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Simple Pendulum Equations

by Ron Kurtus (revised 3 November 2007)

A simple pendulum consists of a weight suspended on a string or wire. 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 or frequency. Although damping effects from air resistance and friction are a factor, they are considered negligible for the basic equations concerning the frequency of the the pendulum, as well as the speed of the bob.

Questions you may have include:

• What are the factors and parameters of pendulum motion?
• What are the equations for frequency and period?
• What is the equation for the speed or velocity of a pendulum?

This lesson will answer those questions. There is a mini-quiz near the end of the lesson.

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Time = 7 min. 15 sec.

Factors and parameters

The major factor involved in the equations for calculating the frequency of a 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 of gravity is in the equation.

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

Frequency and period

The equations for frequency and period are the reciprocals of each other.

Frequency

The equation to calculate the frequency of a simple pendulum, starting at a small angle (a) is

f = (1/2pi)*SQRT(g/L)

That can be rewritten as

f = (1/2π) √(g/L)

where:

• f = frequency in cycles per second (Hertz or Hz)
• pi = 3.14 (it is also written as the Greek letter π)
• SQRT means the square root of what is included in the parentheses (SQRT is also seen as the symbol √)
• g is the acceleration due to gravity (9.8 m/s² or 32 ft/s²)
• L is the length of the rod or wire in meters or feet

Thus, if L = 2 meters and g = 9.8 m/s², f = (1/6.24) √(4.9) = 0.16 * 2.21 = 0.355 Hz. But that doesn't really give you a good feel of how fast the pendulum is swinging. A better way is to look at its period.

Period

The period of the motion for a pendulum is how long it takes to swing back-and-forth. The period T is the reciprocal of the frequency or T = 1/f. The equation for the period of motion is

T = 2π /√(L/g)

Thus, if L = 2 meters, T = 2.8 seconds. That is easier to visualize.

Length

If you wanted to find the length for a given frequency, take the equation f = (1/2π) √(g/L) and solve for L. The result is:

L = g/(4π²f²)

where:

• π² is pi-squared or pi times pi
• f² is frequency-squared or f times f
• 4π²f² is 4 times π² times f²

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

Velocity

Although the velocity of the bob at the bottom of the swing is not a factor in determining frequency, it may be of interest in other calculations. The velocity can be approximated from the gravity equation for a weight dropping from a height. The height is determined by the angle from the vertical that is the starting point of the pendulum's swing. Thus, the velocity at the bottom of the swing is:

v = √{2gL[1-cos(a)]}

where:

• v is the velocity of the weight at the bottom of the swing
• g is the acceleration due to gravity
• L is the length of the wire
• a is the angle from the vertical
• cos(a) is the cosine of angle a

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

Answers to Readers' Questions

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Resources

The following resources provide information on this subject:

Websites

How Pendulum Clocks Work - From How Stuff Works

Physical Science Resources

Books

Top-rated books on Pendulums

Miscellaneous

Mini-quiz to check your understanding

1. If you know the period of a pendulum, what can you easily determine?

2. What happens if you decrease the length of the pendulum rod?

3. How does a pendulum on the Moon compare with one on Earth?

If you got all three correct, you are on your way to becoming a Champion in Physical Science. If you had problems, you had better look over the material again.

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Simple Pendulum Equations