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{{Short description|Concept in classical mechanics}}
{{Classical mechanics|rotational}}▼
A '''rotating frame of reference''' is a special case of a [[non-inertial reference frame]] that is [[rotation|rotating]] relative to an [[inertial reference frame]]. An everyday example of a rotating reference frame is the surface of the [[Earth]]. (This article considers only frames rotating about a fixed axis. For more general rotations, see [[Euler angles#Relationship with physical motions|Euler angles]].)▼
[[File:Corioliskraftanimation.gif|frame|right|In the inertial frame of reference (upper part of the picture), the black ball moves in a straight line. However, the observer (red dot) who is standing in the rotating/non-inertial frame of reference (lower part of the picture) sees the object as following a curved path due to the Coriolis and centrifugal forces present in this frame.]]
▲{{Classical mechanics|rotational}}
▲A '''rotating frame of reference''' is a special case of a [[non-inertial reference frame]] that is [[rotation|rotating]] relative to an [[inertial reference frame]]. An everyday example of a rotating reference frame is the surface of the [[Earth]]. (This article considers only frames rotating about a fixed axis. For more general rotations, see [[Euler angles#
==Fictitious forces==
{{main|Fictitious
All [[non-inertial reference frame]]s exhibit [[fictitious force]]s; rotating reference frames are characterized by three:<ref name=Arnold>{{cite book |title=Mathematical Methods of Classical Mechanics |page=130 |author=Vladimir Igorević Arnolʹd |edition=2nd |isbn=978-0-387-96890-2 |date=1989 |url=https://books.google.com/books?id=Pd8-s6rOt_cC&q=%22additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally%22&pg=PT149 |publisher=Springer}}</ref>
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* the [[Euler force]].
Scientists in a rotating box can measure the [[rotation speed]] and
===Centrifugal force===▼
In [[classical mechanics]],
--><ref name=Marsden>{{cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden |author2=Tudor S. Ratiu |isbn=0-387-98643-X |date=1999 |publisher=Springer |page=251 |url=https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251}}</ref><ref name=Taylor_A>{{cite book |title=Classical Mechanics |url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1 |page=343 |author=John Robert Taylor |isbn=1-891389-22-X |publisher=University Science Books |date=2005}}</ref><ref name=Marion>{{cite book |title=Classical Dynamics of Particles and Systems |author=Stephen T. Thornton |author2=Jerry B. Marion |name-list-style=amp |chapter=Chapter 10 |date=2004 |isbn=0-534-40896-6 |publisher=Brook/Cole |___location=Belmont CA |edition=5th |oclc=52806908}}</ref><ref>{{cite web|url=http://dlmcn.com/circle.html|title=Centrifugal and Coriolis Effects|author=David McNaughton|access-date=2008-05-18}}</ref><ref>{{cite web|title=Frames of reference: The centrifugal force|url=http://www.phy6.org/stargaze/Lframes2.htm|author=David P. Stern|access-date=2008-10-26}}</ref>▼
The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist [[Gaspard-Gustave Coriolis]] in connection with [[hydrodynamics]], and also in the [[Theory of tides|tidal equations]] of [[Pierre-Simon Laplace]] in 1778. Early in the 20th century, the term Coriolis force began to be used in connection with [[meteorology]].▼
Perhaps the most commonly encountered rotating reference frame is the [[Earth]]. Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the [[northern hemisphere]], and to the left in the [[southern hemisphere|southern]]. Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the [[equator]], and to the left of this direction south of the equator. This effect is responsible for the rotation of large [[Cyclone#Structure|cyclones]] <!--Don't add tornadoes here; the Coriolis effect is not directly responsible for tornadoes-->(see [[Coriolis effect#Meteorology|Coriolis effects in meteorology]]).▼
===Euler force===▼
{{main|Euler force}}
In [[classical mechanics]], the
The
|isbn=1-56347-342-9 |date=1999 |publisher=[[American Institute of Aeronautics and Astronautics]] |___location=Reston, VA }}</ref><ref>{{cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden |author2=Tudor S. Ratiu |isbn=0-387-98643-X |date=1999 |publisher=Springer |page=251 |url=https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PP1}}</ref>▼
==Relating rotating frames to stationary frames==
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=== Time derivatives in the two frames ===
Introduce
<math display=block>\frac{\mathrm{d}}{\mathrm{d}t}\hat{\boldsymbol{u}} = \boldsymbol{\Omega} \times \boldsymbol{\hat{u}} \ .</math>
So if <math>R(t)</math> denotes the transformation taking basis vectors of the inertial- to the rotating frame, with matrix columns equal to the basis vectors of the rotating frame, then the cross product multiplication by the rotation vector is given by <math>\boldsymbol{\Omega}\times = R'(t)\cdot R(t)^T</math>.
If <math>\boldsymbol{f}</math> is a vector function that is written as<ref group=note>So <math>f_1, f_2, f_3</math> are <math>\boldsymbol{f}</math>'s coordinates with respect to the rotating basis vector <math>\hat{\boldsymbol{\imath}},\ \hat{\boldsymbol{\jmath}},\ \hat{\boldsymbol{k}}</math> (<math>\boldsymbol{f}</math>'s coordinates with respect to the inertial frame are not used). Consequently, at any given instant, the rate of change of <math>\boldsymbol{f}</math> with respect to these rotating coordinates is <math>\frac{\mathrm{d}f_1}{\mathrm{d}t}\hat{\boldsymbol{\imath}} + \frac{\mathrm{d}f_2}{\mathrm{d}t}\hat{\boldsymbol{\jmath}} + \frac{\mathrm{d}f_3}{\mathrm{d}t}\hat{\boldsymbol{k}}.</math> So for example, if <math>f_1 \equiv 1</math> and <math>f_2 = f_3 \equiv 0</math> are constants, then <math>\boldsymbol{f} \equiv \hat{\boldsymbol{\imath}}</math> is just one of the rotating basis vectors and (as expected) its time rate of change with respect to these rotating coordinates is identically <math>\boldsymbol{0}</math> (so the formula for <math>\frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{f}</math> given below implies that the derivative at time <math>t</math> of this rotating basis vector <math>\boldsymbol{f} \equiv \hat{\boldsymbol{\imath}}</math> is <math>\frac{\mathrm{d}}{\mathrm{d}t} \boldsymbol{i} = \boldsymbol{\Omega}(t) \times \boldsymbol{i}(t)</math>); however, its rate of change with respect to the non-rotating inertial frame will not be constantly <math>\boldsymbol{0}</math> except (of course) in the case where <math>\hat{\boldsymbol{\imath}}</math> is not moving in the inertial frame (this happens, for instance, when the axis of rotation is fixed as the <math>z</math>-axis (assuming standard coordinates) in the inertial frame and also <math>\hat{\boldsymbol{\imath}} \equiv (0, 0, 1)</math> or <math>\hat{\boldsymbol{\imath}} \equiv (0, 0, -1)</math>).</ref>
<math display=block>\boldsymbol{f}(t)=f_1(t) \hat{\boldsymbol{\imath}}+f_2(t) \hat{\boldsymbol{\jmath}}+f_3(t) \hat{\boldsymbol{k}}\ ,</math>
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:<math>\mathbf{v} \ \stackrel{\mathrm{def}}{=}\ \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \ .</math>
The time derivative of a position <math>\boldsymbol{r}(t)</math> in a rotating reference frame has two components, one from the explicit time dependence due to motion of the
:<math>
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\left[\left( \frac{\mathrm{d}\mathbf{r}}{\mathrm{d}t} \right)_{\mathrm{r}} + \boldsymbol\Omega \times \mathbf{r} \right] \ ,
</math>
where subscript <math>\mathrm{i}</math> means the inertial frame of reference, <math>\mathrm{r}</math> the rotating frame of reference, and where the expression, again, <math>\boldsymbol\Omega \times</math> in the bracketed expression on the left is to be interpreted as an [[Operator (mathematics)|operator]] working onto the bracketed expression on the right.
As <math>\boldsymbol\Omega\times\boldsymbol\Omega=\boldsymbol 0</math>, the first time derivatives of <math>\boldsymbol\Omega</math> inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide.
Carrying out the [[Derivative|differentiation]]s and re-arranging some terms yields the acceleration ''relative to the rotating'' reference frame, <math>\mathbf{a}_{\mathrm{r}}</math>
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</math>
where <math>\mathbf{a}_{\mathrm{r}} \ \stackrel{\mathrm{def}}{=}\ \left( \
=== Newton's second law in the two frames ===
When the expression for acceleration is multiplied by the mass of the particle, the three extra terms on the right-hand side result in [[fictitious force]]s in the rotating reference frame, that is, apparent forces that result from being in a [[non-inertial reference frame]], rather than from any physical interaction between bodies.
Using [[Newton's laws of motion|Newton's second law of motion]] <math>\mathbf{F}=m\mathbf{a},</math> we obtain:<ref name=Arnold/><ref name=Lanczos/><ref name=Taylor/><ref name=Landau>{{cite book |title=Mechanics |author=LD Landau |author2=LM Lifshitz |name-list-style=amp |page= 128 |url=https://books.google.com/books?id=e-xASAehg1sC&pg=PA40 |edition=Third |date=1976 |publisher=Butterworth-Heinemann |isbn=978-0-7506-2896-9}}</ref><ref name=Hand/>
* the [[Coriolis force]] <math display="block">
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{{Quotation|Obviously, a rotating frame of reference is a case of a non-inertial frame. Thus the particle in addition to the real force is acted upon by a fictitious force...The particle will move according to Newton's second law of motion if the total force acting on it is taken as the sum of the real and fictitious forces.|HS Hans & SP Pui: ''Mechanics''; p. 341}}
{{Quotation|This equation has exactly the form of Newton's second law, ''except'' that in addition to '''F''', the sum of all forces identified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton's second law in the noninertial frame ''provided'' we agree that in the noninertial frame we must add an extra force-like term, often called the '''inertial force'''. |John R. Taylor: ''Classical Mechanics''; p. 328}}
▲==Centrifugal force==
▲In [[classical mechanics]], '''centrifugal force''' is an outward force associated with [[rotation]]. Centrifugal force is one of several so-called [[pseudo-force]]s (also known as [[inertial force]]s), so named because, unlike [[Fundamental interaction|real forces]], they do not originate in interactions with other bodies situated in the environment of the particle upon which they act. Instead, centrifugal force originates in the rotation of the frame of reference within which observations are made.<!--
▲--><ref name=Marsden>{{cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden |author2=Tudor S. Ratiu |isbn=0-387-98643-X |date=1999 |publisher=Springer |page=251 |url=https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251}}</ref><ref name=Taylor_A>{{cite book |title=Classical Mechanics |url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1 |page=343 |author=John Robert Taylor |isbn=1-891389-22-X |publisher=University Science Books |date=2005}}</ref><ref name=Marion>{{cite book |title=Classical Dynamics of Particles and Systems |author=Stephen T. Thornton |author2=Jerry B. Marion |name-list-style=amp |chapter=Chapter 10 |date=2004 |isbn=0-534-40896-6 |publisher=Brook/Cole |___location=Belmont CA |edition=5th |oclc=52806908}}</ref><ref>{{cite web|url=http://dlmcn.com/circle.html|title=Centrifugal and Coriolis Effects|author=David McNaughton|access-date=2008-05-18}}</ref><ref>{{cite web|title=Frames of reference: The centrifugal force|url=http://www.phy6.org/stargaze/Lframes2.htm|author=David P. Stern|access-date=2008-10-26}}</ref>
▲==Coriolis effect==
▲{{main|Coriolis effect}}
▲The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist [[Gaspard-Gustave Coriolis]] in connection with [[hydrodynamics]], and also in the [[Theory of tides|tidal equations]] of [[Pierre-Simon Laplace]] in 1778. Early in the 20th century, the term Coriolis force began to be used in connection with [[meteorology]].
▲Perhaps the most commonly encountered rotating reference frame is the [[Earth]]. Moving objects on the surface of the Earth experience a Coriolis force, and appear to veer to the right in the [[northern hemisphere]], and to the left in the [[southern hemisphere|southern]]. Movements of air in the atmosphere and water in the ocean are notable examples of this behavior: rather than flowing directly from areas of high pressure to low pressure, as they would on a non-rotating planet, winds and currents tend to flow to the right of this direction north of the [[equator]], and to the left of this direction south of the equator. This effect is responsible for the rotation of large [[Cyclone#Structure|cyclones]] <!--Don't add tornadoes here; the Coriolis effect is not directly responsible for tornadoes-->(see [[Coriolis effect#Meteorology|Coriolis effects in meteorology]]).
▲==Euler force==
▲{{main|Euler force}}
▲In [[classical mechanics]], the '''Euler acceleration''' (named for [[Leonhard Euler]]), also known as '''azimuthal acceleration'''<ref name=Morin>{{cite book |author=David Morin |url=https://archive.org/details/introductiontocl00mori |url-access=registration |quote=acceleration azimuthal Morin. |title=Introduction to classical mechanics: with problems and solutions |page= [https://archive.org/details/introductiontocl00mori/page/469 469] |isbn= 978-0-521-87622-3 |date=2008 |publisher=Cambridge University Press}}</ref> or '''transverse acceleration'''<ref name=Fowles>{{cite book |author=Grant R. Fowles|author2=George L. Cassiday|name-list-style=amp|title=Analytical Mechanics|edition=6th|page=178|date=1999|publisher=Harcourt College Publishers}}</ref> is an [[acceleration]] that appears when a non-uniformly rotating reference frame is used for analysis of motion and there is variation in the [[angular velocity]] of the [[frame of reference|reference frame]]'s axis. This article is restricted to a frame of reference that rotates about a fixed axis.
▲The '''Euler force''' is a [[fictitious force]] on a body that is related to the Euler acceleration by ''' ''F'' ''' = ''m'''a''''', where ''' ''a'' ''' is the Euler acceleration and ''m'' is the mass of the body.<ref name=Battin>{{cite book |title=An introduction to the mathematics and methods of astrodynamics |page=102 |author= Richard H Battin |url=https://books.google.com/books?id=OjH7aVhiGdcC&q=%22Euler+acceleration%22&pg=PA102
▲|isbn=1-56347-342-9 |date=1999 |publisher=[[American Institute of Aeronautics and Astronautics]] |___location=Reston, VA }}</ref><ref>{{cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden |author2=Tudor S. Ratiu |isbn=0-387-98643-X |date=1999 |publisher=Springer |page=251 |url=https://books.google.com/books?id=I2gH9ZIs-3AC&pg=PP1}}</ref>
==Use in magnetic resonance==
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* [[Absolute rotation]]
* [[Centrifugal force (rotating reference frame)]] Centrifugal force as seen from systems rotating about a fixed axis
* [[Coriolis force]] The effect of the Coriolis force on the Earth and other rotating systems
* [[Inertial frame of reference]]
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[[Category:Classical mechanics]]
[[Category:Astronomical coordinate systems]]
[[Category:Rotation]]
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