The simple pendulum: T = 2π·√(L/g)

The simple pendulum is one of the most studied physical systems in classical mechanics: a point mass suspended by an inextensible string of length L, oscillating under gravity g around the equilibrium position. For small oscillations, the period T (time for one full oscillation) is given by Huygens' famous formula: T = 2π·√(L/g). This calculator solves T from L and g, derives L from T and g, or determines g from measured period (a classical method for measuring g in the lab).

Fundamental formulas

• T = 2π·√(L/g) — period from length and gravity
• L = g·(T/(2π))² — length from period and gravity
• g = L·(2π/T)² — gravity from period and length
• f = 1/T — frequency (Hz)
• ω = 2π/T = √(g/L) — angular frequency (rad/s)

Note: the period DOES NOT depend on the mass of the pendulum nor on oscillation amplitude (for small angles). This is the isochronism discovered by Galileo observing the chandelier of Pisa Cathedral (c. 1583).

Physical derivation

For a mass m hanging from a string of length L, the tangential component of gravitational force is F = -m·g·sin(θ), with θ the angle from vertical. For small angles sin(θ) ≈ θ (radians), so:

m·L·d²θ/dt² = -m·g·θ → d²θ/dt² = -(g/L)·θ

Simple harmonic motion equation with ω² = g/L. Period is T = 2π/ω = 2π·√(L/g).

The seconds pendulum

A pendulum with T = 2 s (1 s per simple oscillation) is called seconds pendulum and was historically used in pendulum clocks. Solving for L:

L = g·(2/(2π))² = g/π² ≈ 0.9939 m (at Paris, g=9.81 m/s²)

The slight variation of g with latitude (9.78 at equator, 9.83 at poles) makes this a classical method to determine local gravity. Pendulum clocks needed re-calibration when moved: a Paris → Madrid clock slowed because g decreases southward.

Worked numerical examples

Example 1: child's swing

A 3 m long swing on Earth:

• T = 2π·√(3/9.81) = 3.48 s
• Frequency f = 1/3.48 ≈ 0.287 Hz

Example 2: 1-meter pendulum

• T = 2π·√(1/9.81) = 2.006 s
• Almost seconds pendulum, but not exactly 2 s

Example 3: Foucault pendulum (Panthéon, Paris)

Length 67 m, mass 28 kg:

• T = 2π·√(67/9.81) = 16.42 s
• 1 oscillation per 16 s, deflection observable from Earth's rotation

Example 4: lab measurement of g

Pendulum L = 1 m, measured T = 2.006 s (averaged over 100 oscillations):

• g = L·(2π/T)² = 9.808 m/s²
• Typical precision ±0.5% with good time measurement
• Used in geodesy: g varies with latitude and altitude

Large amplitude correction

T = 2π·√(L/g) holds only for small oscillations (amplitude θ₀ < 15°). For larger amplitudes, true period is:

T = T₀·[1 + (1/4)·sin²(θ₀/2) + (9/64)·sin⁴(θ₀/2) + ...]

Above 30° the error from isochronous model is observable. The pendulum is less "isochronous" than Galileo thought.

The physical (real) pendulum

A real pendulum (rod, disk, rigid object) is not a simple pendulum. The formula becomes:

T = 2π·√(I/(m·g·d))

Where I is moment of inertia about the pivot and d the distance from pivot to center of mass. Generalizes the simple pendulum (where I = m·L² and d = L, recovering T = 2π·√(L/g)).

Example: uniform rod

Rod of length L, pivoted at one end:

• I = (1/3)·m·L²
• d = L/2
• T = 2π·√((1/3)·m·L² / (m·g·L/2)) = 2π·√(2L/(3g))
• Same as simple pendulum of length L' = 2L/3

Foucault pendulum

In 1851, French physicist Léon Foucault hung a 67 m pendulum in the Paris Panthéon (28 kg brass mass) to demonstrate Earth's rotation. The plane of oscillation slowly rotates (clockwise in Northern Hemisphere) because Earth rotates beneath it.

Angular rotation rate of oscillation plane:

ω_Foucault = ω_Earth·sin(φ)

with φ the latitude. At Paris (φ ≈ 48.86°N), plane makes full rotation in about 31 hours 50 minutes. At North Pole would be 24 hours exactly; at equator no rotation.

Practical applications

Mechanical clocks

• Pendulum clocks (Huygens 1656): first 10 s/day precision
• Modern quartz clocks use oscillators instead of pendulums
• Atomic clocks (Cs): 1 oscillation = 9,192,631,770 cesium atom periods

Gravity measurement

• Pendulum gravimeters: historically used in geodesy
• g variations reveal underground deposits (minerals, oil)
• Today replaced by spring or free-fall gravimeters (more precise)

Musical instruments

• Pendulum metronomes: mark time
• Pendulum length sets BPM: BPM = 60·√(g/L)/π

Physics education

• Classic school lab experiment
• Introduces harmonic motion, small oscillations, approximations
• Method to measure g independent of mass

Civil engineering

• Pendulum dampers in skyscrapers (e.g., Taipei 101: 660-ton pendulum against wind)
• Seismic: inverted pendulums to reduce vibrations

Gravity on other planets

Period of a 1 m pendulum varies dramatically in the Solar System:

A pendulum on the Moon beats much slower: Apollo astronauts demonstrated this (hammer and feather fall together in vacuum).

Pendulum energy

During oscillation, mechanical energy transforms between potential and kinetic:

• At maximum amplitude (highest): all PE, no KE. PE = m·g·h = m·g·L·(1-cos(θ₀))
• At lowest point: all KE, no PE. KE = ½·m·v²_max
• Small angles: v_max ≈ L·ω·θ₀ = θ₀·√(g·L)

Without friction, pendulum oscillates forever with constant total energy. In practice, air resistance and pivot friction dissipate energy: amplitude decays exponentially until stopping.

Factors affecting real period

• Amplitude (non-isochronous correction for θ > 15°)
• Temperature: thermal expansion of string changes L
• Atmospheric pressure: buoyancy and drag effects
• Latitude: g varies 9.78 (equator) to 9.83 (poles)
• Altitude: g decreases ~3 μm/s² per meter
• String mass: if not negligible, use physical pendulum
• Pivot friction: damping and slight lag

History of the pendulum

Galileo Galilei (1583, Pisa): observes Cathedral chandelier and discovers isochronism for small amplitudes.

Christian Huygens (1656, Netherlands): builds first pendulum clock; formula T = 2π·√(L/g). Publishes Horologium Oscillatorium (1673).

Isaac Newton (1687, Principia): uses pendulum to verify equivalence of inertial and gravitational mass.

Léon Foucault (1851, Paris): demonstrates Earth rotation with Foucault pendulum. Also installs first gyroscope.

Pendulum gravimeters (19th-20th c.): precision geodetic measurements; now replaced.

FAQ

What is the pendulum period formula?

T = 2π·√(L/g), where L is string length (m) and g gravitational acceleration (m/s²). Result in seconds. Valid only for small oscillations (amplitude < 15°).

Does the period depend on the pendulum mass?

No. Galileo's isochronism: period depends only on L and g, not mass. True for simple pendulums (point mass, massless string) and small oscillations.

Why does the formula require small oscillations?

Because it uses sin(θ) ≈ θ approximation. For θ > 15° the approximation fails and period depends on amplitude (non-isochronism).

How long is a pendulum with period 1 second?

For T = 1 s and g = 9.81: L = g·(T/(2π))² = 0.248 m. For T = 2 s (seconds pendulum): L = 0.994 m.

How do you measure g with a pendulum?

Measure period T of pendulum with known L (better to average over N oscillations). Then g = L·(2π/T)². ±0.5% precision with accurate time measurement. Classic school lab method.

What's the difference between simple and physical pendulum?

Simple pendulum: point mass, massless string: T = 2π·√(L/g). Physical pendulum: rigid body with distributed mass: T = 2π·√(I/(m·g·d)), where I is moment of inertia and d distance from pivot to center of mass.

Why does the Foucault pendulum rotate?

Not the pendulum rotating: the Earth rotates beneath it. The oscillation plane is fixed relative to stars (approximately), while Earth rotates. At latitude φ, plane appears to rotate with ω_Earth·sin(φ): at Paris, full rotation in ~32 hours.

Updated July 2026