PhysicsLAB: Period of a Pendulum

A simple pendulum consists of a string, cord, or wire that allows a suspended mass to swing back and forth. The categorization of "simple" comes from the fact that all of the mass of the pendulum is concentrated in its "bob" - or suspended mass.

As seen in this diagram, the length of the pendulum is measured from the pendulum's point of suspension to the center of mass of its bob. Its amplitude is the string's angular displacement from its vertical or its equilibrium position. If a pendulum is pulled to the right side and released to swing back and forth, its path traces our a sine curve as shown below.

The time required for one complete vibration, for example, from one crest to the next crest, is called the pendulum's period and is measured in seconds.

The formula to calculate this quantity is

where

L is the length of the pendulum in meters
g is the gravitational field strength, or acceleration due to gravity

This quantity at sea level is 9.81 m/sec² and can be calculated as

where

G = 6.67 x 10^-11 nt m²/kg²
M _Earth is the mass of the earth (6.02 x 10²⁴ kg)
R _Earth is the average radius of the earth (6.4 x 10⁶ meters)

Notice in the formula that the mass of a simple pendulum's bob does not affect the pendulum's period; it will however affect the tension in the pendulum's string.

In this related lesson, you will find a derivation of this formula for the period of a simple pendulum that will help you understand the restrictions on its use. It will also explain to you why a simple pendulum is NOT a true representation of simple harmonic motion, SHM. Take a few moments and use this physlet to investigate how the period of a pendulum is impacted by its length and its initial displacement.

The frequency of a pendulum represents the number of vibrations per second. This quantity is measured in hertz (hz) and is the reciprocal of the pendulum's period.

Let's practice a few problems with these formulas.

	What would be the period of a pendulum located at sea level if it is 1.5 meters long?

	If the pendulum's length were to be shortened to one-fourth its original value, what would be its new period?

	How many complete vibrations would this shorter pendulum trace out in one minute if it were to be released with a small initial amplitude?

	At sea level, how long would a pendulum be if it has a frequency of 2 hz?

	The timing mechanism in a grandfather's clock is based on the principles of a simple pendulum. If your clock is gaining time, should you shorten or lengthen its pendulum?

	Would a grandfather clock keep time on the moon?

A physical pendulum could be illustrated by swinging a meter stick about one end or a baseball bat about one end. The formula to calculate the period of a physical pendulum is

where

is the pendulum's moment of inertia measured in kg m²
m is its mass in kilograms
g is the local gravitational field strength or acceleration due to gravity
L is the moment arm or perpendicular distance from the pivot point to object's center of mass measured in meters