Simple Pendulum Calculator

A simple pendulum is a small heavy mass attached to a light string that swings back and forth under the influence of gravity. This calculator computes the period, frequency, and angular velocity of the oscillation based on the length of the string and the local gravitational acceleration. It can also be used in reverse: enter a measured period to find the length, or leave the field for gravity blank to measure the local gravitational acceleration itself.

If you look closely enough, you will notice that pendulum movements occur everywhere: in children on a swing, in a wrecking ball hanging from a rope or in a chandelier swinging through the air. As long as the oscillation always remains within about fifteen degrees of verticality, it follows all the same formulas for a simple pendulum.

What is a simple pendulum?

A simple pendulum is an idealized model. It consists of a point mass, the pendulum bob, attached to a weightless string that can swing freely with one end fixed. When the pendulum bob is pulled sideways and released, gravity pulls it back down toward its lowest point. Due to inertia, the pendulum bob overshoots this lowest point and swings up in the opposite direction. This stable repeating motion of swinging back and forth is called harmonic motion by physicists.

The force that pulls the pendulum back to its central position is determined by the angle theta between the string and the vertical.

For small angles the sine of theta is very close to the value of theta itself (in radians), so that the restoring force is nearly proportional to the displacement of the pendulum body. This proportionality leads to harmonic motion, which is why a pendulum provides stable time measurement.

The formula for a simple pendulum.

The timing of a pendulum is described by three interrelated quantities all derived from the same relationship. The angular velocity gives the speed of the oscillation and is measured in radians per second.

The normal frequency is the number of complete oscillations per second while the period is the time for a complete oscillation. Both can be directly calculated from the angular velocity.

Each symbol represents a particular property of the pendulum or its oscillation.

Here's how to use this calculator:

Enter the length and select its unit. Since the Earth's gravity field is set to Earth by default, entering a length will immediately display period, frequency, and angular velocity. It is not necessary to specify which quantity to calculate; all empty fields are automatically taken into account.

If you want to reverse the calculation, enter the measured period to find length. To measure gravity, leave the field for gravity blank and then enter length and period. The calculator will return the value of g at that location. Each field has its own unit toggle option so period can be entered in seconds or minutes and frequency in hertz or cycles per minute.

Example: A pendulum with a length of one meter

Let's take a pendulum with a length of one meter on Earth as an example. Plug the length and acceleration due to gravity into the formula for period.

A full swing takes just over two seconds. This is why a one meter long pendulum will resemble the classic second hand. A half swing takes almost exactly one second and this was how mechanical clocks measured time for centuries. If you want an accurate half swing, shorten the string slightly to about 0.994 meters.

Note a value that is not included in the formula: mass. If you attach two objects of different masses to a string one meter long, both a bowling ball and a marble will have the same period. Galileo first noticed this when he observed the swinging of a chandelier in Pisa's cathedral. This observation later led to the idea that inertial mass and gravitational mass are equivalent.

How does length affect period?

The period does not depend on the length itself but rather its square root. The square root moderates this change. To double the period, the length of the pendulum must be quadrupled and to triple the period it must be nine times as long. This rule is apparent when looking at some typical lengths on Earth.

How does gravity affect period?

The direction of gravity is opposite to the length. The stronger the gravity, the greater the force pulling down on the weight, making the swing faster and the period shorter. Since the period is inversely proportional to the square root of the gravity, worlds with lower gravity will have slower and longer swings.

A one-meter pendulum that swings every 2.01 seconds on Earth would take about 4.93 seconds on the Moon. The Moon's gravity is about a sixth of Earth's. On Jupiter, where gravity is about two and a half times as high as on Earth, the same pendulum would take about 1.26 seconds for one swing. You can compare results graphically by selecting other planets in the calculator tool.

Large amplitude oscillations and small angle approximation

The simple formula for the period assumes a small amplitude because it uses the angle itself rather than the sine of the angle. This approximation is very accurate for only a few degrees, and can be used up to about ten degrees. Outside this range, the actual period becomes longer since the restoring force does not keep up with the prediction of the linear approximation.

A more accurate formula for the period involves correction terms that depend on the amplitude.

Here the amplitude is given in radians. At a thirty degree angle the actual period is about 1.7 percent longer than the small angle value and the difference becomes more pronounced as the amplitude gets larger. The calculator allows you to turn on the large amplitude correction, apply it to your oscillation and graphically show how the period changes with increasing angles.

Measuring gravity with a pendulum

Since the period links together gravity, length and time measurement, a simple pendulum is surprisingly useful as a gravimeter. If we solve for g in the formula for the period, we get:

Measure the length with a ruler and time several swings, divide by the number of swings to get an accurate period. This will allow you to determine the strength of gravity at that location. Geophysicists use precision pendulums to measure slight changes in gravity at different locations on Earth. Such changes can indicate that dense minerals or oil lie underground. In the calculator tool, leave the field for gravity blank and enter length and measured period to display the calculated value of g.

The difference between a simple pendulum and a physical pendulum.

In a simple pendulum it is assumed that all the mass is concentrated at one point which hangs on a weightless string. Therefore, the period depends only on its length and the acceleration due to gravity. A physical pendulum, on the other hand, is a rigid body with some extent such as a swinging bar or a hanging sign where the mass is distributed throughout the whole body. Its period also depends on how the mass is distributed which is described by a quantity called moment of inertia.

Both types of pendulums are closely related to each other. Any oscillation of a physical pendulum exactly corresponds to the oscillation of a simple pendulum with an appropriate length. If the string is really light and the mass is really small and heavy, then the formulas for the simple pendulum on this page will suffice.

Where pendulum is used

The most familiar use of pendulums is in timekeeping. From the 17th century pendulum clockmakers until the 1930s, swinging pendulums were the world's most accurate clocks. The one-second beat of a meter-long pendulum set the pace for daily life.

The role of pendulums is not limited to timekeeping. A Foucault pendulum is suspended from a high ceiling and can swing freely for hours. The slow rotation of the plane of oscillation demonstrates Earth's rotation. Metronomes, swings, and roller coasters all follow the same principle, while seismographs and active building dampers use this principle to counteract unwanted motion.

Tips for accurate results

When measuring the length, measure to the center of mass of the pendulum bob, not the top. This is because that's where the mass is actually concentrated. To ensure that the formula for small angles holds true, keep the swing under about fifteen degrees or enable the large amplitude correction. Take multiple measurements and divide the time by the number of measurements instead of taking just one measurement. This will allow you to average out reaction times. For an accurate measurement, use the gravitational value for your location. The acceleration due to gravity (g) at Earth's surface can vary by a few tenths of a percent.