The equations for an easy pendulum show exactly how to find the frequency and also period of the movement.

A straightforward pendulum is composed of a suggest mass suspfinished on a string or wire that has negligible mass. If the pendulum weight or bob is pulresulted in a reasonably tiny angle from the vertical and let go, it will swing earlier and forth at a continual duration and also frequency. These requirements allow for the equations to be relatively basic and also are called straightforward harmonic motion.

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If the bob is larger, the wire has actually mass, or the angle is larger, it is referred to as a physical pendulum through complicated equations of motion.

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

Questions you might have include:

What are the components and parameters of pendulum motion? What are the equations for frequency and period? What are the equations for the size of the pendulum string?

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

Factors and parameters

The major aspect involved in the equations for calculating the frequency of a simple pendulum is the length of the rod or wire, offered the initial angle or amplitude of the swing is tiny. The mass or weight of the bob is not a variable in the frequency of the straightforward pendulum, however the acceleration because of gravity is a variable.

Note: This implies that the frequency and also duration would be different on the Moon versus on the Planet.

Knowing the size of the pendulum, you can determine its frequency. Or, if you desire a particular frequency, you have the right to determine the essential size.


Factors and also parameters in a simple pendulum

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

Period equation

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

T = 2π√(L/g)


T is the period in seconds (s) π is the Greek letter pi and is about 3.14 is the square root of what is contained in the parentheses L is the length of the rod or wire in meters or feet g is the acceleration because of 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 secs or rounding off a little to T = 2.8 s.

Frequency equation

The frequency of a pendulum is how many type of 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 deserve to additionally 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 have the right to uncover the size of the rod or wire for a given frequency or period.


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.


Likewise, the size of the wire for a offered duration is:

T = 2π√(L/g)

Square both sides:

T2 = 4π2(L/g)

Solve for L:

L = gT2/4π2


If the pendulum weight or bob of a simple pendulum is pulcaused a relatively little angle and let go, it will swing earlier and forth at a regular frequency. These demands allow for the equations to be reasonably straightforward and also are referred to as basic harmonic movement.

If damping effects from air resistance and also friction are negligible, equations concerning the frequency and also duration of the the pendulum, and also the length of the string have the right to be calculated.

The period equation is: T = 2π√(L/g)

The frequency equation is: f = <√(g/L)>/2π

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

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Resources and also references

Ron Kurtus" Credentials


How Pendulum Clocks Work - From How Stuff Works

Physics Resources


(Notice: The School for Champions might earn comgoals from book purchases)

Top-rated books on Pendulums


Top-rated books on Periodic Motion


Top-rated books on the Physics of Motion


Questions and also comments

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