The Physics of Pendulums — Simple Harmonic Motion — The Complete Guide
A pendulum's period depends only on its length and local gravity — not on mass, and not on amplitude, provided the swing angle stays small. This remarkable independence from mass and amplitude is what made pendulums the basis of the most accurate clocks in the world for roughly three centuries, from Christiaan Huygens' first pendulum clock in 1656 right up until quartz and atomic timekeeping took over in the 20th century. Pendulum Master puts length, mass, and release angle directly under your control so you can discover these relationships through experimentation rather than by memorising a formula.
The Period Formula: T = 2π√(L/g)
For small swing angles — conventionally below about 15° — a pendulum's period is given by T = 2π√(L/g), where L is the length of the string or rod from pivot to bob, and g is the local gravitational acceleration. At Earth's surface (g = 9.81 m/s²), a pendulum exactly 1.0 m long has a period of almost precisely 2.0 seconds — one of the reasons the metre itself was, for a time in the 18th century, seriously proposed as the length of a pendulum with a one-second half-swing.
Shortening the pendulum speeds it up: a pendulum just a quarter of that length, 0.25 m, has a period of exactly 1.0 second. The relationship is a square root, not linear, which is why grandfather clocks with their long pendulums tick noticeably slower than a mantel clock's short one.
This length-dependence is also why precision pendulum clocks need temperature compensation. Almost all pendulum rods expand slightly when warmed, increasing L — and since a longer pendulum has a longer period, a warming clock actually runs measurably slow. Historic precision clocks used compensating pendulums built from two metals with different thermal expansion coefficients, arranged so that one expansion cancels the other and the effective length barely changes with temperature.
Why Mass Doesn't Affect the Period
The restoring force pulling a pendulum bob back toward vertical is the tangential component of gravity, mg sin θ, which for small angles is well approximated as mgθ (with θ in radians). Applying Newton's second law along the arc: ma = −mgθ = −mg(s/L), where s is the arc-length displacement. Notice that mass m appears on both sides of this equation — once in the force (mg) and once in the inertia (ma) — and cancels completely.
What's left is a = −(g/L)s, the defining equation of simple harmonic motion, with angular frequency ω² = g/L and period T = 2π/ω = 2π√(L/g) — mass nowhere in sight. A heavier bob feels a proportionally larger restoring force, but it also has proportionally more inertia resisting acceleration, and the two effects cancel exactly.
This cancellation is a direct consequence of the equivalence principle: the mass that determines how strongly gravity pulls on an object (gravitational mass, in mg) is experimentally indistinguishable from the mass that determines how strongly an object resists acceleration (inertial mass, in ma). Every free-fall and pendulum experiment ever performed confirms this equivalence to extraordinary precision — it's one of the deepest and best-tested principles in all of physics.
Energy Conservation in a Swinging Pendulum
A swinging pendulum continuously trades potential energy for kinetic energy and back again. At the bottom of the swing, all of its energy is kinetic: KE = ½mv_max². At the extremes of the swing, where it momentarily stops before reversing direction, all of its energy is gravitational potential: PE = mgh, where h = L(1 − cos θ) is the height gained above the lowest point.
Equating these two gives the maximum speed directly: v_max = √(2gh). For a 1 m pendulum released from 20°, h = 1 × (1 − cos 20°) ≈ 0.0603 m, giving v_max = √(2 × 9.81 × 0.0603) ≈ 1.087 m/s at the bottom of the swing. At any intermediate displacement x from the bottom, the instantaneous speed follows v = √(v_max² − ω²x²) — another direct consequence of the sinusoidal motion.
This energy exchange is exactly what determines how hard a target-hitting pendulum swings in this game: releasing from a higher angle stores more potential energy, which converts entirely into kinetic energy — and therefore more speed — by the time the bob reaches the bottom of its arc.
Resonance
Every pendulum has a natural frequency, f₀ = (1/2π)√(g/L), at which it "wants" to oscillate. Applying a periodic driving force at exactly this frequency causes the amplitude to grow with each cycle — a phenomenon called resonance. This is precisely how a small child, or an adult pushing a swing, can build a large amplitude from a series of small, well-timed pushes: each push adds energy in phase with the existing motion.
Push at the wrong moment, out of sync with the natural frequency, and the energy you add can just as easily fight the existing motion, keeping amplitude small or even damping it out — destructive interference between the driving force and the oscillation already present.
Resonance isn't just a curiosity — it can be catastrophic at engineering scale. The Tacoma Narrows Bridge collapse in 1940 is a famous (if somewhat oversimplified in popular accounts) example of aeroelastic resonance destroying a structure, and it's precisely why modern bridges and tall buildings are engineered with damping systems to avoid matching the resonant frequencies of wind, foot traffic, or seismic activity. On the opposite end of the scale, resonance is deliberately exploited in quartz watch crystals, which oscillate at a precisely engineered 32,768 Hz to keep time with extraordinary accuracy.
Worked Example — Finding an Unknown Length
Problem: A pendulum is timed swinging through 20 complete oscillations in 36.4 seconds. Find its length, assuming g = 9.81 m/s².
Period: T = 36.4 / 20 = 1.82 s
Rearranging T = 2π√(L/g): L = g(T/2π)² = 9.81 × (1.82/6.283)²
L = 9.81 × 0.0839 ≈ 0.823 m — this timing-and-rearranging technique is exactly how early scientists measured g precisely using nothing more than a string, a mass, and an accurate stopwatch.
Real-World Applications
Seismometers: Many early seismometers used a heavy pendulum with a very long natural period, mounted so that the ground moves beneath it during an earthquake while the pendulum's inertia keeps it nearly stationary — the relative motion between the two is recorded as the seismic trace, a direct application of a pendulum's predictable, mass-independent dynamics.
Foucault's pendulum: A very long, freely swinging pendulum, first demonstrated publicly in Paris in 1851, appears to slowly rotate its plane of swing over the course of a day — not because the pendulum itself twists, but because the Earth rotates beneath it. It remains one of the most visually direct demonstrations that the Earth spins on its axis.
Metronomes and tuning devices: Mechanical metronomes are essentially adjustable pendulums, where sliding a weight up or down the arm changes the effective length L and therefore the tempo, using exactly the T = 2π√(L/g) relationship this game lets you explore directly.