Science

Pendulum Calculator

Calculate the period, frequency, and angular frequency of a simple pendulum from its length and gravity using T = 2π√(L/g).

Key Formulas

T = 2π√(L/g)  |  f = 1/T  |  ω = 2πf = √(g/L)

Calculate

Enter pendulum length and gravity (or use a preset) to calculate period, frequency, and angular frequency.

Period (T)
2.006409 s
Frequency (f)
0.4984028 Hz
Angular Freq (ω)
3.131557 rad/s

Famous Pendulums

Period vs. Length Reference

At g = 9.80665 m/s²

Length (m)Period (s)Frequency (Hz)
0.10.634482331.576088
0.251.0032050.99680559
0.51.4187460.70484799
12.0064090.4984028
1.52.4573390.40694418
22.8374910.352424
33.4752030.28775299
54.4864680.22289251
106.3448230.1576088

About This Tool

The Pendulum Calculator computes the oscillation properties of a simple pendulum using the classic formula T = 2π√(L/g). Enter the pendulum length and gravitational acceleration to instantly see the period (time for one complete swing), frequency (swings per second), and angular frequency (radians per second). The tool includes gravity presets for different celestial bodies, famous pendulum examples, and a reference table showing how period varies with length.

The Simple Pendulum

A simple pendulum consists of a point mass (called a bob) suspended from a fixed pivot by a massless, inextensible string or rod. When displaced from its equilibrium position and released, it swings back and forth under the influence of gravity. For small angular displacements (typically less than 15 degrees), the restoring force is approximately proportional to the displacement, making it a simple harmonic oscillator. This idealized model captures the essential physics of real pendulums remarkably well and has been studied since Galileo first observed the isochronous property of pendulums in Pisa around 1583.

History of Pendulum Science

Galileo noticed that a swinging chandelier in the cathedral of Pisa took the same time for each swing regardless of its amplitude, a property called isochronism. He proposed using pendulums for timekeeping, but it was Christiaan Huygens who built the first pendulum clock in 1656. Huygens also derived the mathematical relationship T = 2π√(L/g) and discovered that perfect isochronism requires a cycloidal path, not a circular arc. Pendulum clocks remained the world's most accurate timepieces for nearly 300 years, until the development of quartz oscillators in the 1930s.

Factors Affecting Accuracy

Several factors cause real pendulums to deviate from the ideal formula. Air resistance damps the oscillation and slightly increases the period. The finite mass and size of the bob shifts the effective length. Temperature changes alter the length of the rod or string (metal rods expand when heated). The amplitude of swing matters: the simple formula assumes small angles, and larger swings have longer periods. Seismic vibrations, air currents, and the flexibility of the support structure all introduce additional perturbations. High-precision pendulum clocks address these issues through vacuum enclosures, temperature-compensating materials, constant-amplitude mechanisms, and vibration isolation.

Pendulums on Other Worlds

Since the period depends on gravitational acceleration, a pendulum swings differently on other celestial bodies. On the Moon, where g = 1.62 m/s² (about one-sixth of Earth's), a pendulum swings about 2.5 times slower. On Jupiter, with g = 24.79 m/s², it would swing about 1.6 times faster. This relationship was historically used to measure gravity at different locations on Earth: by timing a known pendulum, scientists could calculate local g to high precision. Variations in g across Earth's surface reveal information about underground rock density, which is used in geological surveying and mineral exploration.

Modern Applications

While pendulum clocks are no longer the gold standard for timekeeping, pendulums remain important in science and engineering. Seismometers use inverted pendulums to detect ground motion. The Foucault pendulum dramatically demonstrates Earth's rotation in science museums worldwide. Tuned mass dampers (essentially giant pendulums) are installed in skyscrapers to counteract wind-induced swaying; the most famous is the 730-ton damper in Taipei 101. In education, the simple pendulum remains one of the best demonstrations of periodic motion, energy conservation, and the power of mathematical modeling in physics.

Frequently Asked Questions

What is the formula for a simple pendulum's period?
The period of a simple pendulum is T = 2π√(L/g), where L is the length of the pendulum (from the pivot to the center of mass of the bob) and g is the gravitational acceleration (9.81 m/s² on Earth's surface). This formula assumes small oscillation angles (less than about 15°), a massless string or rod, and no air resistance. The remarkable feature is that the period depends only on length and gravity, not on the mass of the bob or the amplitude of swing.
Why doesn't mass affect the pendulum period?
Mass cancels out in the pendulum equation because gravitational force (which drives the pendulum) and inertia (which resists acceleration) both depend on mass in exactly the same way. A heavier bob is pulled harder by gravity but is also harder to accelerate. These effects perfectly cancel, just as they do for free-falling objects. This was one of Galileo's key observations in the late 1500s and helped establish the equivalence principle that later became central to Einstein's general relativity.
How accurate is this formula for large swing angles?
The T = 2π√(L/g) formula is exact only for infinitesimally small angles. For real swings: at 15° the error is about 0.5%, at 30° it's about 1.7%, at 45° it's about 4%, and at 90° it's about 18%. The exact solution involves elliptic integrals. For most practical purposes (clock design, lab experiments), keeping the swing angle below 15° ensures the simple formula is accurate to better than 1%.
How were pendulums used to measure gravity?
Before modern gravimeters, scientists measured local gravitational acceleration by timing pendulum swings. By precisely measuring a pendulum's length and period, g can be calculated as g = 4π²L/T². This technique was used for centuries, from Huygens in the 1600s through geodetic surveys in the 1900s. Variations in g across Earth's surface (due to altitude, latitude, and underground density variations) were mapped using portable pendulums. Modern absolute gravimeters use laser interferometry instead, but the pendulum method illustrates the principle beautifully.
What is a Foucault pendulum?
A Foucault pendulum is a tall pendulum (typically 10-70 meters) that demonstrates Earth's rotation. As it swings back and forth, the plane of oscillation appears to rotate because the Earth is actually rotating beneath it. At the poles, the plane completes one full rotation every 24 hours. At the equator, it doesn't rotate at all. At intermediate latitudes, the rotation rate is 360°×sin(latitude) per day. Léon Foucault first demonstrated this in 1851 at the Panthéon in Paris with a 67-meter pendulum, providing the first direct visual proof of Earth's rotation.
How do grandfather clocks use pendulums?
Grandfather clocks use a pendulum approximately 0.994 meters long, which produces a period of almost exactly 2 seconds (one second per swing in each direction). This precise relationship between length and period was discovered by Christiaan Huygens in 1656 when he invented the pendulum clock. The clock mechanism gives the pendulum a small push each swing to replace energy lost to friction and air resistance. Temperature compensation mechanisms (like mercury or invar steel pendulums) prevent length changes that would affect accuracy. A well-made pendulum clock can keep time to within one second per week.

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