Pendulum Period Calculator – T = 2π√(L/g) | Simple Pendulum

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Length L

Period T

s

Frequency f

Hz

Gravity g

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The Simple Pendulum Explained

A simple pendulum consists of a massless, inextensible string of length L attached to a fixed pivot, with a point mass (bob) at the other end. When displaced from its rest position and released, it oscillates with simple harmonic motion — provided the angle is small.

Its most remarkable property: the period is independent of both mass and amplitude (for small angles). This was first noted by Galileo (~1602) while watching a chandelier swing in Pisa Cathedral.

T = 2π√(L/g)

Small-angle formula

f = 1/T

Frequency in Hz

ω = √(g/L)

Angular frequency

Deriving T = 2π√(L/g)

For small angles, the restoring force is approximately F ≈ −mg·θ (where θ is in radians). Applying Newton's second law for rotational motion and using the small-angle approximation sin(θ) ≈ θ gives:

d²θ/dt² = −(g/L)·θ

This is SHM with ω² = g/L → ω = √(g/L)

T = 2π/ω = 2π√(L/g)

The derivation shows that neither mass m nor amplitude θ appear in the final formula — only string length L and gravitational acceleration g matter.

Small-Angle vs Large-Angle Pendulums

For angles above ~15°, the simple formula underestimates the true period. The first-order correction (Lindstedt series) is:

T_corr = T₀ · (1 + θ²_rad/16 + 11·θ⁴_rad/3072 + ...)

Amplitude θ	Period error vs simple formula
5°	+0.05%
15°	+0.43%
30°	+1.73%
45°	+3.99%
90°	+18.0%

Physical Pendulum

A physical (compound) pendulum is any rigid body swinging around a pivot. Unlike the simple pendulum, its period depends on the mass distribution:

T = 2π · √(I / (m·g·d))

Where I is the moment of inertia about the pivot, m is total mass, and d is the distance from pivot to centre of mass. Examples: a uniform rod swinging from one end has T = 2π√(2L/3g).

Length vs Period Reference Table

Length L	Period T (Earth)	Frequency f	Period T (Moon)
0.1 m	0.635 s	1.575 Hz	1.562 s
0.25 m	1.003 s	0.997 Hz	2.470 s
0.5 m	1.418 s	0.705 Hz	3.492 s
1.0 m	2.006 s	0.499 Hz	4.939 s
2.5 m	3.170 s	0.315 Hz	7.804 s
5.0 m	4.483 s	0.223 Hz	11.035 s
10.0 m	6.342 s	0.158 Hz	15.607 s

Worked Examples

Example 1 — Grandfather clock (L=1m)

Given: L = 1m, g = 9.81

T = 2π√(1/9.81) = 2.006 s

f = 1/2.006 = 0.499 Hz

Example 2 — Find L for 1s tick (T=2s)

Given: T = 2s, g = 9.81

L = g·(T/2π)² = 9.81·(0.3183)²

L = 0.993 m

Example 3 — Moon pendulum 1m

Given: L = 1m, g = 1.62

T = 2π√(1/1.62) = 4.939 s

f = 0.202 Hz

Example 4 — Foucault pendulum (L=67m)

Given: L = 67m, g = 9.81

T = 2π√(67/9.81) = 16.42 s

f = 0.0609 Hz

Frequently Asked Questions

A simple pendulum is an idealized model consisting of a massless string of length L with a point mass (bob) at the end, swinging under gravity with no friction. For small angles (under ~15°), its period is T = 2π√(L/g), independent of mass and approximately independent of amplitude.

No. The period T = 2π√(L/g) contains no mass term. A heavier bob and a lighter bob on the same string length swing with exactly the same period. This was one of Galileo's key discoveries around 1602.

The simple formula T = 2π√(L/g) assumes sin(θ) ≈ θ in radians, which holds well for angles under about 15°. Above this, the actual period is longer. At 30° the error is ~1.7%; at 90° it is ~18%. Use the large-amplitude correction in Advanced mode for precise results at larger angles.

A "seconds pendulum" has a half-period of 1 second (full period T = 2s). Rearranging: L = g(T/2π)² = 9.81×(2/2π)² ≈ 0.9929 m ≈ 99.3 cm. This is why grandfather clocks are tall — the pendulum is nearly 1 metre long.

Period T is the time for one complete oscillation (one full swing back and forth), measured in seconds. Frequency f is the number of complete oscillations per second, measured in hertz (Hz). They are reciprocals: f = 1/T. A pendulum with T = 2s has f = 0.5 Hz.

Period is inversely proportional to the square root of g: T ∝ 1/√g. On the Moon (g = 1.62 m/s²), a 1m pendulum has T ≈ 4.94 s — about 2.46× longer than on Earth (T ≈ 2.01 s). On Jupiter (g = 24.79), the same pendulum has T ≈ 1.26 s.

A physical (compound) pendulum is a rigid body swinging about a pivot point, not a simple point mass on a string. Its period is T = 2π√(I/(mgd)), where I is the moment of inertia about the pivot, m is mass, g is gravity, and d is the distance from pivot to centre of mass.

Léon Foucault's 1851 pendulum (67 m long, 28 kg bob) demonstrated Earth's rotation by showing that the pendulum's swing plane appeared to rotate over the day. The rotation rate at latitude λ is ω = Ω·sin(λ), where Ω = 2π/24h. At Paris (48.9°N), the plane rotates ~11.3°/hour.

Simple Pendulum Period Calculator

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