Equations for a Simple Pendulum

by Ron Kurtus (updated 9 January 2023)

The equations for a simple pendulum show how to find the frequency and period of the motion.

A simple pendulum consists of a point mass suspended on a string or wire that has negligible mass. If the pendulum weight or bob is pulled to a relatively small angle from the vertical and let go, it will swing back and forth at a regular period and frequency. These requirements allow for the equations to be relatively simple and are called simple harmonic motion.

If the bob is larger, the wire has mass, or the angle is larger, it is called a physical pendulum with complex equations of motion.

Although damping effects from air resistance and friction are a factor, they are considered negligible for the basic equations concerning the frequency or period of the pendulum.

Questions you may have include:

What are the factors and parameters of pendulum motion?
What are the equations for frequency and period?
What are the equations for the length of the pendulum string?

This lesson will answer those questions. Useful tool: Units Conversion

Factors and parameters

The major factor involved in the equations for calculating the frequency of a simple pendulum is the length of the rod or wire, provided the initial angle or amplitude of the swing is small. The mass or weight of the bob is not a factor in the frequency of the simple pendulum, but the acceleration due to gravity is a factor.

Note: This means that the frequency and period would be different on the Moon versus on the Earth.

Knowing the length of the pendulum, you can determine its frequency. Or, if you want a specific frequency, you can determine the necessary length.

Factors and parameters in a simple pendulum

(See Demonstration of a Pendulum to see a pendulum in motion)

Period equation

The period of the motion for a pendulum is how long it takes to swing back-and-forth, measured in seconds. The equation for the period of a simple pendulum starting at a small angle α (alpha) is:

T = 2π√(L/g)

where

T is the period in seconds (s)
π is the Greek letter pi and is approximately 3.14
√ is the square root of what is included in the parentheses
L is the length of the rod or wire in meters or feet
g is the acceleration due to gravity (9.8 m/s² or 32 ft/s² on Earth)

Thus, if L = 2 meters:

T = 2 * 3.14 * √(2/9.8) = 6.28 * √(0.204) = 6.28 * 0.4517

T = 2.837 seconds or rounding off a little to T = 2.8 s.

Frequency equation

The frequency of a pendulum is how many back-and-forth swings there are in a second, measured in hertz.

Frequency f is the reciprocal of the period T:

f = 1/T

f = 1/[2π√(L/g)]

The equation can also be rearranged to be:

f = [√(g/L)]/2π

Thus, if L = 2 meters,

f = [√(9.8/2)]/2*3.14

f = [√(4.9)]/6.28 = 2.21/6.28 = 0.353 Hz.

Length of wire

You can find the length of the rod or wire for a given frequency or period.

Frequency

Solve the equation for L:

f = [√(g/L)]/2π

2πf = √(g/L)

Square both sides of the equation:

4π²f² = g/L

Solve for L:

L = g/(4π²f²)

For example, the length of a pendulum that would have a frequency of 1 Hz (1 cycle per second) is about 0.25 meters.

Period

Likewise, the length of the wire for a given period is:

T = 2π√(L/g)

Square both sides:

T² = 4π²(L/g)

Solve for L:

L = gT²/4π²

Summary

If the pendulum weight or bob of a simple pendulum is pulled to a relatively small angle and let go, it will swing back and forth at a regular frequency. These requirements allow for the equations to be relatively simple and are called simple harmonic motion.

If damping effects from air resistance and friction are negligible, equations concerning the frequency and period of the the pendulum, as well as the length of the string can be calculated.

The period equation is: T = 2π√(L/g)

The frequency equation is: f = [√(g/L)]/2π

The length equations are: L = g/(4π²f²) and L = gT²/4π²

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