A simple pendulum is a mass hanging from a massless string of length L is allowed to swing from a central pivot point. As the mass is moved from its center point, gravity pulls the mass down and tension in the string pulls the mass back towards the center point. The mass continues past the center point while the tension force slows it down and pulls it back towards the center point again. This type of motion is known as simple harmonic motion. The time to complete one cycle of harmonic motion is called the period.

The length of a simple pendulum is proportional to the period of the pendulum’s motion. This relationship is expressed by the formula

where

T = period

L = length of the pendulum

g = acceleration due to gravity

### Find the Length of a Pendulum Example Problem

This example problem will show how to use the pendulum formula to find the length of a pendulum for a known period.

**Question:** Grandfather clocks are decorative clocks with a pendulum measuring out the passing of a second. How long of a pendulum is needed to have a period of 1 second?

Use 9.8 m/s^{2} for the acceleration due to gravity.

Start with the period formula from above.

Square both sides to get rid of the radical

Multiply both sides by g

Divide each side by 4π

Plug in the values for the period and gravity.

**L = 0.25 m**

**Answer:** A simple pendulum with a period of 1 second will have a length of 0.25 meters or 25 centimeters.

It is a good idea to write all your units along with your values with these types of problems. This can catch simple math errors when you expect a length for your answer and you happen to have length squared or 1/length. It can save you time in the long run.

If you need further help, check out the Period of a Simple Pendulum example problem and Calculating the Acceleration Due to Gravity Using a Pendulum example.