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Le sommatorie sono nulle in quanto i versori non hanno dipendenza dal tempo
Per esplicitare <math>\mathbf{a}_{AB}</math>, si consideri che l'origine del sistema ''B'' è posta rispetto ad ''A'' in:
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>
leadingda toquesto ala velocityvelocità of the origin ofdell'origine framedi ''B'' asvale:
:<math>\mathbf{v}_{AB} = \frac{d}{dt} \mathbf{X}_{AB} = \mathbf{\Omega \times X}_{AB} \ , </math>
infine l'accelerazione dell'origine di ''B'' vale:
leading to an acceleration of the origin of ''B'' given by:
:<math>\mathbf{a}_{AB} = \frac{d^2}{dt^2} \mathbf{X}_{AB} </math> <math>= \mathbf{ \Omega \ \times } \left( \mathbf{ \Omega \times X}_{AB}\right) </math> <math>= - \omega^2 \mathbf{X}_{AB} \ .</math>
Quindi nel sistema di riferimento ''B'' deve essere introdotta una forza apparente, che ' diretta radialmente in fuori dal centro di rotazione:
Because the first term, which is
::::<math>\mathbf{ F}_{\Omegamathrm{app}} \ \times }= m \left(omega^2 \mathbf{ \Omega \times X}_{AB}\right) \ , </math>
di ampiezza:
is of the same form as the normal centrifugal force expression:
::::<math>|\boldsymbolmathbf{F}_{\Omegamathrm{app}}| \times \left(= m \boldsymbol{\Omega}omega^2 \timesR \mathbf{x}_B \right)\. ,</math>
Nel caso del sistema ruotante la forza centrifuga dipendeva dalla distanza dall'origine di
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:
''B'', nel sitema ruotante dipende dalla distanza del centro di ''B'' dal suo centro di rotazione. Quindi oggetti diversi che si trovano in ''B'' sentono la stessa forza centrifuga.
:<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 |page=307 |url=http://books.google.com/books?id=VS7aS8QS91oC&pg=PA307&dq=centrifugal+revolution+and+rotation+date:1970-2009|isbn=1-4067-4574-X |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> <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> <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> <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> <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> <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=90-5809-361-1 |year=2002 |publisher=Taylor and Francis |page=82 |url=http://books.google.com/books?id=MJkz_IBZZS0C&printsec=frontcover&dq=centrifuge#PPT102,M1 }}</ref><ref name=Nen>{{cite book |title=Consolidation of Soils: Testing and Evaluation: a Symposium |author=Raymond Nen |isbn=0-8031-0446-4 |year=1986 |publisher=ASTM International |page=590 |url=http://books.google.com/books?id=a-BKqGTXA6kC&pg=PA590&dq=radius+centrifuge+effective}}</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 |page=276 |url=http://books.google.com/books?id=YO0KAAAAYAAJ&pg=PA276&dq=rotation+revolution+%22centrifugal+force%22}}</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=0-7506-7403-2 |url=http://books.google.com/books?id=fUBPN8T9bwUC&pg=PA61&dq=spheres+rotating++Coriolis#PPA60,M1
|pages=60–61 }}, and {{cite book |title=Physics for Scientists And Engineers With Modern Physics |author=Douglas C. Giancoli |page=301 |isbn=0-13-149508-9 |year=2007 |publisher=Pearson Prentice-Hall |url=http://books.google.com/books?id=xz-UEdtRmzkC&pg=PA301&dq=spheres+rotating++Coriolis#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>\mathbf{r}(t) =R(t)\mathbf{u}_R = \begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = \begin{bmatrix} R(t)\cos (\omega t + \pi/4) \\ R(t)\sin (\omega t + \pi/4) \end{bmatrix}, </math>
where the added π/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>\mathbf{v}(t) = \frac{dR}{dt} \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix} + \omega R(t) \begin{bmatrix} -\sin(\omega t + \pi/4) \\ \cos (\omega t + \pi/4) \end{bmatrix}</math>
::<math>=\frac{dR}{dt} \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>\mathbf{a}(t) = \frac{d^2 R}{dt^2} \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix} + 2 \frac {dR}{dt} \omega \begin{bmatrix} -\sin(\omega t + \pi/4) \\ \cos (\omega t + \pi/4) \end{bmatrix} - \omega^2 R(t) \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix}</math>
::<math>=2s\omega \begin{bmatrix} -\sin(\omega t + \pi/4) \\ \cos (\omega t + \pi/4) \end{bmatrix} -\omega^2 R(t) \begin{bmatrix} \cos (\omega t + \pi/4) \\ \sin (\omega t + \pi/4) \end{bmatrix}</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=0-7506-6169-0 |publisher=Butterworth-Heinemann |year=2005 |page=5 |url=http://books.google.com/books?id=6aO9aGNBAgIC&pg=PA5&vq=curvature&dq=orbit+%22coordinate+system%22}} and
{{cite book |title=Accelerator physics |author=S. Y. Lee |page= 37 |url=http://books.google.com/books?id=VTc8Sdld5S8C&pg=PA37&vq=curvature&dq=orbit+%22coordinate+system%22|isbn=981-256-182-X |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}_\mathrm{B} - m \boldsymbol\Omega \times (\boldsymbol\Omega \times \mathbf{x}_\mathrm{B} ) - m \frac{d \boldsymbol\Omega}{dt} \times \mathbf{x}_\mathrm{B}.
</math>
In this example, the velocity seen in the rotating frame is:
:<math>\mathbf{v}_\mathrm{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}_\mathrm{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.
== See also ==
{{Col-begin}}
{{Col-1-of-3}}
* [[Newton's laws of motion]]
* [[inertial reference frame]]
* [[non-inertial reference frame]]
* [[rotating reference frame]]
* [[Coriolis force]]
* [[centrifugal force (fictitious)|Centrifugal force]]
* [[Gravity]]
* [[General relativity]]
* [[d'Alembert's principle]] of inertial forces
{{Col-2-of-3}}
* [[Centripetal force]]
* [[Circular motion]]
* [[Uniform circular motion]]
*[[Statics]]
*[[Kinetics (physics)]]
*[[Kinematics]]
*[[Applied mechanics]]
*[[Analytical mechanics]]
*[[Dynamics (physics)]]
{{Col-3-of-3}}
*[[Classical mechanics]]
*[[Generalized force]]
*[[Free motion equation]]
*[[Orthogonal coordinates]]
*[[Curvilinear coordinates]]
*[[Generalized coordinates]]
*[[Frenet-Serret formulas]]
{{Col-end}}
{{Portal|Physics}}
==Notes==
{{Reflist|2}}
==Further reading==
* {{cite book |author=[[Lev D. Landau]] and E. M. Lifshitz|isbn=0-7506-2896-0|edition=3rd Edition |year=1976|title=Mechanics |volume=Vol. 1 |series=[[Course of Theoretical Physics]] |url=http://books.google.com/books?id=LmAV8q_OOOgC&printsec=frontcover&dq=inauthor:lifshitz|publisher= Butterworth-Heinenan|pages= 128–130}}
* {{cite book |author=Keith Symon|year=1971|title=Mechanics|publisher= Addison-Wesley|edition=3rd Edition|isbn=0-201-07392-7}}
* {{cite book |author=Jerry B. Marion |year=1970|title=Classical Dynamics of Particles and Systems|publisher= Academic Press |isbn=0-12-472252-0 }}
* {{cite book|author=Marcel J. Sidi |title=Spacecraft Dynamics and Control: A Practical Engineering Approach |isbn=0-521-78780-7 |year=1997 |publisher=Cambridge University Press |url=http://books.google.com/books?id=xQpZJMtDehQC&pg=RA1-PA101&dq=orbit+%22coordinate+system%22#PRA1-PA88,M1 |nopp=true |pages=Chapter 4.8 }}
<!-- The text referencing these must have been deleted. Kept as a comment in case somebody knows where they are relevant:
# {{note|kandk2}} Kleppner, pages 355-360.
# {{note|fetter}} Alexander Fetter and John Walecka, ''Theoretical Mechanics of particles and continua'', McGraw-Hill, pages 33-39.
# {{note|landau}} [[Lev D. Landau]] and E. M. Lifshitz, (1976) ''Mechanics'', Butterworth-Heinenan, pages 128-130.
#Jerry B. Marion, (1970), ''Classical Dynamics of Particles and Systems'', Academic Press.
#Keith Symon, (1971), ''Mechanics'', Addison-Wesley
-->
==External links==
* [http://www.hcc.hawaii.edu/~rickb/SciColumns/FictForce.04Feb96.html Q and A from Richard C. Brill, Honolulu Community College]
* [http://www-istp.gsfc.nasa.gov/stargaze/Sframes2.htm NASA's David Stern: Lesson Plans for Teachers #23 on ''Inertial Forces'']
* [http://scienceworld.wolfram.com/physics/CoriolisForce.html Coriolis Force]
* [http://mensch.org/physlets/merry.html Motion over a flat surface] Java physlet by Brian Fiedler illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and from a non-rotating point of view.
* [http://mensch.org/physlets/inosc.html Motion over a parabolic surface] Java physlet by Brian Fiedler illustrating fictitious forces. The physlet shows both the perspective as seen from a rotating and as seen from a non-rotating point of view.
{{DEFAULTSORT:Fictitious Force}}
[[Category:Classical mechanics]]
[[Category:Force]]
[[Category:Fictitious forces| ]]
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