Editing Rotating reference frame (section)

=== 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 a [[non-inertial reference frame]], rather than from any physical interaction between bodies.

Using [[Newton's laws of motion|Newton's second law of motion]] <math>\mathbf{F}=m\mathbf{a},</math> we obtain:<ref name=Arnold/><ref name=Lanczos/><ref name=Taylor/><ref name=Landau>{{cite book |title=Mechanics |author=LD Landau |author2=LM Lifshitz |name-list-style=amp |page= 128 |url=https://books.google.com/books?id=e-xASAehg1sC&pg=PA40 |edition=Third |date=1976 |publisher=Butterworth-Heinemann |isbn=978-0-7506-2896-9}}</ref><ref name=Hand/>

* the [[Coriolis force]] <math display="block">
\mathbf{F}_{\mathrm{Coriolis}} = 
-2m \boldsymbol\Omega \times \mathbf{v}_{\mathrm{r}}
</math>
* the [[centrifugal force (fictitious)|centrifugal force]] <math display="block">
\mathbf{F}_{\mathrm{centrifugal}} = 
-m\boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{r})
</math>
* and the [[Euler force]] <math display="block">
\mathbf{F}_{\mathrm{Euler}} = 
-m\frac{\mathrm{d}\boldsymbol\Omega}{\mathrm{d}t} \times \mathbf{r}
</math>
where <math>m</math> 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 <math>\boldsymbol{\Omega} = 0 \ . </math>

For completeness, the inertial acceleration <math>\mathbf{a}_{\mathrm{i}}</math> due to impressed external forces <math>\mathbf{F}_{\mathrm{imp}}</math> can be determined from the total physical force in the inertial (non-rotating) frame (for example, force from physical interactions such as [[Electromagnetism|electromagnetic forces]]) using [[Newton's laws of motion|Newton's second law]] in the inertial frame:
<math display="block">
\mathbf{F}_{\mathrm{imp}} = m \mathbf{a}_{\mathrm{i}}
</math>
Newton's law in the rotating frame then becomes
::<math>\mathbf{F_{\mathrm{r}}} = \mathbf{F}_{\mathrm{imp}} + \mathbf{F}_{\mathrm{centrifugal}} +\mathbf{F}_{\mathrm{Coriolis}} + \mathbf{F}_{\mathrm{Euler}} = m\mathbf{a_{\mathrm{r}}} \ . </math>
In other words, to handle the laws of motion in a rotating reference frame:<ref name=Hand>{{cite book |title=Analytical Mechanics |author =Louis N. Hand |author2 =Janet D. Finch |page=267 |url=https://books.google.com/books?id=1J2hzvX2Xh8C&q=Hand+inauthor:Finch&pg=PA267
|isbn=0-521-57572-9 |publisher=[[Cambridge University Press]] |date=1998 }}</ref><ref name=Pui>{{cite book |title=Mechanics |author=HS Hans |author2=SP Pui |name-list-style=amp |page=341 |url=https://books.google.com/books?id=mgVW00YV3zAC&q=inertial+force+%22rotating+frame%22&pg=PA341 |isbn=0-07-047360-9 |publisher=Tata McGraw-Hill |date=2003  }}</ref><ref name=Taylor2>{{cite book |title=Classical Mechanics |author=John R Taylor |page= 328 |publisher=University Science Books |isbn=1-891389-22-X |date=2005 |url=https://books.google.com/books?id=P1kCtNr-pJsC&pg=PP1}}</ref>
{{Quotation|Treat the fictitious forces like real forces, and pretend you are in an inertial frame.|Louis N. Hand, Janet D. Finch ''Analytical Mechanics'', p. 267}}
{{Quotation|Obviously, a rotating frame of reference is a case of a non-inertial frame. Thus the particle in addition to the real force is acted upon by a fictitious force...The particle will move according to Newton's second law of motion if the total force acting on it is taken as the sum of the real and fictitious forces.|HS Hans & SP Pui: ''Mechanics''; p. 341}}
{{Quotation|This equation has exactly the form of Newton's second law, ''except'' that in addition to '''F''', the sum of all forces identified in the inertial frame, there is an extra term on the right...This means we can continue to use Newton's second law in the noninertial frame ''provided'' we agree that in the noninertial frame we must add an extra force-like term, often called the '''inertial force'''. |John R. Taylor: ''Classical Mechanics''; p. 328}}