Pendulum Calculator

Calculate pendulum period, frequency, and length. Enter length and gravity — get period and frequency for clock design, seismology, and physics experiments.

m
m/s²

Enter your values above to see the results.

Tips & Notes

  • Period depends only on length and gravity — not mass or amplitude (for small angles under 15°). A 1 kg and 100 kg pendulum of the same length swing with identical periods.
  • The 1-meter pendulum on Earth: T = 2π√(1/9.81) = 2.006 seconds. This is close to 2 seconds by design — early clock-makers used L = 0.994 m for a exactly 2-second period (1 second per swing).
  • Gravity varies by location: g = 9.832 m/s² at the poles, 9.780 m/s² at the equator, 9.807 m/s² standard. A clock calibrated at the poles runs faster at the equator — the period increases because g is smaller.
  • The simple pendulum formula T = 2π√(L/g) is accurate within 0.5% for amplitudes under 15° from vertical. Beyond 15°, a correction factor is needed: T ≈ T₀ × (1 + θ²/16) for angle θ in radians.
  • Frequency f = 1/T (Hz). A 0.25-meter pendulum: T = 2π√(0.25/9.81) = 1.003 s, f = 0.997 Hz — nearly 1 oscillation per second, useful as a timing reference.

Common Mistakes

  • Thinking mass affects the period — it does not (for a simple pendulum in the ideal case). This was Galileo's insight: heavy and light pendulums of the same length swing identically.
  • Using degrees instead of radians in the correction formula — the amplitude correction T ≈ T₀ × (1 + θ²/16) requires θ in radians. 15° = 0.2618 radians; correction = 1 + 0.0683/16 = 1.004 (0.4% error).
  • Forgetting to square the length ratio when scaling — to double the period, quadruple the length (period ∝ √L). A 2× period requires L × 4, not L × 2.
  • Using g = 10 m/s² for precision calculations — the 2% error from using 10 instead of 9.81 creates a 1% period error. For clock design or physics experiments, always use g = 9.807 m/s² or the local measured value.
  • Applying the simple pendulum formula to physical pendulums — a physical pendulum (distributed mass, rigid rod) uses T = 2π√(I/mgd), where I is moment of inertia and d is distance from pivot to center of mass.

Pendulum Calculator Overview

The simple pendulum is one of the most elegant and extensively applied systems in physics and engineering. Its period depends only on length and local gravity — not mass, not amplitude (for small angles) — making it a pure length-to-time converter that has served as a timekeeping standard for centuries.

Simple pendulum period formula:

T = 2π × √(L/g) | f = 1/T | L = g × (T/2π)² | Units: T in seconds, L in meters
EX: Grandfather clock pendulum, T = 2 s → L = 9.81 × (2/2π)² = 9.81 × 0.1013 = 0.9940 m ≈ 99.4 cm. Alternatively: 50 cm pendulum → T = 2π × √(0.5/9.81) = 2π × 0.2257 = 1.418 s, f = 0.705 Hz
Pendulum period on different celestial bodies:
T_new / T_Earth = √(g_Earth / g_new) — period scales as inverse square root of gravity
EX: 1 m pendulum on Moon (g=1.62 m/s²) → T = 2π√(1/1.62) = 4.94 s (vs 2.006 s Earth). On Mars (g=3.72 m/s²) → T = 2π√(1/3.72) = 3.26 s
Pendulum length for given period — reference table:
Period (s)Length (m)Frequency (Hz)Application
0.50.06212.0Short clock pendulum
1.00.24851.0Metronome at 60 BPM
2.00.99400.5Grandfather clock (classic)
4.03.9760.25Large tower clock
10.024.850.1Seismograph pendulum
Gravity at different locations — pendulum period comparison (L = 1 m):
Locationg (m/s²)T for L=1mDeviation from standard
North Pole9.8322.004 s−0.1%
Standard (sea level, 45°)9.8072.006 s0%
London9.8122.005 s−0.05%
Equator (sea level)9.7802.009 s+0.15%
Moon1.6204.936 s+146%
Mars3.7203.257 s+62%
The pendulum's independence of mass (for the simple case) is a demonstration of the equivalence principle — the same principle that Einstein generalized into general relativity. Galileo observed it with a chandelier in Pisa around 1602; Newton used it to establish that gravitational and inertial mass are equal to within measurement error. Today, torsion pendulums measure the Newtonian gravitational constant G, and pendulum-like seismographs continue to detect earthquakes worldwide with the same fundamental principle that has governed timekeeping for four centuries.

Frequently Asked Questions

Period T = 2π × √(L/g), where L is pendulum length in meters and g is gravitational acceleration (9.81 m/s² on Earth). Example: L = 0.5 m → T = 2π × √(0.5/9.81) = 2π × √0.051 = 2π × 0.2256 = 1.418 s. Frequency f = 1/T = 0.706 Hz (about 0.7 swings per second). This formula assumes small angles (under 15°) and a massless rod with all mass concentrated at the end.

Rearrange T = 2π√(L/g): L = g × (T/2π)². For T = 1 s: L = 9.81 × (1/2π)² = 9.81 × 0.02533 = 0.2485 m ≈ 24.85 cm. For T = 2 s (1 second each way, like a grandfather clock tick-tock): L = 9.81 × (2/2π)² = 9.81 × 0.1013 = 0.9940 m ≈ 99.4 cm. This is why grandfather clock pendulums are about 1 meter long.

Period T ∝ 1/√g — lower gravity produces longer periods. On the Moon (g = 1.62 m/s²), a 1-meter pendulum: T = 2π√(1/1.62) = 4.94 seconds (vs. 2.006 s on Earth). On Jupiter (g = 24.79 m/s²): T = 2π√(1/24.79) = 1.262 seconds. This principle allows pendulums to measure local gravity — a seismograph pendulum's period changes slightly when massive subsurface structures alter local g. Gravimeters use precision pendulums to map gravitational anomalies for oil and mineral prospecting.

No — for an ideal simple pendulum, mass has no effect on the period. This follows from Galileo's equivalence principle: gravitational force (mg) and inertial resistance (m) both scale with mass, canceling in the equation of motion. The resulting period T = 2π√(L/g) contains no mass term. In the real world, a very light pendulum bob on a heavy rod would deviate from the formula because the rod's mass becomes significant. But for a heavy bob on a light thread, mass truly does not matter.

A simple pendulum assumes all mass is concentrated at a single point (the bob) on a massless, inextensible string. Formula: T = 2π√(L/g). A physical pendulum is a real rigid object rotating about a pivot — its mass is distributed throughout. Formula: T = 2π√(I/mgd), where I is the moment of inertia about the pivot, m is total mass, g = 9.81 m/s², and d is the distance from pivot to center of mass. For a uniform rod pivoting at one end: I = mL²/3 and d = L/2, giving T = 2π√(2L/3g) — about 82% of the simple pendulum period for the same length.

Pendulums have wide engineering applications. Timekeeping: pendulum clocks achieve accuracy of 1 second per day — for centuries the most precise timekeepers. Seismology: seismographs use pendulums to detect ground motion — the inertia of the pendulum mass resists the motion while the frame moves with the earthquake. Gravimetry: precision pendulums measure local gravitational acceleration to detect subsurface density variations. Demolition: wrecking balls are pendulums storing and releasing gravitational potential energy. Structural damping: tuned mass dampers (pendulums inside skyscrapers) reduce building oscillations during earthquakes and wind. Taipei 101 has a 660-tonne steel sphere that acts as a tuned pendulum damper.