How this instrument works
Period here means one complete round trip: bob released, swung across, and returned to where it started. Two quantities famously refuse to appear in T = 2π√(L/g) — bob mass and release angle. Hang a lead weight and a cork on identical one-metre strings and both keep time together, because gravity scales driving force and inertia in equal measure, so m cancels out of every line of working. Galileo noticed isochronism around 1602 and set it down in a letter to Guido Ubaldo del Monte. His eyes were sharper than his instruments; he had nothing accurate enough to time a swing except his own pulse.
Christiaan Huygens turned that observation into machinery, building the first working pendulum clock in 1656 and patenting it twelve months later. The Horologium Oscillatorium he published in 1673 supplied the derivation and proved that the cycloid, not the circle, is exactly isochronous — accuracy jumped from roughly fifteen minutes of drift per day to fifteen seconds. Bobs near 0.994 m beat seconds, and that tidy figure was twice proposed as the natural standard of length: by John Wilkins in 1668, then again during France's metric reform of 1790. It lost, deservedly. In 1672 Jean Richer carried one such clock to Cayenne and watched it fall behind Paris by over two minutes daily, because gravity weakens toward an equator that bulges. Any unit defined this way would have depended on latitude.
Small angles are doing quiet work inside that expression. Restoring torque actually goes as sin θ, and trading sin θ for θ is precisely what makes the closed-form solution possible. Past that swap, period grows with amplitude: release from 10° and true period runs 0.19% long, from 20° about 0.76%, from 30° roughly 1.7% — trivial in the teaching lab, ruinous in the regulator clock, which is why precision escapements keep amplitude stingy. Three further idealisations lurk alongside: massless string, bob compact enough to count as pointlike, and no air drag or pivot friction. Give any real rod or hoop appreciable moment of inertia and you have left this sheet behind for the physical pendulum.
- Type your figure into Pendulum length, in centimetres, metres, feet, or inches; 1 m sits there as the default.
- Measure from pivot to centre of mass of a bob, not to its underside — a 15 mm bob radius overlooked on a 1 m pendulum shifts period by about 0.75%.
- Read Period of one swing, which counts a full there-and-back cycle in seconds. Switch its menu to milliseconds for very short pendulums.
- Frequency appears alongside in hertz, one divided by period — handy for comparing against a metronome mark or a scope trace.
- Keep release angle near 10° or below in real experiments, or your reading starts drifting low at that second decimal place.
Worked example — a one-metre pendulum
Set Pendulum length to 1 m and leave everything else alone. Working: L ⁄ g = 1 ⁄ 9.80665 = 0.1019716, whose square root is 0.3193300, and 2π times that returns Period of one swing = 2.00640929259 s. Frequency follows at 1 ⁄ 2.00640929 = 0.498403 Hz, a shade under half a cycle each second.
That answer sits maddeningly close to two seconds without quite landing there. Trim a bob to 0.9936 m and period becomes 2.000 s exactly — a seconds pendulum, ticking once per second on each traverse, which is why longcase movements were built around roughly a metre of swing and why grandfather cases stand as tall as they do. Those 6.4 missing millimetres are worth 0.32% in period, or about four and a half minutes of drift a day should you ignore them.
Questions
Does a heavier bob swing more slowly?
No — mass cancels completely. Gravity pulls a heavy bob harder, but a heavy bob also resists acceleration in exact proportion, and since both effects scale with m it drops out of the equation of motion. Swap lead for cork on one string and period holds steady, give or take whatever air drag contributes. This is the same cancellation behind free fall, where every object accelerates alike in vacuum. Mass enters only through damping: a light bob bleeds amplitude faster, though it keeps its rhythm while doing so.
Why does amplitude change period when the formula says it cannot?
Because that formula is an approximation. Restoring torque follows sin θ rather than θ, and trading one for the other is what conjures a closed-form answer. True period runs longer by roughly 1 + θ₀²/16 for release angle θ₀ in radians: 0.19% at 10°, 0.76% at 20°, 1.7% at 30°. Huygens sidestepped this in 1673 by hanging his bob between cycloidal cheeks, forcing an exactly isochronous path at any amplitude — elegant geometry, though friction at those cheeks cost him more than it gained.
What length gives a period of exactly one second?
About 24.8 cm at standard gravity. Rearrange to L = g(T/2π)², giving 9.80665 × (1 ⁄ 6.28319)² = 0.2484 m. Notice that quartering length halves period, since T depends on a square root of L — so 1 m at 2.006 s and 24.8 cm at 1.000 s are one relationship seen twice. Horologists mean something else by 'seconds pendulum': theirs is 0.9936 m, beating once per second on each traverse, which is a two-second full cycle.
Does my local gravity change the answer?
Slightly, and measurably. This sheet uses standard gravity, 9.80665 m/s² — a conventional figure the 3rd CGPM pinned down in 1901, not anything measured at your bench. Real g runs near 9.780 at sea level on an equator and 9.832 at either pole — a 0.5% spread, worth 0.26% in period, or nearly four minutes a day on a clock. Pendulums were sensitive enough to expose that variation centuries before anyone could explain it, and Kater's reversible pendulum of 1817 pinned g down to parts per million. Gravimetry stayed pendulum work until free-fall instruments took over.
How should I time swings accurately?
Count many swings, never one. Human reaction error of perhaps 0.2 s wrecks the single 2 s measurement but shrinks to 0.002 s when spread across one hundred cycles. Start and stop at the lowest point of travel, where speed peaks and timing ambiguity is smallest, rather than at the turning point where bobs loiter. Count zero on your first pass, not one. Above all, confirm you are timing the full cycle — releasing, crossing, returning. Timing half of one and calling it T yields g four times too large.
Where does this stop being a simple pendulum?
As soon as mass is spread out instead of concentrated. A swinging rod, hoop, human leg, or wrecking ball on a chain is a physical pendulum, obeying T = 2π√(I/mgd), where I is moment of inertia around the pivot and d is pivot-to-centre-of-mass distance. A uniform rod of length ℓ pivoted at one end reduces to T = 2π√(2ℓ/3g), swinging as though it were a simple pendulum two-thirds as long. Torsion pendulums, coupled pairs, and anything driven near resonance need separate treatment again.