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Explanation of simple pendulum equations - Succeed in Physical Science. Also refer to physics, forces, weight, gravity, bob, amplitude, angle, periodic motion, frequency, period, pi, clocks, Ron Kurtus, School for Champions. Copyright © Restrictions Simple Pendulum Equationsby 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:
This lesson will answer those questions. There is a mini-quiz near the end of the lesson. Useful tools: Metric-English Conversion | Scientific Calculator.
Time = 7 min. 15 sec. Factors and parametersThe 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 periodThe equations for frequency and period are the reciprocals of each other. FrequencyThe 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:
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. PeriodThe 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. LengthIf 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:
For example, the length of a pendulum that would have a frequency of 1 Hz (1 cycle per second) is about 0.25 meters. VelocityAlthough 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:
SummaryIf 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. Feel good by doing your very best ResourcesThe following resources provide information on this subject: WebsitesHow Pendulum Clocks Work - From How Stuff Works BooksMiscellaneousMini-quiz to check your understanding1. 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. What do you think?Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. We will try to get back to you as soon as possible. Share linkFeel free to establish a link from your website to pages in this site. Or use our form to send this link to yourself or a friend. Students and researchersThe Web address of this page is Please include it as a reference in your report, document, or thesis. Where can you go from here?
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