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Claimed by Mary Francis McDaniel
'''Claimed for editing by Elton Leander Pinto 10/29/2018'''


== Introduction ==
'''Edited by Richard Udall June 2019'''


The two major components in Physics II are interactions between electric fields and magnetic fields. This broad subject focuses from a simple topic, electric fields to a much complex idea, electromagnetic radiation. Though this course is very broad in terms of materials that it covers, each topic is very important in understanding the phenomena of electric and magnetic interactions between particles (protons, electrons, dipoles, point charge, capacitor, and, etc), as omitting one concept out of hundreds of concept could lead one approaching the problem differently. This Wiki Page will discuss Chapter 21 of the Matters and Interactions Text Book, 4th edition (Patterns of Field in Space), specifically Gauss's Law and Electric Flux. In order to understand these concepts, one first need to understand the definition of electric field and know each component of Gauss's Law.  
Simple harmonic motion is motion driven by a restorative force which acts to bring the oscillating particle back to its equilibrium position. Prototypical examples of this include a pendulum or a spring which is compressing and extending. However, most oscillating systems that we observe in our day-to-day life are not perfect simple harmonic oscillators. Simple harmonic motion is an approximation that ignores friction and air resistance. Although this is not generally true, simple harmonic motion is a decent approximation of these more complex systems. The more complex and accurate formulation which takes these forces into account is known as damped harmonic motion, and will be considered further in the page [[Iterative Prediction of Spring-Mass System]].  
== Dynamics ==


"Electric Flux" is a quantitative measure of the amount and direction of electric field over an entire surface of a specified object. There are two components in electric flux: direction of the electric field and magnitude of the electric field. These two sums up and give us the value, electric flux, which has a unit of Vm. In order to determine the direction of the electric field of an object, one need to figure out the x,y,z coordinates of the faces of an object and then calculate the normal vector that comes out of the surface. Secondly, to determine the direction of the electric field of an object, one first need to know the number of dimensions of an object (i.e: 6 faces in a rectangular prism) and areas for each face of the object. Finally, one should be able to calculate the electric flux of an object by multiplying the electric field at a location on each surface of the box by corresponding normal vector and multiplying this value by the area of the surface that was just calculated. One must repeat this process the remaining surfaces (faces) and by adding up all electric flux, that will be the electric flux of the object one wanted to calculate. This value is essential because it will be useful for calculating total charged enclosed inside the object later on. The above written method of calculating electric flux may be confusing at first, but knowing the Gauss's Law, being able to apply this Law to the real problem, and by going through the example below should make sure understanding of this concept.
==The Main Idea==


== Energy ==
Simple harmonic motion is a periodic motion, a motion that is repeated over some time interval. This periodic motion has a restoring force, which is a force which attempts to restore the system to its equilibrium position, and which is proportional and opposite in direction to displacement. Thus, the further the system is from its equilibrium position, the larger the force which acts to return it to equilibrium position. As the object gets closer to equilibrium position, the force decreases until is magnitude reaches zero. At the equilibrium, the particle has no forces acting upon it, but it now has a large amount of momentum, and so it passes the equilibrium point going in the opposite direction. As moves in the opposite direction from the equilibrium, the restoring forces increases in the opposite direction until the momentum reaches zero, after which the particle begins moving back towards the equilibrium. This process repeats, which leads to the periodic motion characteristic of the system. Common examples of simple harmonic motion include an undamped spring-mass system, and a pendulum swinging back and forth (in the idealized case where it does not slow down).
The Gauss's Law simplifies definition of "Electric Flux" into a one simple equation.  


[[File:Simple Harmonic Motion Orbit.gif|thumb|Simple Harmonic Motion Orbit]]
===A Mathematical Model===


[[File:Gauss.JPG]]
Simple harmonic motion occurs very frequently in our treatment of physical systems, and is a useful approximation for many because it is often reasonably accurate, and because we have the tools to describe it exactly. We may consider the example of the spring (see the figure) to arrive at a general equation for simple harmonic motion, and tweak its parameters to obtain an equation that is well suited to describe the oscillatory motion of spring-mass systems. If we are to pull the mass a tad bit away from its equilibrium position and then let go, we see that the spring-mass system seems to undergo some kind of periodic motion. To find a mathematical representation for such a motion, we may draw an analogy between this behavior and the mathematics at our disposal.
After a fixed period of time, we will find the system returns to its original state. Use of terminology such as period should remind one of precalculus, and specifically of the sinusoidal functions sine and cosine. Let us suppose then that the position can be described by a cosine function (we choose cosine instead of sine by convention, since in other methods of derivation it is more convenient to choose one over the other). We also know from observing the motion of the object that its position depends upon the frequency and amplitude of oscillations. Hopefully you remember how to parameterize a circle: we define <math> x = R\cos(t) </math> and <math> y = R \sin(t)  </math>, where <math> R </math> is the radius, and we take <math> t </math> from 0 to <math> 2\pi </math>. However, we could just as easily assume that <math> t </math> keeps going past <math> 2\pi </math>, or that it takes on negative values, since it will stay on the circle; we just know that it will trace out a circle over a period of <math> 2\pi </math>. By this same token, we can also choose to give <math> t </math> a coefficient, writing the equations as <math> x = R\cos(2\pi t) </math> and <math> y = R\sin(2\pi t)</math>. Then the circle will be traced out as <math> t </math> goes from 0 to 1. If we were to make the coefficient <math> \pi </math> instead, then <math> t </math> would go from 0 to 2. Thus its period would be twice as large. All of this guides how we will now write out the equation for a spring. We know that the period will affect the argument inside the cosine, and the distance it travels will be determined by the coefficient outside. Calling the maximum distance the system will extend the amplitude, <math> A </math>, and calling the inverse of the period the frequency, <math> f </math>, we have the generic equation


<math>x(t) = A \cos(2\pi f t)</math>         


[http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html Gauss's Law]
By further convention, we'll rewrite this in terms of angular frequency <math> \omega = 2 \pi f </math>, giving


[http://www.colorado.edu/physics/phys1120/phys1120_sp08/notes/notes/Knight27_gauss_lect.pdf Gauss's Law Examples]
<math> x(t) = A \cos(\omega t) </math>


====??====
We may also generalize this to assume that it does not start at maximum extension, by introducing a phase factor <math> \phi </math>


How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]
<math>x(t) = Acos(ωt + φ)</math>


==Examples==
Now let us consider the prominent example of a Hookean spring. The force of an ideal spring-mass system can be found using <math>\vec{F}={-k}*\vec{s}</math> where <math>\vec{s}=\vec{L}-{\vec{L}_{o}}</math>. This equation stems from  Newton's Second Law <ref>https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion</ref> and Hooke's Law<ref>https://en.wikipedia.org/wiki/Hooke%27s_law</ref> which is a first order linear approximation for forces acting on an elastic system, such as a spring. The solution to this differential equation is <math>x=A\cos(\omega t + \phi)</math> where <math>ω=\sqrt{k/m}</math>. Combining this equation with our knowledge of sinusoidal functions, we define the period of oscillation to be <math>T=2π/ω</math>.
 
[[Image:SimpleHarmonicOscillation.png|500px]]
 
<div class="toccolours mw-collapsible mw-collapsed" style="width:700px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Advanced: Derivation with Differential Equations</div>
<div class="mw-collapsible-content">
 
Our differential equation is given by Hooke's Law and Newton's Second Law:
 
<math> x''(t) = -\frac{k}{m}x(t) </math>
 
This is a very nice equation (it is linear with constant coefficients)<ref> http://www.feynmanlectures.caltech.edu/I_21.html </ref>, so we may solve it by creating a linear combination of two known solutions<ref> http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx </ref>. As it happens, we know two functions that can solve this equation: <math>x_1(t) = a\cos(\omega t) </math> and <math> x_2 (t) = b\sin(\omega t)</math>, where <math> \omega^2 = k/m </math>, so that a double derivative gives <math> x_1''(t) = -a \omega^2 \cos(\omega t) = - \frac{k}{m} x_1(t) </math>, and etc. for <math> x_2 </math>. Then any solution may be written as
 
<math> x(t) = a \cos(\omega t) + b\sin(\omega t) </math>
 
Now, to make this a convenient form, lets call <math> A = \sqrt{a^2 + b^2} </math>, and pull it out, giving
 
<math> x(t) = A\biggr{(} \frac{a}{A} \cos(\omega t) + \frac{b}{A}\sin(\omega t) \biggr{)} </math>
 
By construction, these fractions look a lot like the definition of cosine and sine from trigonometry, and so we call them these respectively. Now, we can use<ref> http://www.sosmath.com/trig/Trig5/trig5/trig5.html </ref> to reduce this to
 
<math> x(t) = A \cos(\omega t + \phi) </math>
 
Now, we want to determine the constants. If the initial velocity is zero, this is easy: <math> A = x_i </math> and <math> \phi = 0 </math>. Otherwise, we will need to use the initial value conditions:
 
<math> x(0) = A \cos(\phi) </math>
 
<math> x'(0) = A\omega \sin(\phi) </math>
 
Taking a ratio of these two and solving for <math> \phi </math> will lead to
 
<math> \phi = \arctan\biggr{(}\frac{x'(0)}{\omega x(0)}\biggr{)} </math>
 
Finally, this may be substituted into either equation to solve for <math> A </math>
 
</div></div>


Be sure to show all steps in your solution and include diagrams whenever possible
===A Computational Model===


===Simple===
[https://trinket.io/glowscript/5de153d737?toggleCode=true Simple harmonic motion in vPython] uses glowscript to implement a varying force iterative motion predictor, while [https://colab.research.google.com/drive/1617NfftFpRj7BiZZFJZDI731MHC-J_q4 this] has uses numpy to do the same (here the cell with simp_harm_func is the one we wish to look at). For a detailed description of how varying force computational methods work, see [[Fundamentals of Iterative Prediction with Varying Force]]. The important distinction of simple harmonic motion is that it is defined by the force function


[[File:BBB.PNG]]
<math> F(t,x,v) = -kx </math>


Electric flux on the disk, by using Gauss's Law, is the multiplication between Electric Field normal to the disk's surface and surface area of the disk.
Where by convention we have <math> k >0 </math>.


Electric Field Normal to the Surface: E x sin(40) = 327 V/m x sin(40)
==Examples==


Surface Area of the disk: 3.14 x 0.02 x 0.02 m^2
===Simple===
Problem: A spring has a restoring force of <math> 300 N </math> when it is compressed <math> 0.2 m </math>. What is the spring's constant <math>k</math> in <math>N/m</math>?


Electric Flux on the Disk: 327 V/m x sin(40) x 3.14 x 0.02 x 0.02 m^2 = 0.264 Vm
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Solution</div>
<div class="mw-collapsible-content">


===Middling===
Hooke's law gives
The electric field has been measured to be vertically upward everywhere on the surface of a box 20 cm long, 4 cm high, and 3 cm deep, shown in the figure. All over the bottom of the box E1 = 1100 V/m, all over the sides E2 = 950 V/m, and all over the top E3 = 750 V/m.


[[File:sss.JPG]]
<math>\vec{F}=-k\vec{s}</math>


Since E1, E2, and E3, are all measured to be vertically upward everywhere on the surface of a box, only the bottom surface and the top surface will be focused (multiplying the normal vector of other surfaces than the bottom and top surfaces will result in zero electric flux). The normal vector of the bottom surface is known to be <0,-1,0> and that of the top surface is known to be <0,1,0> assuming vertically upward is in +y direction.
So rearranging gives


E1 = <0, 1100, 0> V/m,
<math>-k=\frac{\vec{F}}{\vec{s}}</math>
E2 = <0, 950, 0> V/m,
E3 = <0, 750, 0> V/m,


E1 corresponds to the bottom surface according to the diagram. Multiplying vector E1 with its normal vector of the bottom surface equals -1100 (dot product), and multiplying this vector by the area equals -1100 V/m x 0.20 m x 0.03 m = -6.6 Vm.
Orienting the direction of extension as the positive direction, we have <math> \vec{F} = 300 N </math> and <math> \vec{s} = -0.2 m </math>


E3 corresponds to the top surface according to the diagram. Multiplying vector E3 with its normal vector of the top surface equals 750 (dot product) and multiplying this vector by the area equals 750 V/m x 0.20 m x 0.03 m = 4.5 Vm.
<math>-k=\frac{300 N}{-.2 m}</math>


Therefore, the sum of the electric flux in this box equals -6.6 Vm + 4.5 Vm = -2.1 Vm
<math>k=\frac{1500N}{m}</math>


To determine the amount of charge enclosed by the box, we use Gauss's Law. Since we know the sum of the electric flux, in order to find q (inside the box) we just have to multiply summed electric flux and epsilon naught (8.85 e-12 unit)
</div></div>


Total Charge: -2.1 x 8.85 x 10 ^ (-12) = -1.8585 x 10 ^ (-11) C
===Middling===
Problem: What is the oscillation period of a spiring with spring constant 15N/m with a 30 kg mass attached?


===Difficult===
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Solution</div>
<div class="mw-collapsible-content">


[[File:AAA.JPG]]
First, we have by definition that


To calculate the electric flux you will have to know the normal vector or the electric field in normal direction of each side of the cube.
<math>T=\frac{2\pi}{\omega}</math>


There are only two sides affecting the net electric flux on this cubical surface, bottom face and right-handed side face (rest will be zero).
and using the formula for <math> \omega </math> gives


Normal Electric Field (Bottom Face): E1 x sin(27) = 400 V/m x sin(27) x 0.35m x 0.35m = 22.2455 Vm
<math>T=2\pi\sqrt{\frac{m}{k}}</math>


Normal Electric Field (Right-Handed Side Face): E1 x sin(27) = 400 V/m x sin(27) x 0.35m x 0.35m = 22.2455 Vm
<math>k=15 (N/m)</math> and <math>m=30kg</math>


Net Electric Field = 44.49 Vm
<math>T=2\pi \sqrt{\frac{30 kg}{15(N/m)}}</math>


Net Charge inside the cubical surface = 44.19 Vm x 8.85 x 10 ^ (-12) = 3.94 x 10 ^ (-10) C
<math>T=8.88 s</math>


==Connectedness==
</div></div>
#How is this topic connected to something that you are interested in?


===Difficult===
Problem: A spring with spring constant 18 N/m has mass of 2 kg is attached to it. The mass is then displaced to x = 2 . How much time does it take for the block to travel to the point x = 1?


Solution:


#How is it connected to your major?
<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Solution</div>
<div class="mw-collapsible-content">
<math>x=A\cos(\omega t)</math>


I am majoring in Materials Science and Engineering and going further on from Electric Flux, there is a way to calculate Electric Flux Density (D). To obtain D, you have to multiply epsilon (permittivity) of the material to the Electric Field, E. Of course, different materials will have different Electric Flux. For example there are three major types of materials: insulator, semi-conductor, and conductor. Depending on their uses, you would want to make sure that the material of the desired product does not contain electric flux due to excess charge on the surface. This is the reason that electric flux and total charge of the closed surfaced objects are useful when as an engineer you want to think what type of properties you would desire for your product.
<math>\frac{x}{A}=\cos\biggr{(}\sqrt{\frac{k}{m}}t\biggr{)}</math>


#Is there an interesting industrial application?
<math>1/2=\cos\biggr{(}\sqrt{\frac{18 (N/m)}{2 kg}}t\biggr{)}</math>


==History==
<math>\arccos(1/2)=3(Hz) \cdot t</math>


Thermodynamics was brought up as a science in the 18th and 19th centuries. However, it was first brought up by Galilei, who introduced the concept of temperature and invented the first thermometer.  G. Black first introduced the word 'thermodynamics'.  Later, G. Wilke introduced another unit of measurement known as the calorie that measures heat.  The idea of thermodynamics was brought up by Nicolas Leonard Sadi Carnot.  He is often known as "the father of thermodynamics".  It all began with the development of the steam engine during the Industrial Revolution.  He devised an ideal cycle of operation.  During his observations and experimentations, he had the incorrect notion that heat is conserved, however he was able to lay down theorems that led to the development of thermodynamics.  In the 20th century, the science of thermodynamics became a conventional term and a basic division of physics.  Thermodynamics dealt with the study of general properties of physical systems under equilibrium and the conditions necessary to obtain equilibrium. 
<math>t=.35 s</math>
</div></div>


== See also ==
==Connectedness==
[A student should expand upon this]


Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?
One interesting application of simple harmonic motion is its ability to approximate the interatomic vibration of molecules which I find very interesting as an Earth and Atmospheric Science major, trying to better understand our environment and how the molecules it is made up of interact. I have always been very intrigued by the composition and interactions of molecules and this approximations brings me another step closer to understanding how our world works. An interesting industrial application of simple harmonic motion is  its approximation of a car running on worn down shock absorbers.


===Further reading===
==History==


Books, Articles or other print media on this topic
Thomas Hooke, an English scientist, discovered what is now known as Hooke's Law in 1660 while working on the springs of watches.<ref> https://www.britannica.com/science/Hookes-law </ref> A remarkable physicist, Hooke was also a pioneer in optics, astronomy, and fluid mechanics. He supported a theory of evolution nearly two hundred years before Darwin (although he did not know about the principle of natural selection), devised the inverse square law which Newton adapted, and accurately described air as individual particles separated by large distances.<ref> https://www.britannica.com/biography/Robert-Hooke </ref> Harmonic motion is thus named because of its connection to music: instruments vibrate, and in doing so produce the sounds we perceive as music.<ref> https://www.britannica.com/science/simple-harmonic-motion </ref> The Fourier series allows for the expression of ''any'' periodic function in terms of sinusoids, meaning that a huge variety of phenomena in physics may be decomposed into simple harmonic motion. Furthermore, for reasons discussed in reference 8, harmonic oscillators are a strong approximation of many systems near equilibrium. The corresponding quantum harmonic oscillator is similarly useful in quantum mechanics. <ref> https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University_Physics_(OpenStax)/Map%3A_University_Physics_III_-_Optics_and_Modern_Physics_(OpenStax)/7%3A_Quantum_Mechanics/7.5%3A_The_Quantum_Harmonic_Oscillator </ref>.


[http://www.phys.utk.edu/daunt/EM/PDF/SJDLecture22.pdf Electric Field and Gauss's Law]
== See also ==
*[[Fundamentals of Iterative Prediction with Varying Force]]
*[[Iterative Prediction of Spring-Mass System]]
*[[Two Dimensional Harmonic Motion]]


===External links===
===External links===
*https://www.khanacademy.org/science/physics/oscillatory-motion/harmonic-motion/v/introduction-to-harmonic-motion
*http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html
*https://en.wikibooks.org/wiki/A-level_Physics_(Advancing_Physics)/Simple_Harmonic_Motion/Mathematical_Derivation
*http://www.feynmanlectures.caltech.edu/I_21.html


Internet resources on this topic
===Further Reading===
*Matter and Interactions, 4th Edition


==References==
==References==
 
<references/>
https://www.grc.nasa.gov/www/k-12/airplane/thermo0.html
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thereq.html
https://www.grc.nasa.gov/www/k-12/airplane/thermo2.html
http://www.phys.nthu.edu.tw/~thschang/notes/GP21.pdf
http://www.eoearth.org/view/article/153532/
 
[[Category:Interactions]]

Latest revision as of 15:24, 29 July 2019

Claimed for editing by Elton Leander Pinto 10/29/2018

Edited by Richard Udall June 2019

Simple harmonic motion is motion driven by a restorative force which acts to bring the oscillating particle back to its equilibrium position. Prototypical examples of this include a pendulum or a spring which is compressing and extending. However, most oscillating systems that we observe in our day-to-day life are not perfect simple harmonic oscillators. Simple harmonic motion is an approximation that ignores friction and air resistance. Although this is not generally true, simple harmonic motion is a decent approximation of these more complex systems. The more complex and accurate formulation which takes these forces into account is known as damped harmonic motion, and will be considered further in the page Iterative Prediction of Spring-Mass System.

The Main Idea

Simple harmonic motion is a periodic motion, a motion that is repeated over some time interval. This periodic motion has a restoring force, which is a force which attempts to restore the system to its equilibrium position, and which is proportional and opposite in direction to displacement. Thus, the further the system is from its equilibrium position, the larger the force which acts to return it to equilibrium position. As the object gets closer to equilibrium position, the force decreases until is magnitude reaches zero. At the equilibrium, the particle has no forces acting upon it, but it now has a large amount of momentum, and so it passes the equilibrium point going in the opposite direction. As moves in the opposite direction from the equilibrium, the restoring forces increases in the opposite direction until the momentum reaches zero, after which the particle begins moving back towards the equilibrium. This process repeats, which leads to the periodic motion characteristic of the system. Common examples of simple harmonic motion include an undamped spring-mass system, and a pendulum swinging back and forth (in the idealized case where it does not slow down).

Simple Harmonic Motion Orbit

A Mathematical Model

Simple harmonic motion occurs very frequently in our treatment of physical systems, and is a useful approximation for many because it is often reasonably accurate, and because we have the tools to describe it exactly. We may consider the example of the spring (see the figure) to arrive at a general equation for simple harmonic motion, and tweak its parameters to obtain an equation that is well suited to describe the oscillatory motion of spring-mass systems. If we are to pull the mass a tad bit away from its equilibrium position and then let go, we see that the spring-mass system seems to undergo some kind of periodic motion. To find a mathematical representation for such a motion, we may draw an analogy between this behavior and the mathematics at our disposal. After a fixed period of time, we will find the system returns to its original state. Use of terminology such as period should remind one of precalculus, and specifically of the sinusoidal functions sine and cosine. Let us suppose then that the position can be described by a cosine function (we choose cosine instead of sine by convention, since in other methods of derivation it is more convenient to choose one over the other). We also know from observing the motion of the object that its position depends upon the frequency and amplitude of oscillations. Hopefully you remember how to parameterize a circle: we define [math]\displaystyle{ x = R\cos(t) }[/math] and [math]\displaystyle{ y = R \sin(t) }[/math], where [math]\displaystyle{ R }[/math] is the radius, and we take [math]\displaystyle{ t }[/math] from 0 to [math]\displaystyle{ 2\pi }[/math]. However, we could just as easily assume that [math]\displaystyle{ t }[/math] keeps going past [math]\displaystyle{ 2\pi }[/math], or that it takes on negative values, since it will stay on the circle; we just know that it will trace out a circle over a period of [math]\displaystyle{ 2\pi }[/math]. By this same token, we can also choose to give [math]\displaystyle{ t }[/math] a coefficient, writing the equations as [math]\displaystyle{ x = R\cos(2\pi t) }[/math] and [math]\displaystyle{ y = R\sin(2\pi t) }[/math]. Then the circle will be traced out as [math]\displaystyle{ t }[/math] goes from 0 to 1. If we were to make the coefficient [math]\displaystyle{ \pi }[/math] instead, then [math]\displaystyle{ t }[/math] would go from 0 to 2. Thus its period would be twice as large. All of this guides how we will now write out the equation for a spring. We know that the period will affect the argument inside the cosine, and the distance it travels will be determined by the coefficient outside. Calling the maximum distance the system will extend the amplitude, [math]\displaystyle{ A }[/math], and calling the inverse of the period the frequency, [math]\displaystyle{ f }[/math], we have the generic equation

[math]\displaystyle{ x(t) = A \cos(2\pi f t) }[/math]

By further convention, we'll rewrite this in terms of angular frequency [math]\displaystyle{ \omega = 2 \pi f }[/math], giving

[math]\displaystyle{ x(t) = A \cos(\omega t) }[/math]

We may also generalize this to assume that it does not start at maximum extension, by introducing a phase factor [math]\displaystyle{ \phi }[/math]

[math]\displaystyle{ x(t) = Acos(ωt + φ) }[/math]

Now let us consider the prominent example of a Hookean spring. The force of an ideal spring-mass system can be found using [math]\displaystyle{ \vec{F}={-k}*\vec{s} }[/math] where [math]\displaystyle{ \vec{s}=\vec{L}-{\vec{L}_{o}} }[/math]. This equation stems from Newton's Second Law [1] and Hooke's Law[2] which is a first order linear approximation for forces acting on an elastic system, such as a spring. The solution to this differential equation is [math]\displaystyle{ x=A\cos(\omega t + \phi) }[/math] where [math]\displaystyle{ ω=\sqrt{k/m} }[/math]. Combining this equation with our knowledge of sinusoidal functions, we define the period of oscillation to be [math]\displaystyle{ T=2π/ω }[/math].

Advanced: Derivation with Differential Equations

Our differential equation is given by Hooke's Law and Newton's Second Law:

[math]\displaystyle{ x''(t) = -\frac{k}{m}x(t) }[/math]

This is a very nice equation (it is linear with constant coefficients)[3], so we may solve it by creating a linear combination of two known solutions[4]. As it happens, we know two functions that can solve this equation: [math]\displaystyle{ x_1(t) = a\cos(\omega t) }[/math] and [math]\displaystyle{ x_2 (t) = b\sin(\omega t) }[/math], where [math]\displaystyle{ \omega^2 = k/m }[/math], so that a double derivative gives [math]\displaystyle{ x_1''(t) = -a \omega^2 \cos(\omega t) = - \frac{k}{m} x_1(t) }[/math], and etc. for [math]\displaystyle{ x_2 }[/math]. Then any solution may be written as

[math]\displaystyle{ x(t) = a \cos(\omega t) + b\sin(\omega t) }[/math]

Now, to make this a convenient form, lets call [math]\displaystyle{ A = \sqrt{a^2 + b^2} }[/math], and pull it out, giving

[math]\displaystyle{ x(t) = A\biggr{(} \frac{a}{A} \cos(\omega t) + \frac{b}{A}\sin(\omega t) \biggr{)} }[/math]

By construction, these fractions look a lot like the definition of cosine and sine from trigonometry, and so we call them these respectively. Now, we can use[5] to reduce this to

[math]\displaystyle{ x(t) = A \cos(\omega t + \phi) }[/math]

Now, we want to determine the constants. If the initial velocity is zero, this is easy: [math]\displaystyle{ A = x_i }[/math] and [math]\displaystyle{ \phi = 0 }[/math]. Otherwise, we will need to use the initial value conditions:

[math]\displaystyle{ x(0) = A \cos(\phi) }[/math]

[math]\displaystyle{ x'(0) = A\omega \sin(\phi) }[/math]

Taking a ratio of these two and solving for [math]\displaystyle{ \phi }[/math] will lead to

[math]\displaystyle{ \phi = \arctan\biggr{(}\frac{x'(0)}{\omega x(0)}\biggr{)} }[/math]

Finally, this may be substituted into either equation to solve for [math]\displaystyle{ A }[/math]

A Computational Model

Simple harmonic motion in vPython uses glowscript to implement a varying force iterative motion predictor, while this has uses numpy to do the same (here the cell with simp_harm_func is the one we wish to look at). For a detailed description of how varying force computational methods work, see Fundamentals of Iterative Prediction with Varying Force. The important distinction of simple harmonic motion is that it is defined by the force function

[math]\displaystyle{ F(t,x,v) = -kx }[/math]

Where by convention we have [math]\displaystyle{ k \gt 0 }[/math].

Examples

Simple

Problem: A spring has a restoring force of [math]\displaystyle{ 300 N }[/math] when it is compressed [math]\displaystyle{ 0.2 m }[/math]. What is the spring's constant [math]\displaystyle{ k }[/math] in [math]\displaystyle{ N/m }[/math]?

Solution

Hooke's law gives

[math]\displaystyle{ \vec{F}=-k\vec{s} }[/math]

So rearranging gives

[math]\displaystyle{ -k=\frac{\vec{F}}{\vec{s}} }[/math]

Orienting the direction of extension as the positive direction, we have [math]\displaystyle{ \vec{F} = 300 N }[/math] and [math]\displaystyle{ \vec{s} = -0.2 m }[/math]

[math]\displaystyle{ -k=\frac{300 N}{-.2 m} }[/math]

[math]\displaystyle{ k=\frac{1500N}{m} }[/math]

Middling

Problem: What is the oscillation period of a spiring with spring constant 15N/m with a 30 kg mass attached?

Solution

First, we have by definition that

[math]\displaystyle{ T=\frac{2\pi}{\omega} }[/math]

and using the formula for [math]\displaystyle{ \omega }[/math] gives

[math]\displaystyle{ T=2\pi\sqrt{\frac{m}{k}} }[/math]

[math]\displaystyle{ k=15 (N/m) }[/math] and [math]\displaystyle{ m=30kg }[/math]

[math]\displaystyle{ T=2\pi \sqrt{\frac{30 kg}{15(N/m)}} }[/math]

[math]\displaystyle{ T=8.88 s }[/math]

Difficult

Problem: A spring with spring constant 18 N/m has mass of 2 kg is attached to it. The mass is then displaced to x = 2 . How much time does it take for the block to travel to the point x = 1?

Solution:

Solution

[math]\displaystyle{ x=A\cos(\omega t) }[/math]

[math]\displaystyle{ \frac{x}{A}=\cos\biggr{(}\sqrt{\frac{k}{m}}t\biggr{)} }[/math]

[math]\displaystyle{ 1/2=\cos\biggr{(}\sqrt{\frac{18 (N/m)}{2 kg}}t\biggr{)} }[/math]

[math]\displaystyle{ \arccos(1/2)=3(Hz) \cdot t }[/math]

[math]\displaystyle{ t=.35 s }[/math]

Connectedness

[A student should expand upon this]

One interesting application of simple harmonic motion is its ability to approximate the interatomic vibration of molecules which I find very interesting as an Earth and Atmospheric Science major, trying to better understand our environment and how the molecules it is made up of interact. I have always been very intrigued by the composition and interactions of molecules and this approximations brings me another step closer to understanding how our world works. An interesting industrial application of simple harmonic motion is its approximation of a car running on worn down shock absorbers.

History

Thomas Hooke, an English scientist, discovered what is now known as Hooke's Law in 1660 while working on the springs of watches.[6] A remarkable physicist, Hooke was also a pioneer in optics, astronomy, and fluid mechanics. He supported a theory of evolution nearly two hundred years before Darwin (although he did not know about the principle of natural selection), devised the inverse square law which Newton adapted, and accurately described air as individual particles separated by large distances.[7] Harmonic motion is thus named because of its connection to music: instruments vibrate, and in doing so produce the sounds we perceive as music.[8] The Fourier series allows for the expression of any periodic function in terms of sinusoids, meaning that a huge variety of phenomena in physics may be decomposed into simple harmonic motion. Furthermore, for reasons discussed in reference 8, harmonic oscillators are a strong approximation of many systems near equilibrium. The corresponding quantum harmonic oscillator is similarly useful in quantum mechanics. [9].

See also

External links

Further Reading

  • Matter and Interactions, 4th Edition

References