Rotating reference frame


Rotating reference frame

A rotating frame of reference is a special case of a non-inertial reference frame that is 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

All non-inertial reference frames exhibit fictitious forces. 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
* the Coriolis forceand, for non-uniformly rotating reference frames,
* the Euler force.

Scientists living in a rotating box can measure the speed and direction of their rotation by measuring these fictitious forces. 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 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 spinning carousel.

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 an angular 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{oldsymbol{i, hat{oldsymbol{j, hat{oldsymbol{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{oldsymbol{i(t) = (cosOmega t, sin Omega t ) where the ("x", "y") components are expressed in the stationary frame. Likewise,:hat{oldsymbol{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{oldsymbol{i(t) = Omega (-sin Omega t, cos Omega t)= Omega hat{oldsymbol{j ; :frac{d}{dt}hat{oldsymbol{j(t) = Omega (-cos Omega t, -sin Omega t)= - Omega hat{oldsymbol{i . This result is the same as found using a vector cross product with the rotation vector oldsymbol{Omega} pointed along the z-axis of rotation oldsymbol{Omega}=(0, 0, Omega), namely,:frac{d}{dt}hat{oldsymbol{u = oldsymbol{Omega imes}hat {oldsymbol{ u , where hat {oldsymbol{ u is either hat{oldsymbol{i or hat{oldsymbol{j.

Time derivatives in the two frames

Introduce the unit vectors hat{oldsymbol{i, hat{oldsymbol{j, hat{oldsymbol{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{oldsymbol{u of the rotating coordinate system abides by the following equation:: frac{d}{dt}hat{oldsymbol{u=oldsymbol{Omega imes hat{u .Then if we have a vector function oldsymbol{f}, : oldsymbol{f}(t)=f_x(t) hat{oldsymbol{i+f_y(t) hat{oldsymbol{j+f_z(t) hat{oldsymbol{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}oldsymbol{f}=frac{df_x}{dt}hat{oldsymbol{i+frac{dhat{oldsymbol{i}{dt}f_x+frac{df_y}{dt}hat{oldsymbol{j+frac{dhat{oldsymbol{j}{dt}f_y+frac{df_z}{dt}hat{oldsymbol{k+frac{dhat{oldsymbol{k}{dt}f_z::=frac{df_x}{dt}hat{oldsymbol{i+frac{df_y}{dt}hat{oldsymbol{j+frac{df_z}{dt}hat{oldsymbol{k+ [oldsymbol{Omega imes} (f_x hat{oldsymbol{i + f_y hat{oldsymbol{j+f_z hat{oldsymbol{k)] ::= left( frac{doldsymbol{f{dt} ight)_r+oldsymbol{Omega imes f}(t) ,where left( frac{doldsymbol{f{dt} ight)_r is the rate of change of oldsymbol{f} as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:::frac{d}{dt}oldsymbol{f} =left [ left(frac{d}{dt} ight)_r + oldsymbol{Omega imes} ight] oldsymbol{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 oldsymbol{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 oldsymbol{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 differentiations 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 forces 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 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 forces. Notice that all three forces vanish when the frame is not rotating, that is, when oldsymbol{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{iNewton'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.


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