A pendulum is a simple tool for demonstrating the law of conservation of energy. It is also used to measure time.

One variable that affects the period of a pendulum is its length. Using a widget below, you can investigate the effect of this variable on the pendulum’s period.

Period of a Pendulum

In a simple pendulum, the interval of time it takes for one complete oscillation (the period) is proportional to the length of the string and to the acceleration due to gravity. The result is a constant, so clocks can be made to run at the correct rate simply by adjusting the string's length.

However, the force of gravity is not a constant at all points on Earth's surface; it varies due to the oblate shape of the planet and because of local differences in air pressure. The result is that a pendulum with a given string length may have a different period in different locations.

Christiaan Huygens proved in 1673 that if any pendulum was turned upside down and hung from a pivot point at its previous center of oscillation, it would have the same period and the same swings as before. Henry Kater used this principle in 1817 to produce a type of reversible pendulum for better measurements of the acceleration due to gravity.

Period of a String

A simple pendulum's period is a function of its length and its amplitude, the width of its swing. It also depends to a small degree on its bob's center of gravity and its mass.

The bob's velocity decreases as it moves away from the equilibrium position and toward the pivot. This is because the restoring force slows it down.

This is the principle behind Galileo's clock and Huygens' pendulum-based chronometer, which was used to define a terrestrial standard of measurement. You can experiment with this yourself by using the widget below, which allows you to enter a value for a pendulum's length and then displays its period. You can also vary the arc angle or angular displacement of the Pendulum, and you'll find that alterations in these variables have only a slight effect on the period. You can also investigate the effects of changing the g value for a given pendulum by using the widget below.

Period of a Bob

The period of a pendulum is independent of its mass and maximum displacement angle, provided the bob's acceleration does not exceed the force due to gravity. For this reason, simple pendulum clocks can be adjusted to the local gravitational acceleration even after a change in distance from the location where they were originally built.

As the bob moves towards its equilibrium position from positions A, B and C, it gains kinetic energy while losing potential energy. This is why the bob will move faster as it gets closer to its equilibrium position.

But as the bob moves away from its equilibrium position, it experiences a restoring force that opposes its motion and slows it down. Consequently, the bob will have its lowest velocity at position G, which is where the bob has its greatest displacement from its equilibrium position. The bob will then gain speed again as it moves towards its position D. This movement is similar to the way that a swing will accelerate as it moves up and down.