Pendulum Calculator

Calculate pendulum period, string length, or local gravity using T = 2π√(L/g).

⏱️ Oscillation📐 T = 2π√(L/g)🕰️ Simple Harmonic Motion
Length (L)
Unit
Gravity g (m/s²)
⚠️ Please enter valid positive numbers.

What Is Pendulum Motion?

A simple pendulum consists of a mass (the bob) suspended by a string or rod of length L, swinging in a gravitational field g. For small oscillation angles (< 15°), the motion is simple harmonic motion (SHM) with period T = 2π√(L/g). The period is the time for one complete swing (out and back). Crucially, for small angles, T is independent of the mass and the amplitude — only L and g matter.

The formula reveals key relationships: doubling the length increases the period by √2 ≈ 1.41 (not 2). Quadrupling the length doubles the period. On the Moon (g = 1.62 m/s²), a 1 m pendulum has T = 2π√(1/1.62) = 4.95 s — nearly 2.5× longer than on Earth. On Jupiter (g = 24.8 m/s²), the same pendulum would swing in T = 1.26 s.

Pendulums were the basis of the world's most accurate clocks for nearly 300 years (1657–1930s). A grandfather clock uses a 1 m pendulum with T ≈ 2 seconds (1 second each way). Temperature compensation was a key engineering challenge because the rod length changes with temperature, shifting the period. Invar alloy (near-zero thermal expansion) solved this in precision clocks.

The simple pendulum formula assumes: small angle (sin θ ≈ θ in radians, valid for θ < 15°), massless string, bob treated as point mass, no air resistance or friction. For large angles, the exact period involves elliptic integrals. For a compound pendulum (distributed mass), replace L with L_effective = I/(mL_cm), where I is the moment of inertia about the pivot.

Formula Reference Table

QuantityFormulaNotes
Period (T)T = 2π · √(L/g)seconds; independent of mass and amplitude
Length (L)L = g·T²/(4π²)string length to pivot (meters)
Gravity (g)g = 4π²L/T²useful for measuring local g
Frequency (f)f = 1/Toscillations per second (Hz)
Angular frequency (ω)ω = 2π/T = √(g/L)rad/s
EnergyE = mgh_max = ½mL²ω²θ²θ in radians, h_max = L(1−cos θ)

3 Worked Examples

Example 1
Grandfather Clock

Design a clock pendulum that ticks once per second (T = 2 s). Find the required length on Earth.

  • T = 2π√(L/g) → T² = 4π²L/g → L = gT²/(4π²)
  • L = 9.8 × 4 / (4 × 9.87) = 39.2 / 39.48
  • L = 0.993 m ≈ 1 meter — matches the classic grandfather clock
✓ L ≈ 0.993 m (≈ 1 meter for a 2-second period clock)
Example 2
Moon Pendulum

A 0.5 m pendulum swings on the Moon (g = 1.62 m/s²). Find period and compare to Earth.

  • T_Moon = 2π√(0.5/1.62) = 2π × √0.3086 = 2π × 0.5555 = 3.49 s
  • T_Earth = 2π√(0.5/9.8) = 2π × 0.2259 = 1.42 s
  • Ratio: 3.49/1.42 = 2.46 — Moon pendulum is 2.46× slower
✓ T_Moon = 3.49 s (vs Earth 1.42 s — 2.46× slower)
Example 3
Measure Local Gravity

A 1.5 m pendulum is timed over 20 complete oscillations taking 49.2 s. Find g.

  • Period: T = 49.2/20 = 2.46 s
  • g = 4π²L/T² = 4 × 9.87 × 1.5 / (2.46²)
  • g = 59.22 / 6.052 = 9.79 m/s² (close to standard 9.8 m/s²)
  • The small difference reflects local geology and latitude
✓ g ≈ 9.79 m/s²

Real-World Applications

🕰️
Clocks and Timekeeping
The pendulum clock (Huygens, 1657) was the first timekeeper accurate to minutes per day. The precise T = 2π√(L/g) relationship made regulation simple: lengthen the rod slightly to slow the clock, shorten to speed it.
⚖️
Measuring Gravity
Pendulums are precision instruments for measuring local gravitational acceleration. Variations in g across Earth's surface (due to altitude, crustal density, latitude) reveal subsurface geology — oil deposits, ore bodies, and density anomalies.
🏗️
Structural Engineering
Pendulum-like oscillations cause problems in tall buildings and long bridges. The Taipei 101 skyscraper has a 660-tonne pendulum tuned mass damper (TMD) designed to swing opposite to wind-induced building oscillation, reducing sway by 40%.
🎭
Stage and Film
Camera stabilizers, Steadicams, and gimbal systems use pendulum-like dynamics for smooth motion. Lighting rigs and overhead cranes are pendulum systems whose natural frequency must be considered in operation.
🌍
Foucault Pendulum
A long pendulum (Foucault, 1851) swings in a fixed plane while Earth rotates beneath it, demonstrating Earth's rotation directly. The pendulum appears to rotate clockwise in the Northern Hemisphere, completing a full rotation in 24h/sin(latitude).

Common Mistakes to Avoid

⚠️
Forgetting the 2π factor

T = 2π√(L/g), not just √(L/g). Forgetting 2π gives a period ~6.28× too small. The factor 2π appears because one full cycle = 2π radians of rotation in the associated circular motion model.

⚠️
Using the formula beyond 15° amplitude

T = 2π√(L/g) assumes sin θ ≈ θ (small angle approximation). At 30°, the error is ~1.7%; at 45°, ~4%; at 90°, ~18%. For large swings, a correction term is needed: T ≈ 2π√(L/g) × (1 + θ²/16 + ...).

⚠️
Including mass in the calculation

The period of a simple pendulum is independent of mass (for a given L and g). Students sometimes try to include bob mass — it cancels out in the derivation.

⚠️
Using length in cm instead of m

T = 2π√(L/g) requires L in meters when g is in m/s². L = 50 cm = 0.50 m, not 50. Using 50 gives T = 2π√(50/9.8) ≈ 14.2 s instead of the correct 1.42 s.

⚠️
Measuring from the support to the string top, not to the bob center

L is the effective pendulum length — from the pivot point to the center of mass of the bob, not just the string length. Add the bob radius or half its height to the string length.

Frequently Asked Questions

Why is the period independent of mass? ▾
In the equation of motion mL(d²θ/dt²) = −mg·sin θ, the mass m cancels from both sides, leaving d²θ/dt² = −(g/L)·sin θ. This shows the oscillation frequency depends only on g/L, not on how heavy the bob is. A gold bob and a plastic bob on the same length string swing with exactly the same period.
What is the small angle approximation and when does it fail? ▾
sin θ ≈ θ (in radians) is accurate to within 1% for θ < 0.24 rad = 14°. For larger angles, the true period is longer than T = 2π√(L/g). At 90° swing (horizontal release), the actual period is about 18% longer. Clock pendulums are designed for small arcs to maintain accuracy.
How was g measured using pendulums historically? ▾
By measuring the period T of a pendulum of known length L and applying g = 4π²L/T². This method was used to compare g at different locations on Earth and discover that Earth is slightly flattened at the poles (where g is larger than at the equator). Modern gravimeters achieve precisions of 1 part in 10⁸.
What is a Foucault pendulum? ▾
A very long pendulum (often 30+ meters) set swinging in a fixed vertical plane. As time passes, the plane of oscillation appears to rotate because Earth rotates under the pendulum. The rotation rate = 15°/hour × sin(latitude). At the poles: 360°/24h. At the equator: no apparent rotation. It was the first direct proof of Earth's rotation (1851).
What is a compound pendulum? ▾
A compound (physical) pendulum has its mass distributed along its length rather than concentrated at one point. Its effective pendulum length is L_eff = I/(mL_cm), where I is the moment of inertia about the pivot and L_cm is the distance from pivot to center of mass. Rods, rulers, and irregularly shaped objects are compound pendulums.
Why do pendulum clocks stop when tilted? ▾
A pendulum clock must hang vertically for the pendulum to swing in its designed plane and with its designed arc. Tilting changes the effective gravity component (g_eff = g·cos(tilt angle)) and can cause the escapement (the mechanism that gives the pendulum a push each swing) to malfunction. This is also why antique clocks have adjustable feet.
What is the relationship between pendulum length and frequency? ▾
Frequency f = 1/T = (1/2π)√(g/L). Doubling frequency requires reducing L to 1/4 of original (f ∝ 1/√L). A pendulum oscillating at 1 Hz (one complete back-and-forth per second) requires L = g/(4π²) = 9.8/39.48 = 0.248 m ≈ 25 cm.
How does a metronome differ from a pendulum? ▾
A metronome uses an inverted pendulum with an adjustable counterweight above the pivot. Moving the weight higher lowers the effective center of mass, reducing the restoring torque and slowing oscillation. Moving it lower speeds the metronome up. The adjustable bob allows musicians to set any tempo between about 40–208 BPM.

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