Pendulum Period & Length Calculator

Pendulum Physics Guide

What should I know about Simple Pendulum Formula?

T = 2π√(L/g), where T is the period in seconds, L is the length in metres, and g is gravitational acceleration (9.81 m/s² on Earth). The period is independent of mass and (for small angles under 15°) independent of amplitude — this is the isochrone property discovered by Galileo. A pendulum of length 0.994m has a period of almost exactly 2 seconds (one second each way), which was the basis of the original metre definition.

What do I need to know about The Seconds Pendulum?

A pendulum with a period of 2 seconds (one beat per second) has a length of approximately 0.994m at sea level in London (g = 9.812 m/s²). This was historically used to define the metre — the French Academy proposed defining the metre as the length of a seconds pendulum, before settling on the one ten-millionth of the Earth's circumference definition. Pendulum clocks use escapement mechanisms to count swings precisely, with pendulum length adjusted by moving the bob to regulate accuracy.

What do I need to know about Effect of Gravity?

The pendulum period depends on local gravity, which varies slightly with latitude and altitude. At the equator (g = 9.780 m/s²), a pendulum runs slightly slower than at the poles (g = 9.832 m/s²). A seconds pendulum calibrated in London would run slow on the equator by approximately 140 seconds (over 2 minutes) per day. On the Moon (g = 1.62 m/s²), the same pendulum would have a period of √(9.81/1.62) ≈ 2.46× longer — beating once every 4.9 seconds instead of 2.

What should I know about Limitations of the Simple Model?

The simple pendulum formula assumes: small amplitude (under 15° for under 0.5% error), massless rigid rod, point mass bob, no air resistance or friction, and uniform gravity. Real pendulums deviate from the formula at large amplitudes, and physical pendulums (distributed mass) require the moment of inertia in the calculation. For large amplitude corrections, the period is: T ≈ T₀(1 + θ²/16 + ...) where θ is in radians.

What should I know about Energy Exchange in a Swinging Pendulum?

As a pendulum swings, it continuously exchanges between two forms of energy while its total mechanical energy (ignoring friction and air resistance) remains constant. At the highest points of the swing (the extremes, where the pendulum momentarily stops before reversing direction), all of its energy is gravitational potential energy, and its kinetic energy is zero. At the lowest point of the swing (the bottom of the arc, directly below the pivot), the pendulum has fallen as far as it will in that swing, converting all of that potential energy into kinetic energy, and this is where the pendulum moves at its fastest speed. At every point in between, the pendulum's energy is some mix of both forms, with the two always summing to the same constant total (assuming no energy losses). This continuous conversion is why a frictionless pendulum would, in principle, swing forever at the same amplitude — real pendulums gradually lose amplitude over time because air resistance and friction at the pivot slowly drain mechanical energy away as heat, not because the energy exchange itself is imperfect.

What's the difference between a Simple Pendulum and a Physical Pendulum?

The formula T = 2π√(L/g) used in this calculator describes an idealised simple pendulum: a single point mass swinging on a massless, rigid string or rod. Most real swinging objects — a metre rule pivoted at one end, a swinging door, a clock's pendulum bob with actual size and shape — are better described as a physical pendulum, where the mass is distributed throughout an extended rigid body rather than concentrated at a single point. A physical pendulum's period is instead T = 2π√(I / (mgd)), where I is the object's moment of inertia about the pivot point, m is its total mass, and d is the distance from the pivot to the object's centre of mass. For a simple pendulum, the moment of inertia reduces to I = mL², which substituted into the physical pendulum formula gives back exactly T = 2π√(L/g) — showing that the simple pendulum formula is really just a special case of the more general physical pendulum equation, valid specifically when essentially all of an object's mass is concentrated at one point at the end of the swing.

What should I know about Foucault's Pendulum?

In 1851, Léon Foucault suspended a long, heavy pendulum from the dome of the Panthéon in Paris and showed its swing plane gradually rotates relative to the floor beneath it — direct, visible proof that the Earth itself is rotating. The rotation rate depends on latitude: a full 360° precession in 24 hours at the poles, no precession at all on the equator, and intermediate rates in between, following a sin(latitude) relationship. It remains one of the simplest and most striking demonstrations of a fundamental physical fact, requiring nothing more exotic than a pendulum, a clock, and patience.

What should I know about Pendulum Length vs Period?

What should I know about a Worked Example: Large-Amplitude Correction?

The simple formula T=2π√(L/g) assumes small-angle swings, but real pendulums released from a larger amplitude run measurably slower than this predicts. The correction formula T ≈ T0(1+θ²/16) lets this error be quantified. Take a 1-metre pendulum (T0=2.006s) released from a 40° amplitude. Converting: 40×(π/180)=0.698 rad. Correction factor: 1+(0.698)²/16 = 1+0.4872/16 = 1.0305. Corrected period: T ≈ 2.006×1.0305 ≈ 2.068s — about 3% longer than the small-angle prediction. For comparison, at just 10° amplitude (0.1745 rad), the correction factor is only 1+(0.1745)²/16 = 1.0019, a 0.2% effect, confirming why the standard small-angle formula is accurate below about 15° but increasingly unreliable at larger swing angles.

What should I know about Damping and the Quality Factor (Q)?

Every real pendulum loses energy over time to air resistance and friction at the pivot, gradually shrinking its swing amplitude even though its period stays almost exactly the same (period is remarkably insensitive to amplitude, within the small-angle approximation). This decay is often described by a quality factor, Q, which measures how many radians of oscillation occur before the energy decays substantially — a high-Q pendulum (a heavy bob on a thin wire, swinging in a near-vacuum) can swing for days before noticeably losing amplitude, while a low-Q pendulum (a lightweight bob swinging in air, or with a stiff, lossy pivot) might visibly die down within a few dozen swings. Clock pendulums are deliberately designed for very high Q, since a clock that loses amplitude slowly keeps more consistent time between the small pushes an escapement mechanism gives it on each swing.

What should I know about Pendulum Motion as Simple Harmonic Motion?

A pendulum swinging through small angles is a classic example of simple harmonic motion (SHM), meaning its angular displacement varies sinusoidally with time: θ(t) = θ_max × cos(ωt), where ω = 2π/T is the angular frequency. This lets you find the pendulum's speed at any point in its swing, not just its period. The maximum speed, reached at the bottom of the swing, is v_max = θ_max × L × ω (for small angles, θ_max in radians). Worked example: a 1-metre pendulum (T=2.006s, so ω=2π/2.006≈3.133 rad/s) released from an angular amplitude of 0.15 radians (about 8.6°). Maximum speed: v_max = 0.15×1×3.133 ≈ 0.470 m/s. This same SHM framework is exactly what connects a swinging pendulum to a mass oscillating on a spring, a vibrating guitar string, and an AC electrical circuit — all governed by the same underlying differential equation, d²x/dt² = −ω²x, just with different physical quantities playing the role of displacement.

What should I know about Air Resistance and Bob Shape?

Although the idealised pendulum formula ignores air resistance entirely, a bob's shape and size do have a small real-world effect, mainly on how quickly its amplitude decays rather than on its period itself. A dense, compact, aerodynamic bob (a small lead or steel sphere) experiences relatively little air drag compared to its weight, so it swings for a long time before noticeably losing amplitude — exactly the design used in scientific pendulum clocks and in classroom demonstrations meant to approximate the idealised simple pendulum as closely as possible. A large, light, or irregularly shaped bob (a sheet of card, a feather on a string) experiences much more drag relative to its weight, damping out far more quickly. This is why physics experiments measuring 'g' using a pendulum specifically recommend a dense, compact, aerodynamic bob — it keeps the real pendulum's behaviour as close as possible to the idealised formula the experiment relies on.