- Rotating reference frame
A rotating frame of reference is a special case of a
non-inertial reference framethat 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.)
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
Coriolis forceand, for non-uniformly rotating reference frames,
Scientists living in a rotating box can measure the speed and direction of their rotation by measuring these
fictitious forces. For example, Léon Foucaultwas able to show the Coriolis forcethat 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 of the rotating reference frame and the coordinates of an
inertial reference framewith the same origin. If the rotation is about the axis with an angular velocityand the two reference frames coincide at time , the transformation from rotating coordinates to inertial coordinates can be written
whereas the reverse transformation is
This result can be obtained from a
Introduce the unit vectors 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"::where the ("x", "y") components are expressed in the stationary frame. Likewise,:Thus the time derivative of these vectors, which rotate without changing magnitude, is::This result is the same as found using a
vector cross productwith the rotation vector pointed along the z-axis of rotation , namely,:where is either or .
Time derivatives in the two frames
Introduce the unit vectors 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 about an axis then each unit vector of the rotating coordinate system abides by the following equation::Then if we have a vector function , : and we want to examine its first dervative we have (using the
chain ruleof 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] :::::where is the rate of change of as observed in the rotating coordinate system. As a shorthand the differentiation is expressed as:::
Relation between velocities in the two frames
A velocity of an object is the time-derivative of the object's position, or
The time derivative of a position 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 , the velocities in the two reference frames are related by the equation
: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
: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
where 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 centrifugal force
* and the
where 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
For completeness, the inertial acceleration due to impressed external forces 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:
:Newton's law in the the rotating frame then becomes::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
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 forceThe effect of the Coriolis force on the Earth and other rotating systems
Inertial frame of reference
Fictitious forceA more general treatment of the subject of this article
* [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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