PENDULUM
The period carries no m — mass cancels before the physics even starts.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A bob on a string of length L, drawn aside by θ₀ and released. Galileo’s question in a cathedral: what sets the rhythm of the swing — the weight, the width, or the string?
Forces
Tension holds the arc; gravity does the pushing. Only the tangential slice −mg sinθ drives the motion, and for small angles sinθ is θ to within a couple of percent at worst. That approximation is the price of admission — posted openly, and cheap.
Solve
Newton along the arc gives mLθ̈ = −mgθ — and the mass cancels before the physics even starts. What remains is the oscillator equation with ω = √(g/L): the pendulum is SHM wearing a string.
Swing
Two full periods. Tension breathes with the swing — lightest at the turning points, heaviest at the bottom where it must both hold the bob and bend its path. The energy bars trade amber for cyan twice each cycle.
Audit
The period carries no m and no θ₀ — double the mass or widen the swing and the clock doesn’t care. The fine print: the true period runs long by θ₀²/16, quoted here to three figures. Honest instruments state their tolerance.