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Simple Pendulum Equations
by Ron Kurtus (revised 17 October 2009)
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 and frequency.
There are equations available to calculate the period and frequency as a function of the length of the wire and the acceleration due to gravity. The weight of the bob is not a factor in the equations.
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.
Useful tools: Metric-English Conversion | Scientific Calculator.
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
(See Pendulum Exhibits Periodic Motion to see a demonstration of a pendulum in motion.
Equations for period and frequency
The period of the motion for a pendulum is how long it takes to swing back-and-forth, measured in seconds. Period is designated as T.
The frequency of a pendulum is how many back-and-forth swings there are in a second, measured in hertz. Frequency is usually designated as f.
The period T is the reciprocal of the frequency. T = 1/f and f = 1/T.
Period equation
The equation for the period of a simple pendulum starting at a small angle (a) is:
T = 2pi*SQRT(L/g)
or
T = 2π√(L/g)
where
- T is the period in seconds (s)
- 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 √)
- 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)
Thus, if L = 2 meters:
T = 2 * 3.14 * √(2/9.8) = 6.28 * √(0.2) = 6.28 * 0.45 =
T = 2.8 seconds (rounding off a little).
Frequency equation
The equation to calculate the frequency of a simple pendulum, starting at a small angle is:
f = √(g/L)/2π
where:
- f = frequency in cycles per second (Hertz or Hz)
- √(g/L)/2π is √(g/L) ÷ 2π
- 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,
f = √(4.9)/6.24 = 0.16 * 2.21 = 0.355 Hz.
Relationship between period and frequency
Check the relationship between T and f:
T = 1/f = 1/0.355 = 2.8 seconds
Length of wire
If you wanted to find the length of the rod or wire for a given frequency, take the equation f = √(g/L)/2π 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
Note that the velocity, as well as the period and frequency are affected by the acceleration due to gravity constant. In other words, a pendulum will swing slower on the Moon than on the Earth, because the gravity on the Moon is less than on the Earth.
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.
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Resources and references
The following resources provide information on this subject:
Websites
How Pendulum Clocks Work - From How Stuff Works
Books
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physics. If you had problems, you had better look over the material again.
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Simple Pendulum Equations