Simple Pendulum Period Calculator

Definition of the simple pendulum and the period formula

The simple pendulum is a model: a point mass m hanging on a massless, inextensible string of length l in a uniform gravitational field. When displaced through an angle θ it feels a restoring force −m·g·sin θ tangent to its path. For small angles sin θ ≈ θ (in radians), so the motion becomes harmonic with angular frequency ω = √(g/l), and the period is T = 2π/ω = 2π√(l/g).

Three properties follow from the form of the formula. The period does not depend on mass, because the mass cancels in the equation of motion (the same reason all bodies fall alike). The period does not depend on amplitude — the isochronism noticed by Galileo. The period grows with the square root of length: a pendulum four times longer swings twice as slowly, and one a hundred times longer ten times as slowly.

Historical note: Galileo, Huygens, and the pendulum clock

According to Vincenzo Viviani, Galileo as a student in Pisa (about 1583) watched a swinging chandelier in the cathedral and, timing it with his pulse, noticed that the period did not depend on the amplitude. He studied the pendulum throughout his life and near its end designed a pendulum clock he did not live to build. He demonstrated the independence of the period from mass experimentally by comparing pendulums with lead and cork bobs.

Christiaan Huygens built the first working pendulum clock in 1656, and in the treatise Horologium oscillatorium (1673) he gave the exact theory: he derived T = 2π√(l/g), showed that Galileo’s isochronism is only approximate and that motion along a cycloid is exactly isochronous, and introduced the center of oscillation for the physical pendulum. The seconds pendulum, about 0.994 m long, was for a time considered as the standard of the meter before a fraction of the meridian was chosen in 1791.

Deriving the formula T = 2π√(l/g)

Step 1. A bob of mass m displaced through an angle θ moves along an arc of length s = l·θ. The bob feels its weight m·g and the string tension; the component of the weight along the path is −m·g·sin θ and acts as the restoring force. Step 2. Newton’s second law along the arc: m·l·θ″ = −m·g·sin θ, that is θ″ + (g/l)·sin θ = 0. The mass cancels already at this stage.

Step 3. For small angles sin θ ≈ θ and the equation takes the harmonic-oscillator form θ″ + ω²θ = 0 with ω² = g/l. Its solution is θ(t) = θ₀·cos(ωt + φ) with period T = 2π/ω = 2π√(l/g). Step 4. For l = 1 m and g = 9.81 m/s²: l/g = 0.10194 s², √0.10194 = 0.31928 s, T = 6.2832·0.31928 = 2.006 s. One swing takes T/2 = 1.003 s, and the frequency is f = 1/T = 0.4985 Hz.

Model assumptions and the amplitude correction

The formula assumes a point mass, a massless and inextensible string, no air resistance, constant g, and a small displacement angle. For a real bob of finite size and a string with mass one must use the physical pendulum: T = 2π√(I/(m·g·d)), where I is the moment of inertia about the pivot and d the distance of the center of mass from the pivot; for a ball of radius R on a string l the relative correction is about ⅕(R/l)². For a ball of radius 2 cm on a 1 m string that is 0.008 %.

The amplitude matters most in practice. The exact period is given by an elliptic integral, expanded as a series: T = T₀·(1 + θ₀²/16 + 11θ₀⁴/3072 + …), where θ₀ is the amplitude in radians. At 10° (0.1745 rad) the correction is 0.19 %, at 20° 0.77 %, at 45° 4.0 %, at 90° 18 %. Pendulum clocks run at amplitudes of a few degrees precisely so that the error is negligible; the calculator’s result corresponds to the limit θ₀ → 0.

The pendulum as a gravimeter and a clock

Inverting the formula, g = 4π²l/T², turns the pendulum into an instrument for measuring the gravitational acceleration. In 1672 Jean Richer found that a pendulum clock regulated in Paris lost about 2.5 minutes a day in Cayenne — the first evidence that g decreases toward the equator and that the Earth is flattened. Kater’s reversible pendulum (1817) measured g to 10⁻⁵ and for more than a century was the basic tool of gravimetry.

A pendulum clock is regulated by length, not by mass: shortening a 0.994 m pendulum by 1 mm shortens the period by 0.05 %, making the clock gain about 43 s per day. That is why clocks have an adjusting nut under the pendulum bob, and precision clocks have temperature compensation (Graham’s mercury pendulum of 1721, Harrison’s gridiron pendulum of 1726), because a steel rod lengthens by 12 μm per meter for every kelvin.

The value of g, other planets, and the Foucault pendulum

The period scales as 1/√g. On the Moon (g = 1.62 m/s²) a 0.5 m pendulum has a period of 3.49 s instead of 1.42 s on Earth, and on Mars (3.71 m/s²) 2.31 s. On Earth itself g varies from 9.780 m/s² at the equator to 9.832 m/s² at the poles, which for a one-meter pendulum gives a period difference of 5 ms — measurable already in the 17th century. The g for any planet comes from the g = GM/r² calculator; the result can be entered in the g field.

The Foucault pendulum (1851) is an ordinary long pendulum, 67 m long in the Paris Panthéon with a period of 16.4 s. Its plane of swing rotates relative to the floor with a period of 24 h/sin φ, where φ is the latitude — about 32 hours in Paris — and so demonstrates the Earth’s rotation. The swing period is still described by T = 2π√(l/g); the great length serves to make the precession clearly visible within an hour.

Sources

The definition, derivation, amplitude correction, and historical note come from physics textbooks and historical sources: C. Huygens, Horologium oscillatorium, Paris 1673; D. Halliday, R. Resnick, J. Walker, Fundamentals of Physics, chapter on oscillations (simple and physical pendulum); H. D. Young, R. A. Freedman, University Physics, chapter “Periodic Motion,” sections “The Simple Pendulum” and “The Physical Pendulum”; R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. I, chapter 21 (The Harmonic Oscillator); L. D. Landau, E. M. Lifshitz, Mechanics, § 11 (the pendulum, elliptic integral); H. Kater, Philosophical Transactions 108 (1818); L. Foucault, Comptes rendus 32 (1851); BIPM, SI Brochure, 9th edition. This page is for educational purposes.