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The '''Coriolis force''' is a force experienced by a object traversing a curved path that is proportional to its speed and also to the sine of the angle between its direction of movement and its axis of rotation. It is one of three such ''inertial forces'' that appear in an accelerating frame of reference due to the acceleration of the frame, the other two being the [[centrifugal force]] and the [[Euler force]]. The mathematical expression for the Coriolis force appeared in an 1835 paper by a French scientist [[Gaspard-Gustave Coriolis]] in connection with the theory of water wheels, and also in the [[Theory of tides|tidal equations]] of [[Pierre-Simon Laplace]] in 1778.
===Quarks===


Although sometimes referred to as an ''apparent'' force, it can have very real effects.
The quarks that may engage one another in reactions are determined by the [[Cabbibo-Kobayashi-Maskawa matrix]]:[http://books.google.com/books?id=f89yg8a1t-EC&pg=PA23&dq=Cabibbo+angle&hl=en&ei=XYZvToTqN7PZiAKM55juBg&sa=X&oi=book_result&ct=result&resnum=2&ved=0CDIQ6AEwATgK#v=onepage&q=Cabibbo%20angle&f=false Morii]
:<math>\begin{pmatrix}
d'\\
s'\\
b'\\
\end{pmatrix} = \begin{pmatrix}
U_{ud}&U_{us}&U_{ub}\\
U_{cd}&U_{cs}&U_{cb}\\
U_{td}&U_{ts}&U_{tb}\\
\end{pmatrix}=\begin{pmatrix}
d\\
s\\
b\\
\end{pmatrix} </math>


==Coriolis effect==
The quarks can be arranged to exhibit right- and left-handedness, subscripts ''L'' and ''R'', to resemble the leptons. The right-handed quarks do not couple to the weak interaction, and are labeled with subscript ''R''. The left-handed quarks corresponding to these right-handed quarks are mixtures of quarks. Thus, the up and down quarks are assembled as:
In [[psychophysical perception]], the '''Coriolis effect''' is a form of nausea induced by the Coriolis force (also referred to as the '''Coriolis illusion''').<ref name=Lewis>


{{cite book
:<math> \tbinom {u}{d}_L \ ; \ \ u_R,\ d_R \ .</math>
|title=Lewis' dictionary of occupational and environmental safety and health
|author=Jeffrey W. Vincoli
|isbn=1566703999
|year=1999
|publisher=CRC Press
|pages=p. 245
|url= http://books.google.com/books?id=7PZ4PjGvlt4C&pg=PA245&dq=nausea+Coriolis&lr=&as_brr=0&sig=MlgYJ08wevi82LUZtx2mFLGvvUg 
}}</ref><ref name=Sanders>
{{cite book
|title=Human Factors in Engineering and Design
|author=Mark S Sanders & Ernest J McCormick
|pages=p. 644
|url= http://books.google.com/books?id=1bK_LSLD9C4C&pg=PT669&dq=nausea+Coriolis&lr=&as_brr=0&sig=raSVrvjpG6AzLSdv9JQN2lrekII
|isbn=0071128263
|year=1993
|edition=7th Edition
|publisher=McGraw-Hill
}}</ref><ref name=Ebenholtz>
{{cite book
|title=Oculomotor Systems and Perception
|author=Sheldon M. Ebenholtz
|url= http://books.google.com/books?id=1W7ePrvrRyYC&pg=PA151&dq=nausea+Coriolis&lr=&as_brr=0&sig=iRxVjBDpQb0s10KW1pVEvyGq3sU
|isbn=0521804590
|year=2001
|publisher=Cambridge University Press
}}</ref><ref name=Mather>
{{cite book |title=Foundations of perception
|author=George Mather
|url=http://books.google.com/books?id=LYA9faq3lt4C&pg=PA73&dq=nausea+Coriolis&lr=&as_brr=0&sig=Izy98Cn_a904vysVnnTarv_XSoo
|isbn=0863778356
|year=2006
|publisher=Taylor & Francis
}}


</ref> The Coriolis effect is a concern of pilots, where it can cause extreme discomfort and disorientation.<ref name=Nicogossian>
The other generations are arranged similarly:


{{cite book
:<math> \tbinom {c}{s}_L \ ; \ \ c_R,\ s_R \ , </math>
|title=Space biology and medicine
:<math> \tbinom {t}{b}_L \ ; \ \ t_R,\ b_R \ . </math>
|author=Arnauld E. Nicogossian
{| class="wikitable" style="margin: 0 auto; text-align:center"
|pages=p. 337
|+'''Right- and left-handed quarks'''
| url= http://books.google.com/books?id=aO6zut2K7lsC&pg=PA337&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U2ODlvCKri-JbJfB-OdyhXyhozbnw
! Symbol
|isbn=1563471809
! Electric charge, ''Q''
|year=1996
! Weak isospin, (''I<sub>W</sub>, I<sub>W3</sub>'')
|publisher=American Institute of Aeronautics and Astronautics, Inc
! Weak hypercharge, (''Y<sub>W</sub>'')
|location=Reston, VA
|-
}}</ref><ref name= Brandt>
| ''u<sub>L</sub>, c<sub>L</sub>, t<sub>L</sub>''
{{cite book
| +2/3
|title=Vertigo: Its Multisensory Syndromes
| (1/2, +1/2)
|author=Thomas Brandt
| +1/3
|pages=p. 416
|-
|url= http://books.google.com/books?id=dFevxJ0mJncC&pg=PA416&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U0_U1pikmdee-bTFvOIqg_rHKmS8A
| ''d<sub>L</sub>, s<sub>L</sub>, b<sub>L</sub>''
|isbn=0387405003
| −1/3
|year=2003
| (1/2, −1/2)
|publisher=Springer
| +1/3
}}</ref><ref name= Ercoline>
|-
{{cite book
| ''u<sub>R</sub>, c<sub>R</sub>, t<sub>R</sub>''
|title=Spatial Disorientation in Aviation
| +2/3
|author=Fred H. Previc, William R. Ercoline
| 0
|pages=p. 249
| +4/3
|url= http://books.google.com/books?id=oYP7m9m2RocC&pg=PA249&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U3_1-RrNgSCpybFHhnt156jhfkY2A
|-
|isbn=1563476541
| ''d<sub>R</sub>, s<sub>R</sub>, b<sub>R</sub>''
|year=2004
| −1/3
|publisher=American Institute of Aeronautics and Astronautics, Inc
| 0
|location=Reston, VA
| −2/3
}}</ref><ref name= Clément>
|-
{{cite book
|}
|title=Fundamentals of Space Medicine
|author=Gilles Clément
|pages=p. 41
|url= http://books.google.com/books?id=Neura4O-taIC&pg=PA41&dq=Coriolis+effect+airplane+nausea&lr=&as_brr=0&sig=ACfU3U16LpzGILe3QVGIeOl5tyYFDAKLrA
|isbn=1402015984
|year=2003
|publisher=Springer}}</ref>


==Mathematical derivation of fictitious forces==
{{anchor|coordinates}}
{{Image|Moving coordinate system.PNG|right|250px| An object located at '''x'''<sub>A</sub> in inertial frame ''A'' is located at location '''x'''<sub>B</sub> in accelerating frame ''B''. The origin of frame ''B'' is located at '''X'''<sub>AB</sub> in frame ''A''. The orientation of frame ''B'' is determined by the unit vectors along its coordinate directions, '''u'''<sub>j</sub> with ''j'' <nowiki>=</nowiki> 1, 2, 3. Using these axes, the coordinates of the object according to frame ''B'' are '''x'''<sub>B</sub>  <nowiki>=</nowiki>  (''x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>'').}}


=== General derivation ===
{{Reflist}}
Many problems require use of noninertial reference frames, for example, those involving satellites<ref name=Isidon>{{cite book |title=Robust Autonomous Guidance: An Internal Model Approach |author=Alberto Isidori, Lorenzo Marconi & Andrea Serrani |pages= p. 61 |url=http://books.google.com/books?id=Mo0YhjLeTA0C&pg=RA2-PA60&dq=orbit+%22coordinate+system%22&lr=&as_brr=0&sig=3RDQgi9CX8h0WcM-VeHeKUc-yuQ#PRA2-PA61,M1
|isbn=1852336951 |publisher=Springer |year=2003 }}</ref><ref name=Ying>{{cite book |title=Advanced Dynamics |author=Shuh-Jing Ying |url=http://books.google.com/books?id=juO1Hj8ocx8C&pg=PA172&dq=orbit+%22coordinate+system%22&lr=&as_brr=0&sig=c_FX8kDfa6ei7t6CdU5mtBjdw78 |pages=p. 172 |isbn=1563472244 |publisher=American Institute of Aeronautics, and Astronautics|location=Reston VA  |year=1997 }}</ref> and particle accelerators.<ref name=Bryant>{{cite book |title=The Principles of Circular Accelerators and Storage Rings |author=Philip J. Bryant & Kjell Johnsen |pages=p. xvii |url=http://books.google.com/books?id=6P-UWLmfD4wC&pg=PR17&dq=orbit+%22coordinate+system%22&lr=&as_brr=0&sig=lqYKO9oGAWWo4sPmJiR9StSjjUk
|isbn=0521355788 |publisher=Cambridge University Press|location=Cambridge UK |year=1993 }}</ref>
Figure 2 shows a particle with [[mass]] ''m'' and [[position (vector)|position]] [[vector (geometric)|vector]] '''x'''<sub>A</sub>''(t)'' in a particular [[inertial frame]] A.  Consider a non-inertial frame B whose origin relative to the inertial one is given by '''X'''<sub>AB</sub>(''t'').  Let the position of the particle in frame B be '''''x'''''<sub>B</sub>(''t'').  What is the force on the particle as expressed in the coordinate system of frame B? <ref name=Fetter>{{cite book |author=Alexander L Fetter & John D Walecka|title=Theoretical Mechanics of Particles and Continua |publisher= Courier Dover Publications |url=http://books.google.com/books?id=olMpStYOlnoC&printsec=frontcover&dq=intitle:theoretical+intitle:mechanics+inauthor:fetter&lr=&as_brr=0#PPA32,M1 |year=2003 |isbn=0486432610 |pages= pp. 33–39}}</ref><ref name=Lim>{{cite book |title=Problems and Solutions on Mechanics: Major American Universities Ph.D. Qualifying Questions and Solutions |author=Yung-kuo Lim & Yuan-qi Qiang |isbn=9810212984 |pages=p. 183|url=http://books.google.com/books?id=93b3cjVJ2l4C&pg=PA183&dq=%22Coriolis+force%22+-snippet+date:1990-2008&lr=&as_brr=0&sig=K-t1sNxH-MSBbnnm5FsrjqlR6tM
|publisher=World Scientific |location=Singapore|year=2001}}</ref>
 
To answer this question, let the coordinate axis in B be represented by unit vectors '''u'''<sub>j</sub> with ''j'' any of { 1, 2, 3 } for the three coordinate axes. Then
 
:<math> \mathbf{x}_{B} = \sum_{j=1}^3 x_j\ \mathbf{u}_j \ . </math>
 
The interpretation of this equation is that '''x'''<sub>B</sub> is the vector displacement of the particle as  expressed in terms of the coordinates in frame B at time ''t''. From frame A the particle is located at:
 
:<math>\mathbf{x}_A =\mathbf{X}_{AB} +  \sum_{j=1}^3 x_j\ \mathbf{u}_j \ . </math>
 
As an aside, the unit vectors { '''u'''<sub>j</sub> } cannot change magnitude, so derivatives of these vectors express only rotation of the coordinate system B. On the other hand, vector '''X'''<sub>AB</sub> simply locates the origin of frame B relative to frame A, and so cannot include rotation of frame B.
 
Taking a time derivative, the velocity  of the particle is:
 
:<math> \frac {d \mathbf{x}_{A}}{dt} =\frac{d \mathbf{X}_{AB}}{dt}+ \sum_{j=1}^3 \frac{dx_j}{dt} \ \mathbf{u}_j + \sum_{j=1}^3 x_j \ \frac{d \mathbf{u}_j}{dt} \ . </math>
 
The second term summation is the velocity of the particle, say '''v'''<sub>B</sub> as measured in frame B. That is:
 
:<math> \frac {d \mathbf{x}_{A}}{dt} =\mathbf{v}_{AB}+ \mathbf{v}_B + \sum_{j=1}^3 x_j \ \frac{d \mathbf{u}_j}{dt} \ . </math>
 
The interpretation of this equation is that the velocity of the particle seen by observers in frame A consists of what observers in frame B call the velocity, namely '''v'''<sub>B</sub>, plus two extra terms related to the rate of change of the frame-B coordinate axes. One of these is simply the velocity of the moving origin '''''v<sub>AB</sub>'''''. The other is a contribution to velocity due to the fact that different locations in the non-inertial frame have different apparent velocities due to rotation of the frame; a point seen from a rotating frame has a rotational component of velocity that is greater the further the point is from the origin.
 
To find the acceleration, another time differentiation provides:
 
:<math> \frac {d^2 \mathbf{x}_{A}}{dt^2} = \mathbf{a}_{AB}+\frac {d\mathbf{v}_B}{dt} + \sum_{j=1}^3 \frac {dx_j}{dt} \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . </math>
 
Using the same formula already used for the time derivative of '''x'''<sub>B</sub>, the velocity derivative on the right is:
 
:<math>\frac {d\mathbf{v}_B}{dt} =\sum_{j=1}^3 \frac{d v_j}{dt} \  \mathbf{u}_j+ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} =\mathbf{a}_B + \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} \ . </math>
 
{{anchor|Eq. 1}}Consequently,
:<math> \frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_{AB}+\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . </math>&ensp;&ensp;&ensp;(Eq. 1)
 
The interpretation of this equation is as follows: the acceleration of the particle in frame A consists of what observers in frame B  call the particle acceleration '''a'''<sub>B</sub>, but in addition there are three acceleration terms related to the movement of the frame-B coordinate axes: one term related to the acceleration of the origin of frame B, namely '''a'''<sub>AB</sub>, and two terms related to rotation of frame B. Consequently, observers in B will see the particle motion as possessing "extra" acceleration, which they will attribute to "forces" acting on the particle, but which observers in A say are "fictitious" forces arising simply because observers in B do not recognize the non-inertial nature of frame B.
 
The factor of two in the Coriolis force arises from two equal contributions: (i) the apparent change of an inertially constant velocity with time because rotation makes the direction of the velocity seem to change (a ''d'''v<sub>B</sub>''' / dt'' term) and (ii) an apparent change in the velocity of an object when its position changes, putting it nearer to or further from the axis of rotation (the change in <big>Σ</big>''x<sub>j</sub>'' <math>\stackrel{\frac{d}{dt}}{}</math>'''u'''<sub>j</sub> due to change in ''x <sub>j</sub>'' ).
 
To put matters in terms of forces, the accelerations are multiplied by the particle mass:
 
:<math>\mathbf{F}_A = \mathbf{F}_B + m\ \mathbf{a}_{AB}+  2m\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} + m\ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . </math>
 
The force observed in frame B, '''F'''<sub>B</sub> = m '''a'''<sub>B</sub> is related to the actual force on the particle, '''F'''<sub>A</sub>, by:
 
:<math>\mathbf{F}_B = \mathbf{F}_A + \mathbf{F}_{\mbox{fictitious}} \ . </math>
 
where:
 
:<math> \mathbf{F}_{\mbox{fictitious}} =-m\ \mathbf{a}_{AB} -2m\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} - m\ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . </math>
 
Thus, we can solve problems in frame B by assuming that Newton's second law holds (with respect to quantities in that frame) and treating '''F'''<sub>fictitious</sub> as an additional force.<ref name="Arnolʹd"/><ref name=Taylor>{{cite book |title=Classical Mechanics |author=John Robert Taylor |location=Sausalito CA |isbn=189138922X |year=2004 |url=http://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1&dq=isbn=189138922X&lr=&as_brr=0&sig=JVfFlMT5TvXh1I64JAFBFq7pA6s#PPA343,M1 |publisher=University Science Books|pages=pp. 343–344}}</ref><ref>Kleppner, pages 62-63</ref>
 
Below are a number of examples applying this result for fictitious forces. More examples can be found in the article on [[centrifugal force]].
 
=== Rotating coordinate systems ===
A common situation in which noninertial reference frames are useful is when the reference frame is rotating. Because such rotational motion is non-inertial, due to the acceleration present in any rotational motion, a fictitious force can always be invoked by using a rotational frame of reference. Despite this complication, the use of fictitious forces often simplifies the calculations involved.
 
To derive expressions for the fictitious forces, derivatives are needed for the apparent time rate of change of vectors that take into account time-variation of the coordinate axes. If the rotation of frame ''B'' is represented by a vector '''Ω''' pointed along the axis of rotation with orientation given by the [[right-hand rule]], and with magnitude given by
 
:<math> |\boldsymbol{\Omega} | = \frac {d \theta }{dt} = \omega (t) \ , </math>
 
then the time derivative of any of the three unit vectors describing frame ''B'' is:<ref name=Taylor/><ref>See for example, {{cite book |author=JL Synge & BA Griffith |title=Principles of Mechanics |publisher=McGraw-Hill |edition=2nd Edition |year=1949 |pages=pp. 348–349}}</ref>
 
:<math> \frac {d  \mathbf{u}_j (t)}{dt} = \boldsymbol{\Omega} \times \mathbf{u}_j (t) \ , </math>
 
and
 
:<math>\frac {d^2  \mathbf{u}_j (t)}{dt^2}= \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j +\boldsymbol{\Omega} \times \frac{d \mathbf{u}_j (t)}{dt} </math>&ensp;<math>= \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j+ \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times  \mathbf{u}_j (t) \right)\ , </math>
 
as is verified using the properties of the [[vector cross product]]. These derivative formulas now are applied to the relationship between acceleration in an inertial frame, and that in a coordinate frame rotating with time-varying angular velocity ω ( ''t'' ). From the previous section, where subscript '''''A''''' refers to the inertial frame and '''''B''''' to the rotating frame, setting '''a'''<sub>AB</sub> = 0 to remove any translational acceleration, and focusing on only rotational properties (see [[#Eq. 1|Eq. 1]]):
 
:<math> \frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} </math>&ensp;<math>+ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ ,</math>
:<math>\mathbf{a}_A=\mathbf{a}_B +\ 2\ \sum_{j=1}^3 v_j \boldsymbol{\Omega} \times \mathbf{u}_j (t)\  </math>&ensp;<math>+\  \sum_{j=1}^3 x_j \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{u}_j  \  + \sum_{j=1}^3 x_j \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times  \mathbf{u}_j (t) \right)\ </math>
::<math>=\mathbf{a}_B </math>&ensp;<math>+ 2\ \boldsymbol{\Omega} \times\sum_{j=1}^3 v_j  \mathbf{u}_j (t) \  </math>&ensp;<math>+ \frac{d\boldsymbol{\Omega}}{dt} \times \sum_{j=1}^3 x_j \mathbf{u}_j  \  </math>&ensp;<math>+\ \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \sum_{j=1}^3 x_j  \mathbf{u}_j (t) \right)\ .</math>
Collecting terms, the result is the so-called ''acceleration transformation formula'':<ref name=Gregory>{{cite book |title=Classical Mechanics: An Undergraduate Text |url=http://books.google.com/books?id=uAfUQmQbzOkC&printsec=frontcover&dq=%22rigid+body+kinematics%22&lr=&as_brr=0#PRA1-PA475,M1
|author=R. Douglas Gregory |year=2006 |isbn=0521826780 |publisher=Cambridge University Press |location=Cambridge UK |pages=Eq. (17.16), p. 475}}</ref>
 
:<math>\mathbf{a}_A=\mathbf{a}_B + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\  </math>&ensp;<math>+ \frac{d\boldsymbol{\Omega}}{dt} \times \mathbf{x}_B  \  + \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \mathbf{x}_B \right)\ .</math>
 
The [[proper acceleration|physical acceleration]] '''''a'''''<sub>A</sub> due to what observers in the inertial frame '''''A''''' call ''real external forces'' on the object is, therefore, not simply the acceleration '''''a'''''<sub>B</sub> seen by observers in the rotational frame '''''B''''' , but has several additional geometric acceleration terms associated with the rotation of '''''B'''''. As seen in the rotational frame, the acceleration '''''a'''''<sub>B</sub> of the particle is given by rearrangement of the above equation as:
:<math>
\mathbf{a}_{B} =
\mathbf{a}_A - 2 \boldsymbol\Omega \times \mathbf{v}_{B} - \boldsymbol\Omega \times (\boldsymbol\Omega \times  \mathbf{x}_B )  - \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_B \ .
</math>
 
The net force upon the object according to observers in the rotating frame is '''F'''<sub>B</sub> = ''m'' '''a'''<sub>B</sub>.  If their observations are to result in the correct force on the object when using Newton's laws, they must consider that the additional force '''F'''<sub>fict</sub> is present, so the end result is '''F'''<sub>B</sub> = '''F'''<sub>A</sub> + '''F'''<sub>fict</sub>. Thus, the fictitious force used by observers in '''''B''''' to get the correct behavior of the object from Newton's laws equals:
 
:<math>
\mathbf{F}_{\mathrm{fict}}  =
- 2 m \boldsymbol\Omega  \times \mathbf{v}_{B} - m \boldsymbol\Omega  \times (\boldsymbol\Omega \times \mathbf{x}_B ) </math>&ensp;<math>\ - m \frac{d \boldsymbol\Omega  }{dt} \times \mathbf{x}_B \ .
</math>
 
Here, the first term is the ''[[Coriolis force]]'',<ref name=Joos>{{cite book |title=Theoretical Physics|author= Georg Joos & Ira M. Freeman|url=http://books.google.com/books?id=vIw5m2XuvpIC&pg=PA233&dq=%22Coriolis+force%22+intitle:Theoretical+intitle:physics&lr=&as_brr=0&sig=wveOPKIvSGTCKQSpw-2jFQRe79M#PPA233,M1
|pages=p. 233 |isbn=0486652270 |publisher=Courier Dover Publications |location=New York|year=1986}}</ref> the second term is the ''[[centrifugal force (fictitious)|centrifugal force]]'',<ref name=Smith>{{cite book |title=Theoretical Mechanics|author=Percey Franklyn Smith & William Raymond Longley|url=http://books.google.com/books?id=qL4EAAAAYAAJ&pg=PA118&dq=%22centrifugal+force%22+intitle:theoretical&lr=&as_brr=0
|pages=p. 118 |year=1910|publisher=Gin|location=Boston}}</ref> and the third term is the ''[[Euler force]]''.<ref name=Lanczos>{{cite book |author=Cornelius Lanczos |title=The Variational Principles of Mechanics |url=http://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103&dq=%22Euler+force%22&lr=&as_brr=0&sig=UV46Q9NIrYWwn5EmYpPv-LPuZd0#PPA103,M1
|pages=p. 103|year=1986|publisher=Courier Dover Publications|isbn=0486650677|location=New York}}</ref><ref name=Marsden>{{cite book |author=Jerold E. Marsden & Tudor.S. Ratiu |year= 1999 |edition=2nd Edition |title=  Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems: Texts in applied mathematics, 17 |publisher= Springer-Verlag|location = NY|url=http://books.google.com/books?id=I2gH9ZIs-3AC&printsec=frontcover&dq=intitle:Introduction+intitle:to+intitle:mechanics+intitle:and+intitle:symmetry&lr=&as_brr=0&sig=u3-zzH2g5eYO9zvYNle-bXbbics#PPA251,M1 |isbn=038798643X|pages= p. 251}}</ref>  When the rate of rotation doesn't change, as is typically the case for a planet, the Euler force is zero.
 
=== Orbiting coordinate systems ===
{{Unreferenced section|date=July 2008}}
[[Image:Orbiter.PNG|thumb|250px|Figure 3: An orbiting but fixed orientation coordinate system ''B'', shown at three different times. The unit vectors '''u'''<sub>j</sub>, j = 1, 2, 3 do ''not'' rotate, but maintain a fixed orientation, while the origin of the coordinate system ''B'' moves at constant angular rate ω about the fixed axis '''Ω'''. Axis '''Ω''' passes through the origin of inertial frame ''A'', so the origin of frame ''B'' is a fixed distance ''R'' from the origin of inertial frame ''A''.]]
As a related example, suppose the moving coordinate system ''B'' rotates in a circle of radius ''R'' about the fixed origin of inertial frame ''A'', but maintains its coordinate axes fixed in orientation, as in Figure 3. The acceleration of an observed body is now (see [[#Eq. 1|Eq. 1]]):
:<math> \frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_{AB}+\mathbf{a}_B </math>&ensp;<math>+ 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} </math>&ensp;<math>+ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\ . </math>
::<math>=\mathbf{a}_{AB}\ +\mathbf{a}_B\ , </math>
where the summations are zero inasmuch as the unit vectors have no time dependence. The origin of system ''B'' is located according to frame ''A'' at:
:<math>\mathbf{X}_{AB} = R \left( \cos ( \omega t) , \ \sin (\omega t) \right)  \ ,</math>
leading to a velocity of the origin of frame ''B'' as:
:<math>\mathbf{v}_{AB} = \frac{d}{dt} \mathbf{X}_{AB} = \mathbf{\Omega \times X}_{AB} \ , </math>
 
leading to an acceleration of the origin of ''B'' given by:
:<math>\mathbf{a}_{AB} = \frac{d^2}{dt^2} \mathbf{X}_{AB} </math>&ensp;<math>= \mathbf{ \Omega \ \times }  \left( \mathbf{ \Omega \times X}_{AB}\right) </math>&ensp;<math>=  - \omega^2 \mathbf{X}_{AB} \ .</math>
Because the first term, which is
::::<math>\mathbf{ \Omega \ \times }  \left( \mathbf{ \Omega \times X}_{AB}\right)\ , </math>
is of the same form as the normal centrifugal force expression:
::::<math>\boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \mathbf{x}_B \right)\ ,</math>
it is a natural extension of standard terminology (although there is no standard  terminology for this case) to call this term a "centrifugal force". Whatever terminology is adopted, the observers in frame ''B''  must introduce a fictitious force, this time due to the acceleration from the orbital motion of their entire coordinate frame, that is radially outward away from the center of rotation of the origin of their coordinate system:
:<math>\mathbf{F}_{\mathrm{fict}} = m \omega^2 \mathbf{X}_{AB} \ , </math>
and of magnitude:
:<math>|\mathbf{F}_{\mathrm{fict}}| = m \omega^2 R \ . </math>
 
Notice that this "centrifugal force" has differences from the case of a rotating frame. In the rotating frame the centrifugal force is related to the distance of the object from the origin of frame ''B'', while in the case of an orbiting frame, the centrifugal force is independent of the distance of the object from the origin of frame ''B'', but instead depends upon the distance of the origin of frame ''B'' from ''its'' center of rotation, resulting in the ''same'' centrifugal fictitious force for ''all'' objects observed in frame ''B''.
 
===Orbiting and rotating===
[[Image:Center-facing orbiting coordinate system.PNG|thumb|250px|Figure 4: An orbiting coordinate system ''B'' similar to Figure 3, but in which unit vectors '''u'''<sub>j</sub>, j = 1, 2, 3 rotate to face the rotational axis, while the origin of the coordinate system ''B'' moves at constant angular rate ω about the fixed axis '''Ω'''.]]
As a combination example, Figure 4 shows a coordinate system ''B'' that orbits inertial frame ''A'' as in Figure 3, but the coordinate axes in frame ''B'' turn so unit vector '''u'''<sub>1</sub> always points toward the center of rotation. This example might apply to a test tube in a centrifuge, where vector '''u'''<sub>1</sub> points along the axis of the tube toward its opening at its top. It also resembles the Earth-Moon system, where the Moon always presents the same face to the Earth.<ref name=Newcomb>However, the Earth-Moon system rotates about its [[Barycentric coordinates (astronomy)|barycenter]], not the Earth's center; see {{cite book |author= Simon Newcomb |title=Popular Astronomy |pages=p. 307 |url=http://books.google.com/books?id=VS7aS8QS91oC&pg=PA307&dq=centrifugal+revolution+and+rotation+date:1970-2009&lr=&as_brr=0&sig=ACfU3U2qOa8MdqbQFXeX56JqvEsiMsu05Q  |isbn=140674574X |year=2007 |publisher=Read Books}}</ref> In this example, unit vector '''u'''<sub>3</sub> retains a fixed orientation, while vectors '''u'''<sub>1</sub>, '''u'''<sub>2</sub> rotate at the same rate as the origin of coordinates. That is,
:<math>\mathbf{u}_1 = (-\cos \omega t ,\ -\sin \omega t )\ ;\  </math>&ensp;<math>\mathbf{u}_2 = (\sin \omega t ,\ -\cos \omega t ) \ . </math>
:<math>\frac{d}{dt}\mathbf{u}_1 = \mathbf{\Omega \times u_1}= \omega\mathbf{u}_2\ ;</math>&ensp;<math> \ \frac{d}{dt}\mathbf{u}_2 = \mathbf{\Omega \times u_2} = -\omega\mathbf{u}_1\ \ .</math>
Hence, the acceleration of a moving object is expressed as (see [[#Eq. 1|Eq. 1]]):
:<math> \frac {d^2 \mathbf{x}_{A}}{dt^2}=\mathbf{a}_{AB}+\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j \ \frac{d \mathbf{u}_j}{dt} </math>&ensp;<math>+\ \sum_{j=1}^3 x_j \ \frac{d^2 \mathbf{u}_j}{dt^2}\  </math>
::<math>=\mathbf{ \Omega \ \times }  \left( \mathbf{ \Omega \times X}_{AB}\right) +\mathbf{a}_B + 2\ \sum_{j=1}^3 v_j\ \mathbf{\Omega \times u_j}</math>&ensp;<math> \  +\  \sum_{j=1}^3 x_j\ \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \mathbf{u}_j \right)\ </math>
::<math>=\mathbf{ \Omega \ \times }  \left( \mathbf{ \Omega \times X}_{AB}\right) + \mathbf{a}_B + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\ </math>&ensp;<math> \  +\  \boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \mathbf{x}_B \right)\ </math>
::<math>=\mathbf{ \Omega \ \times }  \left( \mathbf{ \Omega \times} (\mathbf{ X}_{AB}+\mathbf{x}_B) \right) + \mathbf{a}_B  + 2\ \boldsymbol{\Omega} \times\mathbf{v}_B\  \ ,</math>
where the angular acceleration term is zero for constant rate of rotation.
Because the first term, which is
::::<math>\mathbf{ \Omega \ \times }  \left( \mathbf{ \Omega \times} (\mathbf{ X}_{AB}+\mathbf{x}_B) \right)\ , </math>
is of the same form as the normal centrifugal force expression:
::::<math>\boldsymbol{\Omega} \times \left(  \boldsymbol{\Omega} \times \mathbf{x}_B \right)\ ,</math>
it is a natural extension of standard terminology (although there is no standard  terminology for this case) to call this term the "centrifugal force". Applying this terminology to the example  of a tube in a centrifuge, if the tube is far enough from the center of rotation, |'''X'''<sub>AB</sub>| = ''R'' >> |'''x'''<sub>B</sub>|, all the matter in the test tube sees the same acceleration (the same centrifugal force).  Thus, in this case, the fictitious force is primarily a uniform centrifugal force along the axis of the tube, away from the center of rotation, with a value |'''F'''<sub>Fict</sub>| = ω<sup>2</sup> ''R'', where ''R'' is the distance of the matter in the tube from the center of the centrifuge. It is standard specification of a centrifuge to use the "effective" radius of the centrifuge to estimate its  ability to provided centrifugal force. Thus, a first estimate of centrifugal force in a centrifuge can be based upon the distance of the tubes from the center of rotation, and corrections applied if needed.<ref name=Singh>{{cite book |title=Constitutive and Centrifuge Modelling: Two Extremes |author=Bea K Lalmahomed, Sarah Springman, Bhawani Singh |isbn=9058093611 |year=2002 |publisher=Taylor and Francis |pages=p. 82 |url=http://books.google.com/books?id=MJkz_IBZZS0C&printsec=frontcover&dq=centrifuge&lr=&as_brr=0&sig=ACfU3U3S3gbnFB0z-_rDWX0uL-oJgp1m3g#PPT102,M1  }}</ref><ref name=Nen>{{cite book |title=Consolidation of Soils: Testing and Evaluation: a Symposium |author=Raymond Nen |isbn=0803104464 |year=1986 |publisher=ASTM International |pages=p. 590 |url=http://books.google.com/books?id=a-BKqGTXA6kC&pg=PA590&dq=radius+centrifuge+effective&lr=&as_brr=0&sig=ACfU3U2MFkfhj-3dLbw2pVCW7bTN4TFftw }}</ref>
 
Also, the test tube confines motion to the direction down the length of the tube, so '''v'''<sub>B</sub> is opposite to '''u'''<sub>1</sub> and the Coriolis force is opposite to '''u'''<sub>2</sub>, that is, against the wall of the tube. If the tube is spun for a long enough time, the velocity '''v'''<sub>B</sub> drops to zero as the matter comes to an equilibrium distribution. For more details, see the articles on [[sedimentation]] and the [[Lamm equation]].
 
A related problem is that of centrifugal forces for the Earth-Moon-Sun system, where three rotations appear: the daily rotation of the Earth about its axis, the lunar-month rotation of the Earth-Moon system about their center of mass, and the annual revolution of the Earth-Moon system about the Sun. These three motions influence the [[tides]].<ref name=Appleton>{{cite book |title=The Popular Science Monthly |year=1877 |author=D Appleton |pages=p. 276 |url=http://books.google.com/books?id=YO0KAAAAYAAJ&pg=PA276&dq=rotation+revolution+%22centrifugal+force%22&lr=&as_brr=0  }}</ref>
 
===Crossing a carousel===
{{See also|Coriolis effect#Cannon on turntable|Coriolis effect#Tossed ball on a rotating carousel}}
[[Image:Carousel walk.PNG|thumb |430px |Figure 5: Crossing a rotating carousel walking at constant speed from the center of the carousel to its edge, a spiral is traced out in the inertial frame, while a simple straight radial path is seen in the frame of the carousel.]]
Figure 5 shows another example comparing the observations of an inertial observer with those of an observer on a rotating [[carousel]].<ref name= Giancoli>For a similar example, see {{cite book |title=A Handbook for Wireless/ RF, EMC, and High-Speed Electronics, Part of the EDN Series for Design Engineers |author=Ron Schmitt |year=2002 |publisher=Newnes |isbn=0750674032 |url=http://books.google.com/books?id=fUBPN8T9bwUC&pg=PA61&dq=spheres+rotating++Coriolis&lr=&as_brr=0&sig=p3c4nnM6E1OdmeDZC4QLLsgwljQ#PPA60,M1
|pages=pp. 60–61  }}, and {{cite book |title=Physics for Scientists And Engineers With Modern Physics |author=Douglas C. Giancoli |pages=p. 301 |isbn=0131495089 |year=2007 |publisher=Pearson Prentice-Hall |url=http://books.google.com/books?id=xz-UEdtRmzkC&pg=PA301&dq=spheres+rotating++Coriolis&lr=&as_brr=0&sig=5QoXXsNsx55hcobuv87pbiaoeu8#PPA301,M1  }}</ref> The carousel rotates at a constant angular velocity represented by the vector '''Ω''' with magnitude ω, pointing upward according to the [[right-hand rule]]. A rider on the carousel walks radially across it at constant speed, in what appears to the walker to be the straight line path inclined at 45° in Figure 5 . To the stationary observer, however, the walker travels a spiral path. The points identified on both paths in Figure 5 correspond to the same times spaced at equal time intervals. We ask how two observers, one on the carousel and one in an inertial frame, formulate what they see using Newton's laws.
 
====Inertial observer====
The observer at rest describes the path followed by the walker as a spiral. Adopting the coordinate system shown in Figure 5, the trajectory is described by '''r''' ( ''t'' ):
:<math>\boldsymbol{r}(t) =R(t)\mathbf{u}_R = \left( x(t),\ y(t) \right) = \left( R(t)\cos (\omega t + \pi/4),\ R(t)\sin (\omega t + \pi/4) \right) \ , </math>
where the added <font style="font-family: Times New Roman; font-size:100%; font-style:italic; font-weight:bold;">π</font>/4 sets the path angle at 45° to start with (just an arbitrary choice of direction), '''u'''<sub>R</sub> is a unit vector in the radial direction pointing from the center of the carousel to the walker at time ''t''.  The radial distance ''R''(''t'') increases steadily with time according to:
:<math>R(t) = s t \ ,</math>
with ''s'' the speed of walking. According to simple kinematics, the velocity is then the first derivative of the trajectory:
:<math>\boldsymbol{v}(t) = \frac {d}{dt} R(t) \left( \cos (\omega t + \pi/4),\ \sin (\omega t + \pi/4) \right) + \omega R(t) \left( -\sin((\omega t + \pi/4),\ \cos (\omega t + \pi/4)) \right) \  </math>
::<math>=\frac {d}{dt} R(t) \mathbf{u}_R + \omega R(t) \mathbf{u}_{\theta}\ , </math>
with '''u'''<sub>θ</sub> a unit vector perpendicular to '''u'''<sub>R</sub> at time ''t'' (as can be verified by noticing that the vector [[dot product]] with the radial vector is zero) and pointing in the direction of travel.
The acceleration is the first derivative of the velocity:
:<math>\boldsymbol {a}(t) = \frac{d^2}{dt^2} R(t)\left( \cos (\omega t + \pi/4),\ \sin (\omega t + \pi/4) \right) </math>&ensp;<math>+ 2 \frac {d}{dt} R(t) \omega \left( -\sin((\omega t + \pi/4),\ \cos (\omega t + \pi/4)) \right) </math>&ensp;<math>-\omega^2 R(t)\left( \cos (\omega t + \pi/4),\ \sin (\omega t + \pi/4) \right) \  </math>
::<math>=2s\ \omega \left( -\sin((\omega t + \pi/4),\ \cos (\omega t + \pi/4)) \right)</math>&ensp;<math>-\omega^2 R(t)\left( \cos (\omega t + \pi/4),\ \sin (\omega t + \pi/4) \right) \  </math>
::<math>=2s\ \omega \  \mathbf{u}_{\theta}-\omega^2 R(t)\ \mathbf{u}_R  \ . </math>
The last term in the acceleration is radially inward of magnitude ω<sup>2</sup> ''R'', which is therefore the instantaneous [[centripetal force|centripetal acceleration]] of [[circular motion]].<ref>{{Anchor|Note1}}'''Note''': There is a subtlety here: the distance ''R'' is the instantaneous distance from the rotational axis ''of the carousel''. However, it is not the [[osculating circle|radius of curvature]] ''of the walker's trajectory'' as seen by the inertial observer, and the unit vector '''u'''<sub>R</sub> is not perpendicular to the path. Thus, the designation "centripetal acceleration" is an approximate use of this term. See, for example, {{cite book |title=Orbital Mechanics for Engineering Students |author=Howard D. Curtis |isbn=0750661690 |publisher=Butterworth-Heinemann |year=2005 |pages=p. 5 |url=http://books.google.com/books?id=6aO9aGNBAgIC&pg=PA5&vq=curvature&dq=orbit+%22coordinate+system%22&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U0ucaLloFK3s0GXKHPQUMLxERK7fw }} and
{{cite book |title=Accelerator physics |author=S. Y. Lee |pages= p. 37 |url=http://books.google.com/books?id=VTc8Sdld5S8C&pg=PA37&vq=curvature&dq=orbit+%22coordinate+system%22&lr=&as_brr=0&source=gbs_search_s&sig=ACfU3U1PMaWWVEiZ1Mytg7hyUcIdfz1sFw
|isbn=981256182X |publisher=World Scientific |location=Hackensack NJ |edition=2nd Edition |year=2004 }}</ref>  The first term is perpendicular to the radial direction, and pointing in the direction of travel. Its magnitude is 2''s''ω, and it represents the acceleration of the walker as the edge of the carousel is neared, and the arc of circle traveled in a fixed time increases, as can be seen by the increased spacing between points for equal time steps on the spiral in Figure 5 as the outer edge of the carousel is approached.
 
Applying Newton's laws, multiplying the acceleration by the mass of the walker, the inertial observer concludes that the walker is subject to two forces: the inward, radially directed centripetal force, and another force perpendicular to the radial direction that is proportional to the speed of the walker.
 
====Rotating observer====
The rotating observer sees the walker travel a straight line from the center of the carousel to the periphery, as shown in Figure 5. Moreover, the rotating observer sees that the walker moves at a constant speed in the same direction, so applying Newton's law of inertia, there is ''zero'' force upon the walker. These conclusions do not agree with the inertial observer. To obtain agreement, the rotating observer has to introduce fictitious forces that appear to exist in the rotating world, even though there is no apparent reason for them, no apparent gravitational mass, electric charge or what have you, that could account for these fictitious forces.
 
To agree with the inertial observer, the forces applied to the walker must be exactly those found above. They can be related to the general formulas already derived, namely:
:<math>
\mathbf{F}_{\mathrm{fict}}  =
- 2 m \boldsymbol\Omega  \times \mathbf{v}_{B} - m \boldsymbol\Omega  \times (\boldsymbol\Omega \times \mathbf{x}_B ) </math>&ensp;<math>\ - m \frac{d \boldsymbol\Omega  }{dt} \times \mathbf{x}_B \ .
</math>
In this example, the velocity seen in the rotating frame is:
:<math>\boldsymbol{v}_B = s\  \mathbf{u}_R \ , </math>
with '''u'''<sub>R</sub> a unit vector in the radial direction. The position of the walker as seen on the carousel is:
:<math>\mathbf{x}_B = R(t)\mathbf{u}_R \ , </math>
and the time derivative of '''Ω''' is zero for uniform angular rotation. Noticing that
:<math>\boldsymbol\Omega  \times \mathbf{u}_R =\omega \mathbf{u}_{\theta} \ </math>
and
:<math>\boldsymbol\Omega  \times \mathbf{u}_{\theta} =-\omega \mathbf{u}_R \ ,</math>
we find:
:<math>
\mathbf{F}_{\mathrm{fict}}  = - 2 m \omega s\ \mathbf{u}_{\theta} + m \omega^2 R(t)  \mathbf{u}_R \ .
</math>
To obtain a [[straight-line motion]] in the rotating world, a force exactly opposite in sign to the fictitious force must be applied to reduce the net force on the walker to zero, so Newton's law of inertia will predict a straight line motion, in agreement with what the rotating observer sees. The fictitious forces that must be combated are the [[Coriolis force]] (first term) and the [[centrifugal force]] (second term). (These terms are approximate.<ref>A circle about the axis of rotation is not the [[osculating circle]] of the walker's trajectory, so "centrifugal" and "Coriolis" are approximate uses for these terms. [[#Note1|See note]].</ref>)  By applying forces to counter these two fictitious forces, the rotating observer ends up applying exactly the same forces upon the walker that the inertial observer predicted were needed.
 
Because they differ only by the constant walking velocity, the walker and the rotational observer see the same accelerations. From the walker's perspective, the fictitious force is experienced as real, and combating this force is necessary to stay on a straight line radial path holding constant speed. It's like battling a crosswind while being thrown to the edge of the carousel.
 
===Observation===
Notice that this [[kinematics|kinematical]] discussion does not delve into the mechanism by which the required forces are generated. That is the subject of [[kinetics (physics)|kinetics]]. In the case of the carousel, the kinetic discussion would involve perhaps a study of the walker's shoes and the friction they need to generate against the floor of the carousel, or perhaps the dynamics of skateboarding, if the walker switched to travel by skateboard. Whatever the means of travel across the carousel, the forces calculated above must be realized. A very rough analogy is heating your house: you must have a certain temperature to be comfortable, but whether you heat by burning gas or by burning coal is another problem. Kinematics sets the thermostat, kinetics fires the furnace.
 
 
==References==
<references/>

Latest revision as of 04:07, 22 November 2023


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Quarks

The quarks that may engage one another in reactions are determined by the Cabbibo-Kobayashi-Maskawa matrix:Morii

The quarks can be arranged to exhibit right- and left-handedness, subscripts L and R, to resemble the leptons. The right-handed quarks do not couple to the weak interaction, and are labeled with subscript R. The left-handed quarks corresponding to these right-handed quarks are mixtures of quarks. Thus, the up and down quarks are assembled as:

The other generations are arranged similarly:

Right- and left-handed quarks
Symbol Electric charge, Q Weak isospin, (IW, IW3) Weak hypercharge, (YW)
uL, cL, tL +2/3 (1/2, +1/2) +1/3
dL, sL, bL −1/3 (1/2, −1/2) +1/3
uR, cR, tR +2/3 0 +4/3
dR, sR, bR −1/3 0 −2/3


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