How It Works
The pendulum length calculator inverts T = 2π√(L/g) to give L in terms of the target period and local gravity: L = g(T/2π)². It tells you how long a simple-pendulum string must be to produce a chosen tick rate. The result is the effective length — the distance from the pivot to the centre of mass of the bob — so a long thin rod with a heavy bob has a slightly larger effective length than the visible string.
Example Problem
How long does a simple pendulum need to be on Earth (g = 9.81 m/s²) to have a period of exactly 2 seconds — the classic grandfather-clock tick?
- Identify knowns: T = 2 s, g = 9.81 m/s².
- Compute T / 2π = 2 / 6.2832 ≈ 0.31831 s.
- Square the ratio: (0.31831)² ≈ 0.10132 s².
- Multiply by gravity: L = 9.81 × 0.10132 ≈ 0.9940 m.
- So the required pendulum length is just under one metre — about 99.4 cm.
Real clock movements use a slightly shorter rod (closer to 0.994 m) and tune to exact pitch with a small adjustment nut at the bottom of the bob.
Key Concepts
L = g(T/2π)² is the inverse of the small-angle simple-pendulum equation. It depends on g, so the same target period gives different required lengths on different planets — a 2-second pendulum on the Moon would be over 6× shorter than on Earth. The formula assumes the bob is a point mass and the string is massless; for a real bob with finite size, use the physical-pendulum formula instead.
Applications
- Designing a pendulum clock to tick at a chosen rate (seconds, half-seconds, or fractional ticks).
- Sizing a Foucault pendulum: a 25 m wire gives a period of about 10 s, slow enough to clearly show Earth's rotation.
- Tuning resonance: matching a swing's natural length to a driving frequency to maximize amplitude gain.
- Reverse-engineering historical clocks where the case dimensions imply the original pendulum length.
- Estimating the effective length of a wrecking ball's swing for crane operators planning lift cycles.
Common Mistakes
- Forgetting to square T/2π — squaring the period itself rather than the ratio gives the wrong answer by a factor of (2π)².
- Measuring length to the bottom of the bob instead of to its centre of mass; the effective length is pivot-to-COM.
- Using a fixed g = 9.81 m/s² at a high-altitude or polar location where local gravity differs by several parts per thousand.
- Applying the small-angle formula when the pendulum will swing through angles much larger than ~15°.
- Forgetting that the formula gives the simple-pendulum length — a stiff rod swinging about an end follows the physical-pendulum equation, not this one.
Frequently Asked Questions
How do you calculate the length of a pendulum?
Use L = g(T/2π)². Divide the target period T by 2π, square the result, then multiply by the local gravitational acceleration g. The output is the required string length in metres.
What is the formula for pendulum length?
L = g(T/2π)². It is the inverse of T = 2π√(L/g), the small-angle simple-pendulum period equation.
How long is a pendulum with a period of 1 second?
On Earth (g = 9.81 m/s²), a 1-second pendulum is L = 9.81 × (1/(2π))² ≈ 0.248 m, or about 24.8 cm.
How long is a 2-second (grandfather-clock) pendulum?
On Earth, L = 9.81 × (2/(2π))² ≈ 0.994 m — just under one metre. This is why grandfather clocks are about a metre tall in the pendulum case.
Does the formula work for any swing angle?
Only for small swings (below about 15°). For larger amplitudes the true period is longer than the small-angle prediction, so the formula gives a length that is slightly too long for the desired period.
How does the required length change with gravity?
L scales linearly with g for the same target period: doubling g doubles the required length. A 2-second pendulum on the Moon (g = 1.62 m/s²) is only 0.164 m long, versus 0.994 m on Earth.
Reference: Lindeburg, Michael R. 1992. Engineer In Training Reference Manual. Professional Publication, Inc. 8th Edition.
Worked Examples
Horology
How long should a grandfather clock pendulum be for a 2-second period?
- L = g(T / 2π)²
- L = 9.81 × (2 / 6.2832)²
- L = 9.81 × 0.10132
- L ≈ 0.994 m — almost exactly one metre.
Clock movements typically include a fine-pitch adjustment nut so the pendulum can be tuned to drift no more than a few seconds per day.
Earth Sciences
How long is a Foucault pendulum with a 10 s period (good for showing Earth's rotation)?
- L = g(T / 2π)²
- L = 9.81 × (10 / 6.2832)²
- L = 9.81 × 2.5330
- L ≈ 24.85 m — about the height of an eight-story building.
Long Foucault pendulums minimise air-drag corrections; the original 1851 installation at the Panthéon used a 67 m wire.
Lunar Physics
What length pendulum would tick once per second on the Moon?
- L = g_moon(T / 2π)² with g_moon = 1.62 m/s²
- L = 1.62 × (1 / 6.2832)²
- L = 1.62 × 0.025330
- L ≈ 0.0410 m — just over 4 cm.
The same 1-second period pendulum on Earth requires about 0.248 m of string — six times longer.
Related Calculators
- Pendulum Period Calculator — the forward direction: T = 2π√(L/g)
- Pendulum Hub (Period, Frequency, Physical) — the full pendulum calculator with all solve-for options
- Gravity Calculator — compute local gravitational acceleration before solving for length
- Circular Motion Calculator — another periodic-motion calculator (rotational period)
- Length Converter — convert pendulum length between metres, feet, and inches
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