Pendulum Calculator

Free Online Pendulum Calculator

This Pendulum Calculator calculates the period, frequency, length, or gravitational acceleration of an ideal simple pendulum. Select the unknown, enter the other values, and the calculator applies the small-angle pendulum equation T = 2π√(L/g). Every answer includes the formula substitution, copyable solution steps, and a live labelled pendulum diagram.

It is useful for school and university physics, science laboratories, clock experiments, and basic oscillation revision. The model treats the bob as a point mass on a light, inextensible string and assumes a small swing angle. Under those conditions, pendulum period is controlled by length and gravity, not by the bob's mass.

Simple Pendulum Formula

The simple pendulum period formula is T = 2π√(L/g). T is period in seconds, L is the distance from the pivot to the centre of the bob in metres, and g is gravitational acceleration in metres per second squared. One period is one full back-and-forth oscillation, returning to the same location with the same direction of motion.

The formula can be rearranged to find length: L = g(T/2π)². To estimate gravitational acceleration, use g = 4π²L/T². These rearrangements make a pendulum a practical classroom tool for measuring time and local gravity.

How to Calculate Pendulum Period

Enter pendulum length and gravitational acceleration, then select Period. For a 1 m pendulum on Earth, with g ≈ 9.80665 m/s², T = 2π√(1/9.80665) ≈ 2.006 s. The period is close to two seconds, so one movement from one side to the other takes roughly one second.

Length must be measured from the pivot point to the centre of mass of the bob, not merely to the top of the bob or end of the string. Use metres and seconds for the cleanest calculation. A longer pendulum has a longer period because it takes more time to swing through its arc.

Pendulum Frequency and Period

Frequency is the number of complete swings per second and is measured in hertz. It is the reciprocal of period: f = 1/T. A pendulum with a 2-second period has a frequency of 0.5 Hz. The calculator displays both values because they express the same motion in two useful ways.

If pendulum length is increased by a factor of four, period doubles rather than quadruples, because period depends on the square root of length. Consequently frequency becomes half as large. This square-root relationship is a valuable check for pendulum questions and experiments.

Pendulum Length Calculator

If you know a desired period, rearrange the formula to L = g(T/2π)². A pendulum clock designed for a 2-second full period needs a length close to 1 m on Earth. This is why traditional seconds pendulums are approximately one metre long.

The calculated length is an ideal value. A real clock also depends on temperature, the bob's centre of mass, suspension details, air drag, and exact swing amplitude. Nonetheless, the formula provides an excellent first design estimate and a clear demonstration of how physics connects geometry to timing.

Using a Pendulum to Calculate Gravity

A pendulum can estimate local gravity with g = 4π²L/T². Measure length carefully, time many full oscillations, divide by the number of cycles to get period, then enter length and period. Timing twenty or more swings reduces the influence of hand reaction time compared with timing one swing.

On Earth, g is near 9.81 m/s² but varies slightly by altitude, latitude, and local geology. On the Moon, g is about 1.62 m/s², so the same pendulum swings much more slowly. Its period is larger by the square root of the inverse gravity ratio.

Mass, Amplitude, and the Small-Angle Condition

In the ideal formula, bob mass does not affect period. A heavy and light bob on strings of equal length swing together if air resistance and buoyancy are negligible. This sometimes surprises students, but the larger gravitational force on a heavier bob is exactly balanced by its greater inertia.

Swing angle does matter when it becomes large. The familiar formula assumes small angles, often below about 10 degrees for high accuracy. At wider amplitudes, a real pendulum has a slightly longer period than the simple formula predicts. Keep the release angle small and avoid pushing the bob during experimental measurements.

Applications of Pendulums

Pendulums have been used in clocks, metronomes, gravimeters, seismometers, amusement rides, and scientific demonstrations. They provide a visible example of periodic motion and energy exchange: gravitational potential energy is highest at the ends of the swing, while kinetic energy is greatest at the centre.

Modern precision clocks use quartz and atomic oscillators, but pendulum principles remain important in sensing and education. Seismometers use related inertial systems to detect ground movement, while gravimeters can use precisely controlled oscillators to study small changes in gravitational acceleration.

Common Pendulum Calculation Mistakes

Do not use the bob mass in the simple pendulum period formula. Do not confuse a half swing with a full period. Also ensure length is in metres and gravity is in m/s². A period measured from several cycles must be divided by the number of complete oscillations before it is entered.

The simple formula does not apply exactly to a rigid physical pendulum, a spring–mass system, or a large-angle swing. For a compound pendulum, mass distribution matters. For an accelerating support, changing gravity, damping, or driving forces, a more advanced model is required.

How to Use This Pendulum Calculator

Choose period, length, or gravity. Enter only the required positive values with consistent units. Review the highlighted answer, related frequency, and copyable working. The diagram highlights the correct pivot-to-bob-centre length, the downward gravitational field, and the calculated period.

For best results, use the equation as an ideal small-angle model and report measured inputs with sensible significant figures. It gives fast, clear answers while also showing why pendulum motion changes from Earth to the Moon or when the string length is adjusted.

Pendulum Calculator FAQ Summary

Use T = 2π√(L/g) to calculate simple pendulum period. Period depends on length and gravity, not mass, for small swings. Frequency is 1/T. Rearrange the formula to calculate length or gravitational acceleration. Measure string length to the bob's centre and time many oscillations for a reliable experiment.