- Rotating reference frame
A

**rotating frame of reference**is a special case of anon-inertial reference frame that is rotating relative to aninertial reference frame . An everyday example of a rotating reference frame is the surface of theEarth . (This article considers only frames rotating about a fixed axis. For more general rotations, see Euler angles.)**Fictitious forces**All

non-inertial reference frame s exhibitfictitious force s. Rotating reference frames are characterized by three fictitious forcescite book |title=Mathematical Methods of Classical Mechanics |page=p. 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&lr=&as_brr=0&sig=ACfU3U1qRbkvn6x7FcBsHO8Bp4Ty95XbZw#PPT150,M1 |publisher=Springer]* the centrifugal force

* theCoriolis force and, for non-uniformly rotating reference frames,

* theEuler force .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 theCoriolis force that results from the Earth's rotation using theFoucault pendulum . If the Earth were to rotate a thousand-fold faster (making each day only ~86 seconds long), these fictitious forces could be felt easily by humans, as they are on a spinningcarousel .**Relating rotating frames to stationary frames**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 coordinates of the position of a particle in a rotating frame and the 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 a comparison of Newton's second law as formulated in the frames identifies the fictitious forces.

**Relation between positions in the two frames**To derive these fictitious forces, it's helpful to be able to convert between the coordinates $left(\; x^\{prime\},y^\{prime\},z^\{prime\}\; ight)$ of the rotating reference frame and the coordinates $left(\; x,\; y,\; z\; ight)$ of an

inertial reference frame with the same origin. If the rotation is about the $z$ axis with anangular velocity $Omega$ and the two reference frames coincide at time $t=0$, the transformation from rotating coordinates to inertial coordinates can be written:$x\; =\; x^\{prime\}\; cosOmega\; t\; -\; y^\{prime\}\; sinOmega\; t$:$y\; =\; x^\{prime\}\; sinOmega\; t\; +\; y^\{prime\}\; cosOmega\; t$

whereas the reverse transformation is

:$x^\{prime\}\; =\; x\; cosleft(-Omega\; t\; ight)\; -\; y\; sinleft(\; -Omega\; t\; ight)$:$y^\{prime\}\; =\; x\; sinleft(\; -Omega\; t\; ight)\; +\; y\; cosleft(\; -Omega\; t\; ight)$

This result can be obtained from a

rotation matrix .Introduce the unit vectors $hat\{\backslash boldsymbol\{i,\; hat\{\backslash boldsymbol\{j,\; hat\{\backslash boldsymbol\{k$ 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"::$hat\{\backslash boldsymbol\{i(t)\; =\; (cosOmega\; t,\; sin\; Omega\; t\; )$where the ("x", "y") components are expressed in the stationary frame. Likewise,:$hat\{\backslash boldsymbol\{j(t)\; =\; (-sin\; Omega\; t,\; cos\; Omega\; t\; )\; .$Thus the time derivative of these vectors, which rotate without changing magnitude, is:$frac\{d\}\{dt\}hat\{\backslash boldsymbol\{i(t)\; =\; Omega\; (-sin\; Omega\; t,\; cos\; Omega\; t)=\; Omega\; hat\{\backslash boldsymbol\{j\; ;$:$frac\{d\}\{dt\}hat\{\backslash boldsymbol\{j(t)\; =\; Omega\; (-cos\; Omega\; t,\; -sin\; Omega\; t)=\; -\; Omega\; hat\{\backslash boldsymbol\{i\; .$This result is the same as found using a

vector cross product with the rotation vector $\backslash boldsymbol\{Omega\}$ pointed along the z-axis of rotation $\backslash boldsymbol\{Omega\}=(0,\; 0,\; Omega)$, namely,:$frac\{d\}\{dt\}hat\{\backslash boldsymbol\{u\; =\; \backslash boldsymbol\{Omega\; imes\}hat\; \{\backslash boldsymbol\{\; u\; ,$where $hat\; \{\backslash boldsymbol\{\; u$ is either $hat\{\backslash boldsymbol\{i$ or $hat\{\backslash boldsymbol\{j$.**Time derivatives in the two frames**Introduce the unit vectors $hat\{\backslash boldsymbol\{i,\; hat\{\backslash boldsymbol\{j,\; hat\{\backslash boldsymbol\{k$ 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 $Omega$ about an axis $oldsymbol\; \{Omega\}$ then each unit vector $hat\{\backslash boldsymbol\{u$ of the rotating coordinate system abides by the following equation::$frac\{d\}\{dt\}hat\{\backslash boldsymbol\{u=\backslash boldsymbol\{Omega\; imes\; hat\{u\; .$Then if we have a vector function $\backslash boldsymbol\{f\}$, :$\backslash boldsymbol\{f\}(t)=f\_x(t)\; hat\{\backslash boldsymbol\{i+f\_y(t)\; hat\{\backslash boldsymbol\{j+f\_z(t)\; hat\{\backslash boldsymbol\{k\; ,$ and we want to examine its first dervative we have (using the

chain rule of differentiation):cite book |url=http://books.google.com/books?as_q=&num=10&btnG=Google+Search&as_epq=The+author+likes+to+call+it+the+%22Euler+force%2C%22+in+view&as_oq=&as_eq=&as_brr=0&lr=&as_vt=&as_auth=&as_pub=&as_sub=&as_drrb=c&as_miny=&as_maxy=&as_isbn= |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 |page=Chapter 4, §5] cite book |title=Classical Mechanics |author=John R Taylor |page= p. 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:189138922X&sig=ACfU3U0kWmspY7W8eh9g1e6AqiMP83uSGw#PPA342,M1] :$frac\{d\}\{dt\}\backslash boldsymbol\{f\}=frac\{df\_x\}\{dt\}hat\{\backslash boldsymbol\{i+frac\{dhat\{\backslash boldsymbol\{i\}\{dt\}f\_x+frac\{df\_y\}\{dt\}hat\{\backslash boldsymbol\{j+frac\{dhat\{\backslash boldsymbol\{j\}\{dt\}f\_y+frac\{df\_z\}\{dt\}hat\{\backslash boldsymbol\{k+frac\{dhat\{\backslash boldsymbol\{k\}\{dt\}f\_z$::$=frac\{df\_x\}\{dt\}hat\{\backslash boldsymbol\{i+frac\{df\_y\}\{dt\}hat\{\backslash boldsymbol\{j+frac\{df\_z\}\{dt\}hat\{\backslash boldsymbol\{k+\; [\backslash boldsymbol\{Omega\; imes\}\; (f\_x\; hat\{\backslash boldsymbol\{i\; +\; f\_y\; hat\{\backslash boldsymbol\{j+f\_z\; hat\{\backslash boldsymbol\{k)]$::$=\; left(\; frac\{d\backslash boldsymbol\{f\{dt\}\; ight)\_r+\backslash boldsymbol\{Omega\; imes\; f\}(t)\; ,$where $left(\; frac\{d\backslash boldsymbol\{f\{dt\}\; ight)\_r$ is the rate of change of $\backslash boldsymbol\{f\}$ as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:::$frac\{d\}\{dt\}\backslash boldsymbol\{f\}\; =left\; [\; left(frac\{d\}\{dt\}\; ight)\_r\; +\; \backslash boldsymbol\{Omega\; imes\}\; ight]\; \backslash boldsymbol\{f\}\; .$**Relation between velocities in the two frames**A velocity of an object is the time-derivative of the object's position, or

:$mathbf\{v\}\; stackrel\{mathrm\{def\{=\}\; frac\{dmathbf\{r\{dt\}$

The time derivative of a position $\backslash boldsymbol\{r\}(t)$ 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 $\backslash boldsymbol\{r\}(t)$, the velocities in the two reference frames are related by the equation

:$mathbf\{v\_i\}\; stackrel\{mathrm\{def\{=\}\; frac\{dmathbf\{r\{dt\}\; =\; left(\; frac\{dmathbf\{r\{dt\}\; ight)\_\{mathrm\{r\; +\; oldsymbolOmega\; imes\; mathbf\{r\}\; =\; mathbf\{v\}\_\{mathrm\{r\; +\; oldsymbolOmega\; imes\; mathbf\{r\}\; ,$where subscript "i" means the inertial frame of reference, and "r" means the rotating frame of reference.

**Relation between accelerations in the two frames**Acceleration is the second time derivative of position, or the first time derivative of velocity

:$mathbf\{a\}\_\{mathrm\{i\; stackrel\{mathrm\{def\{=\}\; left(\; frac\{d^\{2\}mathbf\{r\{dt^\{2\; ight)\_\{mathrm\{i\; =\; left(\; frac\{dmathbf\{v\{dt\}\; ight)\_\{mathrm\{i\; =\; left\; [\; left(\; frac\{d\}\{dt\}\; ight)\_\{mathrm\{r\; +\; oldsymbolOmega\; imes\; ight]\; left\; [left(\; frac\{dmathbf\{r\{dt\}\; ight)\_\{mathrm\{r\; +\; oldsymbolOmega\; imes\; mathbf\{r\}\; ight]\; ,$where subscript "i" means the inertial frame of reference.Carrying out the

differentiation s and re-arranging some terms yields the acceleration in the "rotating" reference frame:$mathbf\{a\}\_\{mathrm\{r\; =\; mathbf\{a\}\_\{mathrm\{i\; -\; 2\; oldsymbolOmega\; imes\; mathbf\{v\}\_\{mathrm\{r\; -\; oldsymbolOmega\; imes\; (oldsymbolOmega\; imes\; mathbf\{r\})\; -\; frac\{doldsymbolOmega\}\{dt\}\; imes\; mathbf\{r\}$

where $mathbf\{a\}\_\{mathrm\{r\; stackrel\{mathrm\{def\{=\}\; left(\; frac\{d^\{2\}mathbf\{r\{dt^\{2\; ight)\_\{mathrm\{r$ is the apparent acceleration in the rotating reference frame.

**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 anon-inertial reference frame , rather than from any physical interaction between bodies.Using Newton's second law of motion

**"F**"$=$"m"**"a**", we obtain:cite book |title=Mechanics |author=LD Landau and LM Lifshitz |page= p. 128 |url=http://books.google.com/books?id=e-xASAehg1sC&pg=PA40&dq=isbn:9780750628969&sig=ACfU3U2LCcLQRZqYxDOXTg_9Ks_zp_qorg#PPA128,M1 |edition=Third Edition |year=1976 |isbn=978-0-7506-2896-9]* the

Coriolis force :$mathbf\{F\}\_\{mathrm\{Coriolis\; =\; -2m\; oldsymbolOmega\; imes\; mathbf\{v\}\_\{mathrm\{r$

* the centrifugal force

:$mathbf\{F\}\_\{mathrm\{centrifugal\; =\; -moldsymbolOmega\; imes\; (oldsymbolOmega\; imes\; mathbf\{r\})$

* and the

Euler force :$mathbf\{F\}\_\{mathrm\{Euler\; =\; -mfrac\{doldsymbolOmega\}\{dt\}\; imes\; mathbf\{r\}$

where $m$ 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 $\backslash boldsymbol\{Omega\}\; =\; 0\; .$For completeness, the inertial acceleration $mathbf\{a\}\_\{mathrm\{i$ due to impressed external forces $mathbf\{F\}\_\{mathrm\{imp$ can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as electromagnetic forces) using Newton's second law in the inertial frame:

:$mathbf\{F\}\_\{mathrm\{imp\; =\; m\; mathbf\{a\}\_\{mathrm\{i$Newton's law in the the rotating frame then becomes::$mathbf\{F\_r\}\; =\; mathbf\{F\}\_\{mathrm\{imp\; +mathbf\{F\}\_\{mathrm\{centrifugal\; +mathbf\{F\}\_\{mathrm\{Coriolis+mathbf\{F\}\_\{mathrm\{Euler\; =\; mmathbf\{a\_r\}\; .$In other words, to handle the laws of motion in a rotating reference frame:cite book |title=Analytical Mechanics |author =Louis N. Hand, Janet D. Finch |page=p. 267 |url=http://books.google.com/books?id=1J2hzvX2Xh8C&pg=PA267&vq=fictitious+forces&dq=Hand+inauthor:Finch&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U33emV_6eJZihu3M6IZKurSt85_eg

isbn=0521575729 |publisher=Cambridge University Press |year=1998 ] cite book |title=Mechanics |author=HS Hans & SP Pui |page=P. 341 |url=http://books.google.com/books?id=mgVW00YV3zAC&pg=PA341&dq=inertial+force+%22rotating+frame%22&lr=&as_brr=0&sig=ACfU3U1--cWJ02SuFZwp4Y6Uyoe4hbGFmQ |isbn=0070473609 |publisher=Tata McGraw-Hill |year=2003 ] cite book |title=Classical Mechanics |author=John R Taylor |page= p. 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:189138922X&sig=ACfU3U0kWmspY7W8eh9g1e6AqiMP83uSGw#PPA328,M1]**References and notes****ee also***

Centrifugal force (rotating reference frame) Centrifugal force as seen from systems rotating about a fixed axis

*Centrifugal force (planar motion) Centrifugal force exhibited by a particle in planar motion as seen by the particle itself and by observers in a co-rotating frame of reference

*Coriolis force The effect of the Coriolis force on the Earth and other rotating systems

*Inertial frame of reference

*Non-inertial frame

*Fictitious force A more general treatment of the subject of this article**External links*** [

*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.

*Wikimedia Foundation.
2010.*

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