Simple Harmonic Motion
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[edit] Simple Harmonic Motion and Your Life
You may never have heard of simple harmonic motion, but chances are you have experienced its effects countless times. Have you ever been on a swing or jumped up and down on the edge of a diving board? Then you're well acquainted with simple harmonic motion (or SHM for short).
[edit] The Basics
A swing is a perfect example of SHM. Imagine you just got onto a swing and you haven’t started moving yet, the swing is as close to the ground as possible. The swing is at equilibrium meaning that it is in its natural state. If you begin to move back and fourth on the swing it begins to move away from the ground and away from equilibrium. The distance between where the swing is now and where it was origionally is called the displacement. Since the force bringing you back towards equilibrium (the ground) is directly proportional (meaning that as one element increases or decreases the other element increases/decreases as well [1]) to your displacement from equilibrium, this motion is said to be simple harmonic.
This image is from http://www.sxc.hu/photo/522560
Simple Harmonic Motion is a way to describe motion that repeats over and over again. Simple harmonic motion occurs when the relationship between the mass (in the swing example that would be you) and time vary sinusoidally. A sinusoidal relationship is one that can be graphed as a sine wave[2]. The x-axis (the horizontal axis) represents the progress of time, while the y-axis (the vertical axis) represents the displacement of the swing from equilibrium.
[edit] Graphic Representations
It is easy to identify a graphic representation of SHM because it looks like a repeating pattern of identical waves.[[3]] The height of the wave is called the amplitude. Amplitude tells you the distance from equilibrium to the maximum point of motion. In the swing example the maximum point of motion would be when you have swung up as high as you can go and are about to start going back down again. The distance from the top of one wave to the top of the next is called the wavelength. Riding on a swing, one wavelength would be the distance from where you are up as high as you can be on one side to where you are as high up as you can be on the other side. The time it takes for one wavelength to occur is known as the period. The number of wavelengths that occur over a given amount of time is called the frequency.
Here is a graph of what SHM looks like:
[edit] Springs
Another place you can find SHM in your life is by picking up a common spring, that is what Robert Hooke (1635-1703) a noted physicist did way back in the 17th century. If you were to hold one end of a spring so that it dangled in midair and then attach a weight to the end of the spring, the spring would bounce up and down. This is a perfect example of simple harmonic motion. When the spring is simply hanging in the air it is at equilibrium, then when a weight is attached onto it, the spring is pulled down and stretched (uncoiled) as far as possible by the weight. However, the spring does not want to be uncoiled, it liked being comfortably coiled up at equilibrium. The spring then attempts to bounce back up to equilibrium where again, the weight of the mass drags it down. This cycle repeats over and over again. Robert Hooke noticed this pattern and worked his observations into a handy equation called Hooke's Law[4]
[edit] Hooke's Law
The word "law" when used by physicists means an equation. Physicists use the word law because a law will always hold true no matter what the circumstances are, this is also true for any mathematical equation. Hooke's Law is based upon SHM functioning in a spring. Robert Hooke picked variables (symbols that can represent anything) and designated them to the various elements of the equation. He chose the letter F to represent the force that the spring exerts, he chose k to represent the spring constant (the stiffness of the spring), and he chose x to represent the displacement ( the distance the spring is being pulled). Hooke's Law looks like this: F=kx
[edit] Pendulums
The movement of a pendulum is another common application of SHM. A pendulum [5] is comprised of an object called a bob attached to the end of a string. When the string and the bob are hanging straight downwards the only force that is acting upon them is gravity, the pendulum is now at equilibrium. If you were to take hold of the bob and pull it upwards and then let go, the pendulum would swing back down towards equilibrium and then back up again on the opposite side. This motion can be described by the equation
T= 2π√(l/g)
Where T is equal to the period, l is the length of the spring, and g is acceleration due to gravity [6]
[edit] Sample Problems
[edit] Questions
1. A mass is executing simple harmonic motion with amplitude A. What is the total distance traveled by the mass during one period of the oscillation?
2. What is the length of a pendulum with a period of 1.00 second?
3. Would it be practical to make a pendulum with a period of 10.0 seconds? explain.
4. On a planet with an unknown value of g, the period of a 0.65 m long pendulum is 2.8 seconds. What is g for this planet?
5. What is the frequency of the motion?
[edit] Solutions
1. 4A. The answer is 4 Amplitude because as you can see in the graph below, 4 amplitude lengths comprise one period.
4amplitude_graph.jpg)
2. 0.248 m The first step is to write out the equation you will be using to solve this problem: T= 2π√(l/g). Next, isolate the variable that you are solving for, the length of the string. Now the equation reads l=g(T/2π)^2 Plug the known values into the equation: l=(9.81 m/s^2)(1.00 s/6.28 s)^2 and solve!
3. If you had a way to use a pendulum thats over 75 feet long! To solve this problem you need to use the same equation as in problem 2: T= 2π√(l/g). Next, isolate the variable that you are solving for, the length of the string. Now the equation reads l=g(T/2π)^2 Plug the known values into the equation: l=(9.81 m/s^2)(10.0 s/6.28 s)^2 and solve!
4. 3.3 m/s^2 To solve this problem you need to use the same equation as in problems 2 and 3: T= 2π√(l/g). Next, isolate the variable that you are solving for, the value of g. Now the equation reads g= l(2π/T)^2 Plug the known values into the equation: (0.65 m)(6.28/2.8 s)^2 and solve!
5. 0.5 Hz The first step to solving this problem is to identify what we know about frequency (f), we know that period=1/f and that period is the time it takes for one wavelength to occur, aka, the time between any two crests. When you measure the period you get 2 seconds. Now insert that into the equation and you have 2s=1/f. Here we are solving for f so let's isolate f: f=1/2s. And solve!
[edit] Resources and References
[edit] Resources
1. http://www.ucmp.berkeley.edu/history/hooke.html This site has a more in depth chronology of Robert Hooke’s life and contributions to the scientific community. [7]
2. http://coursedocs.slcc.edu/phys/tvanausdal/2210/2210welcome.htm [8] This page has many mini quizzes spanning a variety of physics topics including SHM.
3. http://www.roberthooke.org.uk/ This link is useful if you want to learn some facts about the life of Robert Hooke.[9]
4. http://www.physics247.com/solved_problems/vibrations_waves_doppler.php This website has a few practice problems on Simple Harmonic Motion and other related topics. [10]
5. Barron's Regents Review for Physics, by Miriam A. Lazar, third Edition. Barron's Educational Series, Inc. 2004 This book covers all regents level physics topics and includes many practice problems.
6. Glencoe Physics- Principles and Problems, by Paul Zitzewitz. Glencoe/McGraw-Hill, 1998. This textbook is a perfect resource for any physics student looking for concise definitions and diagrams concerning basic physics concepts.
[edit] References
1. http://physics.mtsu.edu/~wmr/shm.htm [11] This site has a an introduction to the basics of SHM as well as a helpful animation of the process.
2. http://library.advanced.org/16600/intermediate/simpleharmonicmotion.shtml [12] This page has information on springs and how they utilize SHM.
3. http://www.phy.ntnu.edu.tw/ntnujava/viewtopic.php?t=236 [13] This link has a very neatly made animation of SHM and uniform circular motion.
4. http://www.bookrags.com/sciences/physics/harmonic-motion-wop.html [14] This website discusses all the terms necessary to understanding SHM.
5. http://webphysics.davidson.edu/Applets/animator4/demo_hook.html [15] This site has an animation illustrating Hooke’s Law.
6. http://theoryx5.uwinnipeg.ca/physics/shm/node5.html [16] This page discusses the basic equations related to the functioning of pendulums.
7. http://physics.kenyon.edu/EarlyApparatus/Mechanics/Pendulum/Pendulum.html [17] This source goes in depth into the history and development of pendulums.
[edit] Image Credits
1. http://www.sxc.hu/photo/522560 [18] Stock xchng is a stock photo library of over 110,000 royalty-free images
2. http://www.allfree-clipart.com/ [19] All Free Clipart has thousands of royalty-free animations, drawings, diagrams, and photos.
