Simple Harmonic Motion - Revision history

Richard.Udall: /* See also */

2019-07-29T20:24:20Z

Richard.Udall: /* See also */

2019-07-24T20:15:43Z

Richard.Udall: /* Connectedness */

2019-07-24T20:14:37Z

Connectedness

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	==Connectedness==		==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.		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.

Richard.Udall: /* A Mathematical Model */

2019-07-24T20:12:21Z

A Mathematical Model

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	===A Mathematical Model===		===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 ~~fig~~) 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.		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		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

Richard.Udall at 20:11, 24 July 2019

2019-07-24T20:11:59Z

← Older revision		Revision as of 16:11, 24 July 2019
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	'''Claimed for editing by Elton Leander Pinto 10/29/2018'''		'''Claimed for editing by Elton Leander Pinto 10/29/2018'''

	'''Edited by Richard Udall ~~July~~ 2019'''		'''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]].		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]].
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	==The Main Idea==		==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, a force ~~that is always working~~ to ~~return~~ the system to equilibrium position, ~~that~~ is proportional and opposite to displacement. ~~Due to this as~~ the system ~~gets farther~~ from equilibrium position the larger the force is to return it to equilibrium position. ~~When no friction or air resistance is present the system will continue to oscillate as the restoring force decreases as~~ the object gets closer to equilibrium position ~~until~~ the force reaches zero, ~~but at which point~~ the particle ~~continues due to its initial~~ momentum ~~until~~ passes equilibrium ~~position~~. ~~Then~~ the restoring forces increases in the opposite direction until the momentum ~~is changed enough to change the direction of~~ the particle ~~and~~ the process repeats. ~~A common example~~ of simple harmonic motion is an undamped spring-mass system, ~~or one that~~ does not ~~proceed to rest due to friction or another dissipative force~~.		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).

	[[File:Simple Harmonic Motion Orbit.gif\|thumb\|Simple Harmonic Motion Orbit]]		[[File:Simple Harmonic Motion Orbit.gif\|thumb\|Simple Harmonic Motion Orbit]]

Richard.Udall at 20:03, 24 July 2019

2019-07-24T20:03:18Z

← Older revision		Revision as of 16:03, 24 July 2019
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	'''Edited by Richard Udall July 2019'''		'''Edited by Richard Udall July 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]].		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]].

Richard.Udall at 19:58, 24 July 2019

2019-07-24T19:58:13Z

← Older revision		Revision as of 15:58, 24 July 2019
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	'''Claimed for editing by Elton Leander Pinto 10/29/2018'''		'''Claimed for editing by Elton Leander Pinto 10/29/2018'''

			'''Edited by Richard Udall July 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]].		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]].

Richard.Udall at 19:57, 24 July 2019

2019-07-24T19:57:47Z

← Older revision		Revision as of 15:57, 24 July 2019
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	'''Claimed for editing by Elton Leander Pinto 10/29/2018'''		'''Claimed for editing by Elton Leander Pinto 10/29/2018'''
			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]].
	~~This~~ is ~~a type of periodic~~ motion ~~in which, from a physical point of view,~~ a restorative force acts to ~~always~~ bring ~~back~~ the oscillating particle back to its equilibrium position. ~~An example~~ of ~~a system exhibiting~~ this ~~kind of behavior includes~~ a pendulum ~~oscillating in~~ a ~~medium that provides minimal resistance to motion~~. 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 ~~assumption can often~~ not ~~be made in everyday calculations~~, simple harmonic motion ~~can approximate these otherwise complicated situations, fairly well. This~~ is ~~because of the effect of damping on the motion~~ of systems ~~that reduces the amplitude of oscillation,~~ and ~~hence we get what~~ is known as damped ~~oscillations. Another interesting idea that is relevant when we talk about these systems is the concept of resonance, and how it can be exploited to bring about astonishing results. Simple~~ harmonic motion ~~can be used to estimate many systems including spring-mass systems~~ and ~~the swinging of a pendulum in certain instances which~~ will be ~~explained in~~ further ~~detail below. We will explore these ideas~~ in the ~~following section~~.

	==The Main Idea==		==The Main Idea==

Richard.Udall: /* The Main Idea */

2019-06-25T17:08:20Z

The Main Idea

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	<math> x''(t) = -\frac{k}{m}x(t) </math>		<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) = - k/m x_1(t) </math>, and etc. for <math> x_2 </math>. Then any solution may be written as		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>		<math> x(t) = a \cos(\omega t) + b\sin(\omega t) </math>

Richard.Udall: /* A Mathematical Model */

2019-06-25T17:07:19Z

A Mathematical Model

← Older revision		Revision as of 13:07, 25 June 2019
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	<math> x''(t) = -\frac{k}{m}x(t) </math>		<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>. Then any solution may be written as		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) = - 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>		<math> x(t) = a \cos(\omega t) + b\sin(\omega t) </math>

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	== See also ==		== See also ==
			*[[Fundamentals of Iterative Prediction with Varying Force]]
	[[Fundamentals of Iterative Prediction with Varying Force]]		*[[Iterative Prediction of Spring-Mass System]]
			*[[Two Dimensional Harmonic Motion]]
	[[Iterative Prediction of Spring-Mass System]]

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

	http://www.feynmanlectures.caltech.edu/I_21.html

	===Further Reading===		===Further Reading===
	Matter and Interactions, 4th Edition		*Matter and Interactions, 4th Edition

	==References==		==References==
	<references/>		<references/>

← Older revision		Revision as of 16:15, 24 July 2019
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	http://www.feynmanlectures.caltech.edu/I_21.html		http://www.feynmanlectures.caltech.edu/I_21.html

			===Further Reading===
			Matter and Interactions, 4th Edition

	==References==		==References==
	<references/>		<references/>