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{{Short description|To-and-fro periodic motion in science and engineering}}
{{Use American English|date=April 2019}}
[[File:Simple Harmonic Motion Orbit.gif|thumb|upright=1.5|Simple harmonic motion shown both in real space and [[phase space]]. The [[orbit (dynamics)|orbit]] is [[periodic function|periodic]]. (Here the [[velocity]] and [[position (vector)|position]] axes have been reversed from the standard convention to align the two diagrams)]]
{{Classical mechanics}}

In [[mechanics]] and [[physics]], '''simple harmonic motion''' (sometimes abbreviated as '''{{abbr|SHM|simple harmonic motion}}''') is a special type of [[periodic function|periodic]] [[motion]] an object experiences by means of a [[restoring force]] whose magnitude is directly [[proportionality (mathematics)|proportional]] to the distance of the object from an equilibrium position and acts towards the equilibrium position. It results in an [[oscillation]] that is described by a [[sinusoid]] which continues indefinitely (if uninhibited by [[friction]] or any other [[dissipation]] of [[energy]]).<ref>{{cite web |date=2024-09-30 |title=Simple harmonic motion {{!}} Formula, Examples, & Facts {{!}} Britannica |website=britannica.com |language=en |url=https://www.britannica.com/science/simple-harmonic-motion |access-date=2024-10-11}}</ref>

Simple harmonic motion can serve as a [[mathematical model]] for a variety of motions, but is typified by the oscillation of a [[mass]] on a [[spring (device)|spring]] when it is subject to the linear [[elasticity (physics)|elastic]] restoring force given by [[Hooke's law]]. The motion is [[sinusoidal]] in time and demonstrates a single [[resonance|resonant]] frequency. Other phenomena can be modeled by simple harmonic motion, including the motion of a [[pendulum|simple pendulum]], although for it to be an accurate model, the [[net force]] on the object at the end of the pendulum must be proportional to the displacement (and even so, it is only a good approximation when the angle of the swing is small; see [[small-angle approximation]]). Simple harmonic motion can also be used to model [[molecular vibration]]. A mass-spring system is a classic example of simple harmonic motion.

Simple harmonic motion provides a basis for the characterization of more complicated periodic motion through the techniques of [[Fourier analysis]].

==Introduction==
The motion of a [[particle]] moving along a straight line with an [[acceleration]] whose direction is always toward a [[fixed point (mathematics)|fixed point]] on the line and whose magnitude is proportional to the displacement from the fixed point is called simple harmonic motion.<ref>{{cite web |title=Simple Harmonic Motion – Concepts |url=https://www.webassign.net/question_assets/ncsucalcphysmechl3/lab_7_1/manual.html}}</ref>

In the diagram, a [[harmonic oscillator|simple harmonic oscillator]], consisting of a weight attached to one end of a spring, is shown. The other end of the spring is connected to a rigid support such as a wall. If the system is left at rest at the [[mechanical equilibrium|equilibrium]] position then there is no net [[force]] acting on the mass. However, if the mass is displaced from the equilibrium position, the spring [[exertion|exerts]] a restoring [[elasticity (physics)|elastic]] force that obeys [[Hooke's law]].

Mathematically,
<math display="block"> \mathbf{F}=-k\mathbf{x}, </math>
where {{math|'''F'''}} is the restoring elastic force exerted by the spring (in [[International System of Units|SI]] units: [[newton (unit)|N]]), {{math|''k''}} is the [[Hooke's law|spring constant]] ([[newton (unit)|N]]·m<sup>−1</sup>), and {{math|'''x'''}} is the [[displacement (vector)|displacement]] from the equilibrium position (in [[metre]]s).

For any simple mechanical harmonic oscillator:
*When the system is displaced from its equilibrium position, a restoring force that obeys Hooke's law tends to restore the system to equilibrium.

Once the mass is displaced from its equilibrium position, it experiences a net restoring force. As a result, it [[acceleration|accelerates]] and starts going back to the equilibrium position. When the mass moves closer to the equilibrium position, the restoring force decreases. At the equilibrium position, the net restoring force vanishes. However, at {{math|''x'' {{=}} 0}}, the mass has [[momentum]] because of the acceleration that the restoring force has imparted. Therefore, the mass continues past the equilibrium position, compressing the spring. A net restoring force then slows it down until its [[velocity]] reaches zero, whereupon it is accelerated back to the equilibrium position again.

As long as the system has no [[energy]] loss, the mass continues to oscillate. Thus simple harmonic motion is a type of [[frequency|periodic]] motion. If energy is lost in the system, then the mass exhibits [[damped oscillator|damped oscillation]].

Note if the real space and [[phase space]] plot are not co-linear, the phase space motion becomes elliptical. The area enclosed depends on the amplitude and the maximum momentum.

==Dynamics==
In [[Newtonian mechanics]], for one-dimensional simple harmonic motion, the equation of motion, which is a second-order linear [[ordinary differential equation]] with '''constant''' coefficients, can be obtained by means of [[Newton's second law]] and [[Hooke's law]] for a [[mass]] on a [[spring (device)|spring]].

<math display="block"> F_\mathrm{net} = m\frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -kx,</math>
where {{mvar|m}} is the [[Mass#Inertial mass|inertial mass]] of the oscillating body, {{mvar|x}} is its [[displacement (vector)|displacement]] from the [[mechanical equilibrium|equilibrium]] (or mean) position, and {{math|''k''}} is a constant (the [[Hooke's law#Formal definition|spring constant]] for a mass on a spring).

Therefore,
<math display="block"> \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = -\frac{k}{m}x</math>

Solving the [[differential equation]] above produces a solution that is a [[sine wave|sinusoidal function]]:
<math> x(t) = c_1\cos\left(\omega t\right) + c_2\sin\left(\omega t\right),</math> where <math display="inline"> {\omega} = \sqrt{{k}/{m}}.</math>
The meaning of the constants <math> c_1</math> and <math> c_2</math> can be easily found: setting <math> t=0</math> on the equation above we see that <math> x(0) = c_1</math>, so that <math> c_1</math> is the initial position of the particle, <math> c_1=x_0</math>; taking the derivative of that equation and evaluating at zero we get that <math> \dot{x}(0) = \omega c_2</math>, so that <math> c_2</math> is the initial speed of the particle divided by the angular frequency, <math> c_2 = \frac{v_0}{\omega}</math>. Thus we can write:
<math display="block"> x(t) = x_0 \cos\left(\sqrt{\frac{k}{m}} t\right) + \frac{v_0}{\sqrt{\frac{k}{m}}}\sin\left(\sqrt{\frac{k}{m}} t\right).</math>

This equation can also be written in the form:
<math display="block"> x(t) = A\cos\left(\omega t - \varphi\right),</math>
where
* <math> A = \sqrt{{c_1}^2 + {c_2}^2} </math>
* <math>\tan \varphi = \frac{c_2}{c_1}, </math>
* <math>\sin \varphi = \frac{c_2}{A}, \; \cos \varphi = \frac{c_1}{A} </math>
or equivalently
* <math> A = |c_1 + c_2i|, </math>
* <math>\varphi = \arg(c_1 + c_2i) </math>
In the solution, {{math|''c''<sub>1</sub>}} and {{math|''c''<sub>2</sub>}} are two constants determined by the initial conditions (specifically, the initial position at time {{math|1=''t'' = 0}} is {{math|''c''<sub>1</sub>}}, while the initial velocity is {{math|''c''<sub>2</sub>''ω''}}), and the origin is set to be the equilibrium position.{{Cref2|A}} Each of these constants carries a physical meaning of the motion: {{math|''A''}} is the [[amplitude]] (maximum displacement from the equilibrium position), {{math|1=''ω'' = 2''πf''}} is the [[angular frequency]], and {{math|''φ''}} is the initial [[phase (waves)|phase]].{{Cref2|B}}

Using the techniques of [[calculus]], the [[velocity]] and [[acceleration]] as a function of time can be found:
<math display="block"> v(t) = \frac{\mathrm{d} x}{\mathrm{d} t} = - A\omega \sin(\omega t-\varphi),</math>
*Speed: <math> {\omega} \sqrt {A^2 - x^2} </math>
*Maximum speed: {{math|1=''v'' = ''ωA''}} (at equilibrium point)
<math display="block"> a(t) = \frac{\mathrm{d}^2 x}{\mathrm{d}t^2} = - A \omega^2 \cos( \omega t-\varphi).</math>
*Maximum acceleration: {{math|''Aω''<sup>2</sup>}} (at extreme points)

By definition, if a mass {{math|''m''}} is under SHM its acceleration is directly proportional to displacement.
<math display="block"> a(x) = -\omega^2 x.</math>
where
<math display="block"> \omega^2=\frac{k}{m}</math>

Since {{math|1=''ω'' = 2''πf''}},
<math display="block">f = \frac{1}{2\pi}\sqrt{\frac{k}{m}},</math>
and, since {{math|1=''T'' = {{sfrac|1|''f''}}}} where {{math|''T''}} is the time period,
<math display="block">T = 2\pi \sqrt{\frac{m}{k}}.</math>

These equations demonstrate that the simple harmonic motion is [[wikt:isochronous|isochronous]] (the period and frequency are independent of the amplitude and the initial phase of the motion).

==Energy==
Substituting {{math|''ω''<sup>2</sup>}} with {{math|''{{sfrac|k|m}}''}}, the [[kinetic energy]] {{math|''K''}} of the system at time {{math|''t''}} is
<math display="block"> K(t) = \tfrac12 mv^2(t) = \tfrac12 m\omega^2A^2\sin^2(\omega t - \varphi) = \tfrac12 kA^2 \sin^2(\omega t - \varphi),</math>
and the [[potential energy]] is
<math display="block">U(t) = \tfrac12 k x^2(t) = \tfrac12 k A^2 \cos^2(\omega t - \varphi).</math>
In the absence of friction and other energy loss, the total [[mechanical energy]] has a constant value
<math display="block">E = K + U = \tfrac12 k A^2.</math>

==Examples==
[[Image:Animated-mass-spring.gif|right|frame|An undamped [[spring–mass system]] undergoes simple harmonic motion.]]
The following physical systems are some examples of [[harmonic oscillator|simple harmonic oscillator]].

===Mass on a spring===
A mass {{math|''m''}} attached to a spring of spring constant {{math|''k''}} exhibits simple harmonic motion in [[closed space]]. The equation for describing the period:
<math display="block"> T= 2 \pi\sqrt\frac{m}{k}</math>
shows the period of oscillation is independent of the amplitude, though in practice the amplitude should be small. The above equation is also valid in the case when an additional constant force is being applied on the mass, i.e. the additional constant force cannot change the period of oscillation.

===Uniform circular motion===
Simple harmonic motion can be considered the one-dimensional [[projection (mathematics)|projection]] of [[uniform circular motion]]. If an object moves with angular speed {{math|''ω''}} around a circle of radius {{math|''r''}} centered at the [[origin (mathematics)|origin]] of the {{math|''xy''}}-plane, then its motion along each coordinate is simple harmonic motion with amplitude {{math|''r''}} and angular frequency {{math|''ω''}}.

=== Oscillatory motion ===
The motion of a body in which it moves to and from a definite point is also called [[oscillatory motion]] or vibratory motion. The time period is able to be calculated by
<math display="block"> T= 2 \pi\sqrt\frac{l}{g}</math>
where l is the distance from rotation to the object's center of mass undergoing SHM and g is acceleration due to gravity. This is analogous to the mass-spring system.

===Mass of a simple pendulum===
{{Infobox physical quantity
| image = ลูกตุ้มธรรมชาติ.gif
| caption = A [[pendulum]] making 25 complete [[oscillation]]s in 60 s, a frequency of 0.41{{overline|6}} [[Hertz]]
{{ubl
}}
}}
In the [[small-angle approximation]], the [[pendulum (mechanics)#Small-angle approximation|motion of a simple pendulum]] is approximated by simple harmonic motion. The period of a mass attached to a pendulum of length {{math|''l''}} with gravitational acceleration <math>g</math> is given by
<math display="block"> T = 2 \pi \sqrt\frac{l}{g}</math>

This shows that the period of oscillation is independent of the amplitude and mass of the pendulum but not of the acceleration due to [[gravity]], <math>g</math>, therefore a pendulum of the same length on the Moon would swing more slowly due to the Moon's lower gravitational field strength. Because the value of <math>g</math> varies slightly over the surface of the earth, the time period will vary slightly from place to place and will also vary with height above sea level.

This approximation is accurate only for small angles because of the expression for [[angular acceleration]] {{math|''α''}} being proportional to the sine of the displacement angle:

<math display="block">-mgl \sin\theta =I\alpha,</math>

where {{math|''I''}} is the [[moment of inertia]]. When {{math|''θ''}} is small, {{math|sin ''θ'' ≈ ''θ''}} and therefore the expression becomes

<math display="block">-mgl \theta =I\alpha</math>

which makes angular acceleration directly proportional and opposite to {{math|''θ''}}, satisfying the definition of simple harmonic motion (that net force is directly proportional to the displacement from the mean position and is directed towards the mean position).

===Scotch yoke===
{{Main|Scotch yoke}}
A Scotch yoke mechanism can be used to convert between rotational motion and linear reciprocating motion. The linear motion can take various forms depending on the shape of the slot, but the basic yoke with a constant rotation speed produces a linear motion that is simple harmonic in form.
[[File:Scotch yoke animation.gif|thumb|200px|Scotch yoke animation]]

==See also==
{{Columns-list|colwidth=18em|
* [[Circle group]]
* [[Complex harmonic motion]]
* [[Damping ratio]]
* [[Harmonic oscillator]]
* [[Isochronous timing]]
* [[Lorentz oscillator model#Dielectric function|Lorentz oscillator model]]
* [[Newtonian mechanics]]
* [[Pendulum (mathematics)|Pendulum]]
* [[Rayleigh–Lorentz pendulum]]
* [[Small-angle approximation]]
* [[String vibration]]
* [[Uniform circular motion]]
}}

==Notes==
{{Cnote2 Begin|liststyle=upper-alpha|colwidth=40em}}
{{Cnote2|A|The choice of using a cosine in this equation is a convention. Other valid formulations are:
<math display="block"> x(t) = A\sin\left(\omega t +\varphi'\right),</math>
where
<math display="block"> \tan \varphi' = \frac{c_1}{c_2},</math>
since {{math|cos ''θ'' {{=}} sin({{sfrac|π|2}} − ''θ'')}}.}}
{{Cnote2|B|The maximum displacement (that is, the amplitude), {{math|''x''<sub>max</sub>}}, occurs when {{math|cos(''ωt'' ± ''φ'') {{=}} 1}}, and thus when {{math|''x''<sub>max</sub> {{=}} ''A''}}.}}
{{Cnote2 End}}<br>

==References==
{{Reflist}}
*{{cite book |last1=Fowles |first1=Grant R. |last2=Cassiday |first2=George L. |date=2005 |title=Analytical Mechanics |edition=7th |publisher=Thomson Brooks/Cole |isbn=0-534-49492-7}}
*{{cite book |last=Taylor |first=John R. |date=2005 |title=Classical Mechanics |publisher=University Science Books |isbn=1-891389-22-X}}
*{{cite book |last1=Thornton |first1=Stephen T. |last2=Marion |first2=Jerry B. |date=2003 |title=Classical Dynamics of Particles and Systems |edition=5th |publisher=Brooks Cole |isbn=0-534-40896-6}}
*{{cite book |last=Walker |first=Jearl |date=2011 |title=Principles of Physics |edition=9th |location=Hoboken, New Jersey |publisher=Wiley |isbn=978-0-470-56158-4}}

==External links==
{{Commons category|Simple harmonic motion}}
*[http://hyperphysics.phy-astr.gsu.edu/hbase/shm.html Simple Harmonic Motion] from [[HyperPhysics]]
*[http://www.phy.hk/wiki/englishhtm/SpringSHM.htm Java simulation of spring-mass oscillator]
*[https://www.geogebra.org/m/jwq5gucu Geogebra applet for spring-mass, with 3 attached PDFs on SHM, driven/damped oscillators, spring-mass with friction]

[[Category:Classical mechanics]]
[[Category:Motion (physics)]]
[[Category:Pendulums]]