Animated walk through pendulum geometry, equilibrium pass (KE max), turning point (PE max), and the breakdown of the small-angle approximation at large amplitudes.
A free, animated simple pendulum you can read here or embed on any website, from Scrollchart.
A simple pendulum is the canonical example of energy conservation in mechanics. As the bob swings, kinetic and potential energy continuously trade places. At the bottom of the arc all the energy is kinetic and the bob is moving fastest. At the turning points all the energy is potential and the bob is momentarily at rest. The string does no work because tension is always perpendicular to motion.
For small angles the motion is simple harmonic with period T = 2π·sqrt(L/g). The period depends only on length and gravity, not on mass or amplitude. This is why pendulum clocks work and why a child on a swing takes the same time per swing whether they swing high or low (within the small-angle limit).
Newton's second law along the arc gives m·d²s/dt² = -m·g·sin(θ), where s is arc length and θ is angular displacement. For small θ, sin(θ) ≈ θ, and the equation reduces to d²θ/dt² = -(g/L)·θ — the simple harmonic oscillator equation with angular frequency ω = sqrt(g/L) and period T = 2π/ω = 2π·sqrt(L/g).
At larger amplitudes, sin(θ) and θ diverge and the period grows. The exact period is T(θ_0) = 4·sqrt(L/g) · K(sin(θ_0/2)), where K is the complete elliptic integral of the first kind. Numerically this means the period is about 1.7% longer at 30 degrees, 7% longer at 60 degrees, and unbounded as θ_0 approaches 180 degrees (the inverted pendulum, which never falls back).
Good for
Classical mechanics teaching
Pendulum-clock articles
Energy conservation visualizations
Source & accuracy
This simple pendulum is an editorial illustration built to represent the concept accurately. Where it shows figures, they are typical or representative values chosen to make the relationship clear, not a single underlying dataset. The diagram and its explainer are reviewed and maintained centrally, and updated over time as understanding improves.
Small angle approximation and its breakdown
For small swings, a pendulum behaves as a simple harmonic oscillator with a period that depends only on its length and gravity, independent of the mass or amplitude. This remarkable property held for centuries before the small angle approximation's limits became clear. At large amplitudes, the curved path of the pendulum bob deviates significantly from the sinusoidal motion predicted for small angles, and the period increases measurably. The approximation fails because the restoring force no longer equals the full gravitational component at large angles; geometric effects cause the bob to spend disproportionate time near the turning points.
Energy exchange during swing
A pendulum converts between gravitational potential energy (maximum at the highest points) and kinetic energy (maximum at the lowest point). At the turning points where the bob momentarily stops, all energy is potential. As it passes through equilibrium, potential energy reaches its minimum and velocity peaks. This exchange occurs continuously, with the total mechanical energy remaining constant in the absence of damping. The precise balance between height gained and speed attained determines the trajectory, making the pendulum a natural demonstration of energy conservation in mechanical systems.
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Reference
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A free, embeddable, animated simple pendulum for any website.
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