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| {{Classical mechanics|cTopic=Core topics}}
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| 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]].)
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| ==Fictitious forces==
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| {{main|Fictitious forces}}
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| All [[non-inertial reference frame]]s exhibit [[fictitious force]]s. Rotating reference frames are characterized by three fictitious forces<ref name=Arnold>{{cite book |title=Mathematical Methods of Classical Mechanics |page=130 |author=Vladimir Igorević Arnolʹd |edition=2nd Edition |isbn=978-0-387-96890-2 |year=1989 |url=http://books.google.com/books?id=Pd8-s6rOt_cC&pg=PT149&dq=%22additional+terms+called+inertial+forces.+This+allows+us+to+detect+experimentally%22#PPT150,M1 |publisher=Springer}}</ref>
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| * the [[centrifugal force (fictitious)|centrifugal force]]
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| * the [[Coriolis force]]
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| and, for non-uniformly rotating reference frames,
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| * the [[Euler force]].
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| Scientists living in a rotating box can measure the speed and direction of their rotation by measuring these [[fictitious force]]s. For example, [[Léon Foucault]] was able to show the [[Coriolis force]] that results from the Earth's rotation using the [[Foucault pendulum]]. If the Earth were to rotate many times faster, these fictitious forces could be felt by humans, as they are when on a spinning [[carousel]].
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| ==Relating rotating frames to stationary frames==
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| The following is a derivation of the formulas for accelerations as well as fictitious forces in a rotating frame. It begins with the relation between a particle's coordinates in a rotating frame and its coordinates in an inertial (stationary) frame. Then, by taking time derivatives, formulas are derived that relate the velocity of the particle as seen in the two frames, and the acceleration relative to each frame. Using these accelerations, the fictitious forces are identified by comparing Newton's second law as formulated in the two different frames.
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| === Relation between positions in the two frames ===
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| To derive these fictitious forces, it's helpful to be able to convert between the coordinates <math>\left( x',y',z' \right)</math> of the rotating reference frame and the coordinates <math>\left( x, y, z \right)</math> of an [[inertial reference frame]] with the same origin. If the rotation is about the <math>z</math> axis with an [[angular velocity]] <math>\Omega</math> and the two reference frames coincide at time <math>t=0</math>, the transformation from rotating coordinates to inertial coordinates can be written
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| : <math>x = x'\cos\left(\Omega t\right) - y'\sin\left(\Omega t\right)</math>
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| : <math>y = x'\sin\left(\Omega t\right) + y'\cos\left(\Omega t\right)</math>
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| whereas the reverse transformation is
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| : <math>x' = x\cos\left(-\Omega t\right) - y\sin\left( -\Omega t \right)</math>
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| : <math>y' = x\sin\left( -\Omega t \right) + y\cos\left( -\Omega t \right)</math>
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| This result can be obtained from a [[rotation matrix]].
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| Introduce the unit vectors <math>\hat{\boldsymbol{\imath}},\ \hat{\boldsymbol{\jmath}},\ \hat{\boldsymbol{k}}</math> representing standard unit basis vectors in the rotating frame. The time-derivatives of these unit vectors are found next. Suppose the frames are aligned at ''t = ''0 and the ''z''-axis is the axis of rotation. Then for a counterclockwise rotation through angle ''Ωt'':
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| : <math>\hat{\boldsymbol{\imath}}(t) = (\cos\Omega t,\ \sin \Omega t ) </math>
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| where the (''x'', ''y'') components are expressed in the stationary frame. Likewise, | |
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| : <math>\hat{\boldsymbol{\jmath}}(t) = (-\sin \Omega t,\ \cos \Omega t ) \ .</math>
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| Thus the time derivative of these vectors, which rotate without changing magnitude, is
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| : <math>\frac{d}{dt}\hat{\boldsymbol{\imath}}(t) = \Omega (-\sin \Omega t, \ \cos \Omega t)= \Omega \hat{\boldsymbol{\jmath}} \ ; </math>
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| : <math>\frac{d}{dt}\hat{\boldsymbol{\jmath}}(t) = \Omega (-\cos \Omega t, \ -\sin \Omega t)= - \Omega \hat{\boldsymbol{\imath}} \ . </math>
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| This result is the same as found using a [[vector cross product]] with the rotation vector <math>\boldsymbol{\Omega}</math> pointed along the z-axis of rotation <math>\boldsymbol{\Omega}=(0,\ 0,\ \Omega)</math>, namely,
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| : <math>\frac{d}{dt}\hat{\boldsymbol{u}} = \boldsymbol{\Omega \times}\hat {\boldsymbol{ u}} \ , </math>
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| where <math>\hat {\boldsymbol{ u}}</math> is either <math>\hat{\boldsymbol{\imath}}</math> or <math> \hat{\boldsymbol{\jmath}}</math>.
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| === Time derivatives in the two frames ===
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| Introduce the unit vectors <math>\hat{\boldsymbol{\imath}},\ \hat{\boldsymbol{\jmath}},\ \hat{\boldsymbol{k}}</math> representing standard unit basis vectors in the rotating frame. As they rotate they will remain normalized. If we let them rotate at the speed of <math> \Omega </math> about an axis <math>\boldsymbol {\Omega}</math> then each unit vector <math>\hat{\boldsymbol{u}}</math> of the rotating coordinate system abides by the following equation:
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| :<math> \frac{d}{dt}\hat{\boldsymbol{u}}=\boldsymbol{\Omega \times \hat{u}} \ .</math>
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| Then if we have a vector function <math>\boldsymbol{f}</math>,
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| :<math> \boldsymbol{f}(t)=f_x(t) \hat{\boldsymbol{\imath}}+f_y(t) \hat{\boldsymbol{\jmath}}+f_z(t) \hat{\boldsymbol{k}}\ , </math>
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| and we want to examine its first dervative we have (using the [[product rule]] of differentiation):<ref name=Lanczos>{{cite book |url=http://books.google.com/books?num=10&btnG=Google+Search |title=The Variational Principles of Mechanics |author=Cornelius Lanczos |year=1986 |isbn=0-486-65067-7 |publisher=Dover Publications |edition=Reprint of Fourth Edition of 1970 |nopp=true |pages=Chapter 4, §5}}</ref><ref name=Taylor>{{cite book |title=Classical Mechanics |author=John R Taylor |page= 342 |publisher=University Science Books |isbn=1-891389-22-X |year=2005 |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn=1-891389-22-X#PPA342,M1}}</ref>
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| :<math>\frac{d}{dt}\boldsymbol{f}=\frac{df_x}{dt}\hat{\boldsymbol{\imath}}+\frac{d\hat{\boldsymbol{\imath}}}{dt}f_x+\frac{df_y}{dt}\hat{\boldsymbol{\jmath}}+\frac{d\hat{\boldsymbol{\jmath}}}{dt}f_y+\frac{df_z}{dt}\hat{\boldsymbol{k}}+\frac{d\hat{\boldsymbol{k}}}{dt}f_z</math>
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| ::<math>=\frac{df_x}{dt}\hat{\boldsymbol{\imath}}+\frac{df_y}{dt}\hat{\boldsymbol{\jmath}}+\frac{df_z}{dt}\hat{\boldsymbol{k}}+[\boldsymbol{\Omega \times} (f_x \hat{\boldsymbol{\imath}} + f_y \hat{\boldsymbol{\jmath}}+f_z \hat{\boldsymbol{k}})]</math>
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| ::<math>= \left( \frac{d\boldsymbol{f}}{dt}\right)_r+\boldsymbol{\Omega \times f}(t)\ ,</math>
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| where <math>\left( \frac{d\boldsymbol{f}}{dt}\right)_r</math> is the rate of change of <math>\boldsymbol{f}</math> as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:
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| ::<math>\frac{d}{dt}\boldsymbol{f} =\left[ \left(\frac{d}{dt}\right)_r + \boldsymbol{\Omega \times} \right] \boldsymbol{f} \ . </math>
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| This result is also known as the Transport Theorem in analytical dynamics, and is also sometimes referred to as the Basic Kinematic Equation.<ref>{{cite web|last=Corless|first=Martin|title=Kinematics|url=https://engineering.purdue.edu/AAE/Academics/Courses/aae203/2003/fall/aae203F03supp.pdf|work=Aeromechanics I Course Notes|publisher=Purdue University|accessdate=18 July 2011|page=213}}</ref>
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| === Relation between velocities in the two frames ===
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| A velocity of an object is the time-derivative of the object's position, or
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| :<math>\mathbf{v} \ \stackrel{\mathrm{def}}{=}\ \frac{d\mathbf{r}}{dt}</math>
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| 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 particle itself, and another from the frame's own rotation. Applying the result of the previous subsection to the displacement <math>\boldsymbol{r}(t)</math>, the [[velocity|velocities]] in the two reference frames are related by the equation
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| :<math> | |
| \mathbf{v_i} \ \stackrel{\mathrm{def}}{=}\ \frac{d\mathbf{r}}{dt} =
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| \left( \frac{d\mathbf{r}}{dt} \right)_{\mathrm{r}} +
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| \boldsymbol\Omega \times \mathbf{r} =
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| \mathbf{v}_{\mathrm{r}} + \boldsymbol\Omega \times \mathbf{r} \ ,
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| </math>
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| where subscript ''i'' means the inertial frame of reference, and ''r'' means the rotating frame of reference.
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| === Relation between accelerations in the two frames ===
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| Acceleration is the second time derivative of position, or the first time derivative of velocity
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| :<math>
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| \mathbf{a}_{\mathrm{i}} \ \stackrel{\mathrm{def}}{=}\
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| \left( \frac{d^{2}\mathbf{r}}{dt^{2}}\right)_{\mathrm{i}} =
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| \left( \frac{d\mathbf{v}}{dt} \right)_{\mathrm{i}} =
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| \left[ \left( \frac{d}{dt} \right)_{\mathrm{r}} +
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| \boldsymbol\Omega \times
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| \right]
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| \left[
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| \left( \frac{d\mathbf{r}}{dt} \right)_{\mathrm{r}} +
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| \boldsymbol\Omega \times \mathbf{r}
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| \right] \ ,
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| </math>
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| where subscript ''i'' means the inertial frame of reference.
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| Carrying out the [[derivative|differentiation]]s and re-arranging some terms yields the acceleration in the ''rotating'' reference frame
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| :<math>
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| \mathbf{a}_{\mathrm{r}} =
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| \mathbf{a}_{\mathrm{i}} -
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| 2 \boldsymbol\Omega \times \mathbf{v}_{\mathrm{r}} -
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| \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{r}) -
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| \frac{d\boldsymbol\Omega}{dt} \times \mathbf{r}
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| </math>
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| where <math>\mathbf{a}_{\mathrm{r}} \ \stackrel{\mathrm{def}}{=}\ \left( \frac{d^{2}\mathbf{r}}{dt^{2}} \right)_{\mathrm{r}}</math> is the apparent acceleration in the rotating reference frame, the term <math>-\boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{r})</math> represents [[centrifugal acceleration]], and the term <math>-2 \boldsymbol\Omega \times \mathbf{v}_{\mathrm{r}}</math> is the [[coriolis effect]].
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| === Newton's second law in the two frames ===
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| 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.
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| 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 and LM Lifshitz |page= 128 |url=http://books.google.com/books?id=e-xASAehg1sC&pg=PA40&dq=isbn=9780750628969#PPA128,M1 |edition=Third Edition |year=1976 |isbn=978-0-7506-2896-9}}</ref><ref name=Hand/>
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| * the [[Coriolis force]]
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| :<math>
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| \mathbf{F}_{\mathrm{Coriolis}} =
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| -2m \boldsymbol\Omega \times \mathbf{v}_{\mathrm{r}}
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| </math>
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| * the [[centrifugal force (fictitious)|centrifugal force]]
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| :<math>
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| \mathbf{F}_{\mathrm{centrifugal}} =
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| -m\boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{r})
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| </math>
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| * and the [[Euler force]]
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| :<math>
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| \mathbf{F}_{\mathrm{Euler}} =
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| -m\frac{d\boldsymbol\Omega}{dt} \times \mathbf{r}
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| </math>
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| where <math>m</math> is the mass of the object being acted upon by these [[fictitious force]]s. Notice that all three forces vanish when the frame is not rotating, that is, when <math>\boldsymbol{\Omega} = 0 \ . </math>
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| For completeness, the inertial acceleration <math>\mathbf{a}_{\mathrm{i}}</math> due to impressed external forces <math>\mathbf{F}_{\mathrm{imp}}</math> can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as [[electromagnetism|electromagnetic forces]]) using [[Newton's laws of motion|Newton's second law]] in the inertial frame:
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| :<math>
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| \mathbf{F}_{\mathrm{imp}} = m \mathbf{a}_{\mathrm{i}}
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| </math>
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| Newton's law in the rotating frame then becomes
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| ::<math>\mathbf{F_r} = \mathbf{F}_{\mathrm{imp}} +\mathbf{F}_{\mathrm{centrifugal}} +\mathbf{F}_{\mathrm{Coriolis}}+\mathbf{F}_{\mathrm{Euler}} = m\mathbf{a_r} \ . </math>
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| In other words, to handle the laws of motion in a rotating reference frame:<ref name=Hand>{{cite book |title=Analytical Mechanics |author =Louis N. Hand, Janet D. Finch |page=267 |url=http://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA267&vq=fictitious+forces&dq=Hand+inauthor:Finch | |
| |isbn=0-521-57572-9 |publisher=Cambridge University Press |year=1998 }}</ref><ref name=Pui>{{cite book |title=Mechanics |author=HS Hans & SP Pui |page=341 |url=http://books.google.com/books?id=mgVW00YV3zAC&pg=PA341&dq=inertial+force+%22rotating+frame%22 |isbn=0-07-047360-9 |publisher=Tata McGraw-Hill |year=2003 }}</ref><ref name=Taylor2>{{cite book |title=Classical Mechanics |author=John R Taylor |page= 328 |publisher=University Science Books |isbn=1-891389-22-X |year=2005 |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn=1-891389-22-X#PPA328,M1}}</ref>
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| {{Quotation|Treat the fictitious forces like real forces, and pretend you are in an inertial frame.|Louis N. Hand, Janet D. Finch ''Analytical Mechanics'', p. 267}}
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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}}
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| {{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}}
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| ==Centrifugal force==
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| {{main|Centrifugal force (rotating reference frame)}}
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| 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.<!--
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| --><ref>{{cite book |title=Physics |author=Robert Resnick & David Halliday |page=121 |year=1966 |url=http://books.google.com/books?q=%22cannot+associate+them+with+any+particular+body+in+the+environment+of+the+particle%22+inauthor%3ADavid+inauthor%3AHalliday&btnG=Search+Books |publisher=Wiley |isbn=0-471-34524-5 }}</ref><!--
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| --><ref name=Marsden>{{cite book |title=Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems |author=Jerrold E. Marsden, Tudor S. Ratiu |isbn=0-387-98643-X |year=1999 |publisher=Springer |page=251 |url=http://books.google.com/books?id=I2gH9ZIs-3AC&pg=PA251&vq=Euler+force&dq=isbn=0-387-98643-X}}</ref><ref name=Taylor_A>{{cite book |title=Classical Mechanics |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn=1-891389-22-X#PPA343,M1 |page=343 |author=John Robert Taylor |isbn=1-891389-22-X |publisher=University Science Books |year=2005}}</ref><ref name=Marion>{{cite book |title=Classical Dynamics of Particles and Systems |author=Stephen T. Thornton &
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| Jerry B. Marion |page=Chapter 10 |year=2004 |isbn=0-534-40896-6 |publisher=Brook/Cole |location=Belmont CA |edition=5th |url=http://worldcat.org/oclc/52806908&referer=brief_results |nopp=true}}</ref><ref>{{cite web|url=http://dlmcn.com/circle.html|title=Centrifugal and Coriolis Effects|author=David McNaughton|accessdate=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|accessdate=2008-10-26}}</ref>
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| ==Coriolis effect==
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| {{main|Coriolis effect}}
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| [[Image:Corioliskraftanimation.gif|frame|right|Figure 1: In the inertial frame of reference (upper part of the picture), the black object moves in a straight line. However, the observer (red dot) who is standing in the rotating frame of reference (lower part of the picture) sees the object as following a curved path.]]
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| 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]].
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| 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]]).
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| ==Euler force==
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| {{main|Euler force}}
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| In [[classical mechanics]], the '''Euler acceleration''' (named for [[Leonhard Euler]]), also known as '''azimuthal acceleration'''<ref name=Morin>{{cite book |author=David Morin |url=http://books.google.com/books?id=Ni6CD7K2X4MC&pg=PA469&dq=acceleration+azimuthal+inauthor:Morin |title=Introduction to classical mechanics: with problems and solutions |page= 469 |isbn= 0-521-87622-2 |year=2008 |publisher=Cambridge University Press}}</ref> or '''transverse acceleration'''<ref name=Fowles>{{cite book |author=Grant R. Fowles and George L. Cassiday|title=Analytical Mechanics, 6th ed.|page=178|year=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.
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| 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=http://books.google.com/books?id=OjH7aVhiGdcC&pg=PA102&vq=Euler&dq=%22Euler+acceleration%22 | |
| |isbn=1-56347-342-9 |year=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, Tudor S. Ratiu |isbn=0-387-98643-X |year=1999 |publisher=Springer |page=251 |url=http://books.google.com/books?id=I2gH9ZIs-3AC&pg=PP1&dq=isbn:038798643X#PPA251,M1}}</ref>
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| ==Use in magnetic resonance==
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| It is convenient to consider [[magnetic resonance]] in a frame that rotates at the [[Larmor frequency]] of the spins. This is illustrated in the animation below. The [[rotating wave approximation]] may also be used.
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| [[File:Animated Rotating Frame.gif|Animation showing the rotating frame. The red arrow is a spin in the [[Bloch sphere]] which precesses in the laboratory frame due to a static magnetic field. In the rotating frame the spin remains still until a resonantly oscillating magnetic field drives magnetic resonance.]]
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| ==References==
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| {{Reflist|2}}
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| ==See also==
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| * [[Absolute rotation]]
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| * [[Centrifugal force (rotating reference frame)]] Centrifugal force as seen from systems rotating about a fixed axis
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| * [[Mechanics of planar particle motion]] Fictitious forces exhibited by a particle in planar motion as seen by the particle itself and by observers in a co-rotating frame of reference
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| * [[Coriolis force]] The effect of the Coriolis force on the Earth and other rotating systems
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| * [[Inertial frame of reference]]
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| * [[Non-inertial frame]]
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| * [[Fictitious force]] A more general treatment of the subject of this article
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| ==External links==
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| * [http://www.youtube.com/watch?v=49JwbrXcPjc Animation clip] showing scenes as viewed from both an inertial frame and a rotating frame of reference, visualizing the Coriolis and centrifugal forces.
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| [[Category:Frames of reference]]
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| [[Category:Classical mechanics]]
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| [[Category:Celestial coordinate system]]
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| [[Category:Surveying]]
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| [[Category:Rotation]]
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