: heavy and light bobs on equal strings swing in unison, because restoring force and inertia both scale with mass and cancel — exactly as in free fall. Amplitude doesn't appear either, as long as the swing stays small (the approximation sin θ ≈ θ is what keeps the motion harmonic; past ~20° the real period runs slightly long). On Earth a 1 m pendulum gives — one second out, one second back. This near-coincidence was once proposed as a definition of the metre. **All forms:** , . **Limiting case:** the formula is the small-angle limit () — at the true period is already longer, and never appears at all. **Connect it:** for small swings the restoring force is — Hooke's law in disguise with ; substitute into and the pendulum formula falls out.
Thinking a heavier bob or wider swing changes the period — for small swings it depends only on length and g, not on mass or amplitude.