Editing Coriolis force (section)

==Formula==
{{See also|Fictitious force}}

In [[Newtonian mechanics]], the equation of motion for an object in an [[Inertial frame of reference|inertial reference frame]] is:
<math display="block">\mathbf{F} = m\mathbf{a}</math>
where <math>\mathbf F</math> is the vector sum of the physical forces acting on the object, <math>m</math> is the mass of the object, and <math>\mathbf a</math> is the acceleration of the object relative to the inertial reference frame.

Transforming this equation to a [[Rotating reference frame|reference frame rotating]] about a fixed axis through the origin with [[angular velocity]] <math> \boldsymbol{\omega} </math> having variable rotation rate, the equation takes the form:<ref name="Persson1998"/><ref>{{cite book |title=Classical Mechanics: With introduction to Nonlinear Oscillations and Chaos |last=Bhatia |first=V.B. |publisher=Narosa Publishing House |year=1997 |page=201 |isbn=978-81-7319-105-3}}</ref>
<math display="block">\begin{align}
\mathbf{F'} &=
\mathbf{F} - m\frac{\mathrm{d} \boldsymbol{\omega}}{\mathrm{d}t}\times\mathbf{r}' - 2m \boldsymbol{\omega}\times \mathbf{v}' - m\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}')  \\
& = m\mathbf{a}'
\end{align}</math>
where the ''prime'' (') variables denote coordinates of the rotating reference frame (not a derivative) and:
* <math>\mathbf F</math> is the vector sum of the physical forces acting on the object
* <math> \boldsymbol{\omega} </math> is the [[angular velocity]] of the rotating reference frame relative to the inertial frame
* <math>\mathbf r '</math> is the position vector of the object relative to the rotating reference frame
* <math>\mathbf v '</math> is the velocity of the object relative to the rotating reference frame
* <math> \mathbf a '</math> is the acceleration of the object relative to the rotating reference frame

The fictitious forces as they are perceived in the rotating frame act as additional forces that contribute to the apparent acceleration just like the real external forces.<ref name=Silverman>
{{cite book |title=A Universe of Atoms, an Atom in the Universe|author=Silverman, Mark P.|url=https://books.google.com/books?id=-Er5pIsYe_AC&pg=PA249| page=249| isbn=9780387954370| year=2002| location = Berlin, Germany | publisher=Springer| edition=2nd}}</ref><ref>Taylor (2005). p. 329.</ref><ref>{{cite book |last1=Lee |first1=Choonkyu |last2=Min |first2=Hyunsoo |title=Essential Classical Mechanics |date=17 April 2018 |publisher=World Scientific Publishing Company |isbn=978-981-323-466-6 |url=https://books.google.com/books?id=_WdhDwAAQBAJ&q=coriolis+frame+of+reference&pg=PT506 |access-date=13 March 2021 |language=en}}</ref> The fictitious force terms of the equation are, reading from left to right:<ref name=Lanczos_A>
{{cite book|url=https://books.google.com/books?id=ZWoYYr8wk2IC&pg=PA103|title=The Variational Principles of Mechanics |author=Lanczos, Cornelius |year=1986 |orig-year = 1970 | isbn=9780486650678 | location = Mineola, NY |publisher=Dover Publications|edition=4th (reprint) |page=Chapter 4, §5 |no-pp=true}}</ref>
* [[Euler force]], <math>-m \frac{\mathrm{d}\boldsymbol{\omega}}{\mathrm{d}t} \times\mathbf{r}' </math>
* Coriolis force, <math>-2m ( \boldsymbol{\omega} \times \mathbf{v}') </math>
* [[centrifugal force]], <math>-m\boldsymbol{\omega} \times (\boldsymbol{\omega} \times \mathbf{r}')
</math>

As seen in these formulas the Euler and centrifugal forces depend on the position vector <math>\mathbf r '</math> of the object, while the Coriolis force depends on the object's velocity <math>\mathbf v '</math> as measured in the rotating reference frame. As expected, for a non-rotating [[inertial frame of reference]] <math>(\boldsymbol\omega=0)</math> the Coriolis force and all other fictitious forces disappear.<ref name=Tavel>{{cite book|title=Contemporary Physics and the Limits of Knowledge|page=93 |quote=Finally, noninertial forces, like centrifugal and Coriolis forces, can be eliminated by jumping into a reference frame that moves with constant velocity, the frame that Newton called inertial.|author=Tavel, Morton |url=https://books.google.com/books?id=SELS0HbIhjYC&q=Einstein+equivalence+laws+physics+frame&pg=PA95 |isbn=9780813530772 | location = New Brunswick, NJ | publisher=[[Rutgers University Press]] |year=2002}}</ref>

===Direction of Coriolis force for simple cases===

As the Coriolis force is proportional to a [[cross product]] of two vectors, it is perpendicular to both vectors, in this case the object's velocity and the frame's rotation vector. It therefore follows that:
* if the velocity is parallel to the rotation axis, the Coriolis force is zero. For example, on Earth, this situation occurs for a body at the equator moving north or south relative to the Earth's surface. (At any latitude other than the equator, however, the north–south motion would have a component perpendicular to the rotation axis and a force specified by the ''inward'' or ''outward'' cases mentioned below).
* if the velocity is straight inward to the axis, the Coriolis force is in the direction of local rotation. For example, on Earth, this situation occurs for a body at the equator falling downward, as in the Dechales illustration above, where the falling ball travels further to the east than does the tower. Note also that heading north in the northern hemisphere would have a velocity component toward the rotation axis, resulting in a Coriolis force to the east (more pronounced the further north one is).
* if the velocity is straight outward from the axis, the Coriolis force is against the direction of local rotation. In the tower example, a ball launched upward would move toward the west.
* if the velocity is in the direction of rotation, the Coriolis force is outward from the axis. For example, on Earth, this situation occurs for a body at the equator moving east relative to Earth's surface. It would move upward as seen by an observer on the surface. This effect (see Eötvös effect below) was discussed by Galileo Galilei in 1632 and by Riccioli in 1651.<ref name=Graney2015>{{cite book| last=Graney|first=Christopher M.| title=Setting Aside All Authority: Giovanni Battista Riccioli and the Science Against Copernicus in the Age of Galileo| year=2015|publisher=University of Notre Dame Press|location=Notre Dame, IN| pages=115–125| url=https://books.google.com/books?id=6r_nrQEACAAJ| isbn=9780268029883}}</ref>
* if the velocity is against the direction of rotation, the Coriolis force is inward to the axis. For example, on Earth, this situation occurs for a body at the equator moving west, which would deflect downward as seen by an observer.