Pendulum Calculator

Understanding the Simple Pendulum

A simple pendulum consists of a point mass (the bob) suspended from a fixed pivot by a massless, inextensible string. When displaced from its equilibrium position and released, it oscillates back and forth under the influence of gravity. The motion is a form of simple harmonic motion for small angles of displacement, and the period T = 2π√(L/g) depends only on the pendulum length and local gravitational acceleration, not on the mass of the bob or the amplitude of the swing.

This calculator lets you solve for the period, length, or gravitational acceleration of a simple pendulum. Enter any two known values and the third is computed automatically, along with the frequency and angular frequency. The default gravity value is 9.80665 m/s² (standard gravity on Earth), but you can change it to model pendulums on the Moon, Mars, or any other environment.

Period, Frequency, and Angular Frequency

The period (T) is the time for one complete back-and-forth oscillation. The frequency (f = 1/T) is the number of oscillations per second, measured in hertz (Hz). The angular frequency (ω = 2πf = √(g/L)) expresses the rate of oscillation in radians per second and appears naturally in the mathematical description of simple harmonic motion. A one-meter pendulum on Earth has a period of approximately 2.006 seconds, which is why grandfather clocks use pendulums close to one meter long to achieve a two-second period.

Historical Significance

Galileo Galilei first studied pendulum motion in the late 1500s, reportedly observing the swing of a chandelier in the Pisa Cathedral. Christiaan Huygens built the first practical pendulum clock in 1656, revolutionizing timekeeping. Pendulums were later used to measure the Earth's gravitational field with high precision, revealing that gravity varies slightly with latitude and altitude. Jean Foucault's famous pendulum demonstration in 1851 proved the rotation of the Earth by showing that the swing plane of a long pendulum slowly rotates relative to the floor.

Limitations and Real-World Factors

The simple pendulum formula assumes small oscillation angles (typically less than 15°), a massless string, no air resistance, and a point mass. Real pendulums experience air drag, have strings with finite mass, and may swing at large angles where the period increases beyond the small-angle prediction. Physical (compound) pendulums distribute mass along their length, requiring a modified formula using the moment of inertia. Despite these limitations, the simple pendulum remains an excellent model for understanding oscillatory motion and is widely used in education, engineering, and geophysics.

Frequently Asked Questions

What is the period formula for a simple pendulum?

T = 2π√(L/g), where L is the string length in meters and g is gravitational acceleration. The period depends only on length and gravity, not on mass.

Does mass affect the period?

No. The mass of the bob cancels out in the derivation. Heavy and light bobs on identical strings swing with the same period, just as objects of different masses fall at the same rate.

What is the frequency of a pendulum?

Frequency f = 1/T, the number of complete oscillations per second in hertz. A 1-meter pendulum on Earth has f of approximately 0.499 Hz.

How does gravity affect a pendulum?

Stronger gravity shortens the period (faster swing). On the Moon (g = 1.62 m/s²), a pendulum swings about 2.5 times slower than on Earth.

When does the simple pendulum formula break down?

At large angles (above roughly 15°), air resistance, string mass, or when the bob is not a point mass. For large angles, the true period is longer than T = 2π√(L/g) predicts.