ABSTRACT

The research work examines the existence and linear stability of equilibrium points in the perturbed Robe’s circular restricted three-body problem under the assumption that the hydrostatic equilibrium figure of the first primary is an oblate spheroid. The problem is perturbed in the sense that small perturbations given to the Coriolis and centrifugal forces are being considered. Results of the analysis found two axial equilibrium points on the line joining the centre of both primaries. It is further observed that under certain conditions, points on the circle within the first primary are also equilibrium points. And a special case where the density of the fluid and that of the infinitesimal mass are equal (D = 0) is discussed. The linear stability of this configuration is examined; it is observed that the axial equilibrium points ?1,0,0 and ?11+?2 ,0,0 are conditionally stable, while the circular points 1+r?os?,??in?,0 are unstable.

TABLE OF CONTENTS

Cover page ………………………………………………………………………………………………… i
Fly leaf …………………………………………………………………………………………………….. ii
Title page …………………………………………………………………………………………………. ii
Declaration ………………………………………………………………………………………………. iii
Certification …………………………………………………………………………………………….. iv
Acknowledgement ……………………………………………………………………………………… v
Dedication ……………………………………………………………………………………………….. vi
Abstract ………………………………………………………………………………………………….. vii
Table of Content ……………………………………………………………………………………… viii
List of Figures ………………………………………………………………………………………….. xi
List of Table …………………………………………………………………………………………….. xi
List of Notations ……………………………………………………………………………………….. xi
CHAPTER ONE: GENERAL INTRODUCTION ……………………………………….. 1
1.1 Introduction ………………………………………………………………………………………. 1
1.2 Statement of the Problem ……………………………………………………………………… 2
1.3 Significance / Justification of the Study ………………………………………………… 2
1.4 Research Methodology ………………………………………………………………………… 3
1.5 Aim and Objectives of the Study ……………………………………………………………. 3
1.6 Preliminary Ideas ……………………………………………………………………………… 4
1.6.1 Circular restricted three-body problem …………………………………………………. 4
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1.6.2 Equations of motion of two-body problem ……………………………………………. 5
1.6.3 Equations of relative motion ………………………………………………………………. 6
1.7 Theoretical Framework ………………………………………………………………………. 7
1.7.1 The buoyancy force ………………………………………………………………………….. 7
1.7.2 Robe’s restricted three-body problem …………………………………………………. 14
1.7.3 Perturbation …………………………………………………………………………………… 16
1.7.4 Coriolis and centrifugal forces…………………………………………………………… 17
1.7.5 Oblate spheroid ………………………………………………………………………………. 18
1.7.6 Moment of Inertia …………………………………………………………………………… 18
1.8 Linear Stability ………………………………………………………………………………… 21
CHAPTER TWO: LITERATURE REVIEW ……………………………………………. 22
CHAPTER THREE: EQUATIONS OF MOTION …………………………………….. 29
3.1 Introduction …………………………………………………………………………………… 29
3.2. Mathematical Formulations of the Problems ……………………………………….. 29
3.3 The Case (??=??) ……………………………………………………………………………. 32
3.3 Conclusion …………………………………………………………………………………….. 34
CHAPTER FOUR: LOCATIONS AND LINEAR STABILITY OF EQUILIBRIUM POINTS………………………………………………………………………… 35
4.1 Introduction ……………………………………………………………………………………. 35
4.2 The Equilibrium Points ………………………………………………………………………. 35
4.2.1 Axial equilibrium points………………………………………………………………….. 36
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4.2.2 Locations of circular points ……………………………………………………………. 45
4.3 Linear Stability of The Equilibrium Points …………………………………………….. 46
4.3.1 Variational equations ……………………………………………………………………… 47
4.3.2 Stability of axial equilibrium point (?1,0,0) ……………………………………… 50
4.3.2.1 Critical Mass ……………………………………………………………………………… 65
4.3.2.2 Range of stability ……………………………………………………………………….. 78
4.3.3 Stability of the axial equilibrium point (?11 +?2 ,0,0) ………………………… 82
4.3.4 Stability of the circular equilibrium point ……………………………………………. 91
4.4 Conclusion …………………………………………………………………………………….. 97
CHAPTER FIVE: SUMMARY, CONCLUSION AND RECOMMENDATIONS …………………………………………………………………………. 99
5.1 Introduction …………………………………………………………………………………… 99
5.2 Summary ………………………………………………………………………………………. 99
5.3 Conclusion ……………………………………………………………………………………. 100
5.4 Recommendations ………………………………………………………………………….. 100
REFERENCES …………………………………………………………………………………….. 101
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LIST OF FIGURES
Figure 1.1 A cylinder of height h and radius R fully immersed in a fluid …………….. 8
Figure 1.2 The Robe’s restricted three-body problem ……………………………………… 15
LIST OF TABLE
Table 1: numerical computation of the critical mass ?? …………………………………… 81
LIST OF NOTATIONS
?1 mass of the first primary…………………………………….…………………29
?2 mass of the second primary ………………………………………………………………. 29
?3 mass of the infinitesimal body …………………………………………………………… 29
?1 density of the fluid …………………………………………………………………………… 29
?3 density of the infinitesimal body ………………………………………………………… 29
? oblateness coefficient……………………………………….…………………30
? mean motion…………………………………………………………. ……….30
? potential due to the first primary………………….………………………….. 30
?’ potential due to the second primary…………………………………. ………30
V potential that explain the combine force upon the infinitesimal mass ………….. 30
ϕ perturbation in Coriolis force ………………………………………………………………. 30
ѱ perturbation in centrifugal force…………………………………………………………… 30
? mass ratio ………………………………………………………………………………………… 30
ξ, η, ζ, small displacements ……………………………………………………………………….. 48
? constant …………………………………………………………………………………………… 50
Δ discriminant…………………………………………………………………….

CHAPTER ONE

GENERAL INTRODUCTION
1.1 Introduction
In the scientific exploration of space by man, space dynamics is very important and this later rely on celestial mechanics- which is a branch of science devoted to the movement of artificial celestial bodies such as artificial satellite, while space dynamics is a branch of astronomy which deals with the nature and motions of stellar phenomena such as planets, asteroids and stars. The most celebrated problem of space dynamics is the problem of three bodies, known as the three-body problem (3BP). This is defined in terms of three bodies with arbitrary masses and free to move in space attracting one another according to Newton’s law of gravitation. An example of 3BP is the Sun-Earth-Moon system, when they are considered as point masses; they form the main problem of the lunar theory. (Singh, J 2012) The R3BP describes the motion of an infinitesimal mass moving under the gravitational effects of two finite masses, called primaries, which move in circular orbits around the center of mass on the account of their mutual attraction and the infinitesimal mass not influencing the motion of the primaries. The R3BPs have produced many significant results by well-known mathematicians and scientists in an attempt to understand and predict the motion of natural bodies. A system made up of the Sun and Jupiter as primaries and Trojan asteroid assuming the role of the infinitesimal mass in the Sun-Jupiter system is a typical example of the R3BP.
A special case of R3BP referred to as Robe’s problem was introduced by Robe (1977), where the infinitesimal mass is embedded in the first primary which is filled with
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homogenous incompressible fluid of known density, the second primary is a point mass located outside an orbiting shell. The infinitesimal mass is under the gravitational attraction of the primaries as well as buoyancy force due to the fluid. Robe’s models provides an insight into the study of the oscillations of the Earth’s inner core taking into cognizance the attraction of the Moon; and the motion of an artificial Earth satellite located inside another satellite. This Robe’s problem has been varied by the introduction of perturbing forces by many researchers like Hallan and Rana (2001b), Hallan and Mangang (2007), Singh and Sandah (2012), Singh and Mohammed (2013) to mention a few.
1.2 Statement of the Problem
Hallan and Mangang (2007) studied the existence and linear stability of equilibrium points in the Robe’s circular restricted three-body problem. They assumed one of the primaries of mass ?1 to be a rigid spherical shell filled with homogenous, incompressible fluid of density ?1 in the shape of an oblate spheroid, the second primary ?2 is a point mass located outside the shell and moving around the mass ?1 in a Keplerian orbit; the infinitesimal mass ?3 is a small solid sphere of density ?3 moving inside the shell and is subject to the attraction of ?2 and the buoyancy force due to the fluid of the first primary. It will be interesting to see what happens when perturbations are included in their study. In this present thesis, we examine the effect of small perturbations in Coriolis and centrifugal forces on the existence and linear stability of the equilibrium points. Thus our study will be a generalization of Hallan and Mangang (2007).
1.3 Significance / Justification of the Study
In reality, we found that several heavenly bodies are sufficiently oblate. Earth, Jupiter, Saturn and Ragulus are oblate. The participating bodies in the classical R3BP are assumed to be strictly spherical in shape. The minor planets and meteoroids have
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irregular shapes. Large perturbations from the two-body orbit are caused by the lack of sphericity, oblateness or triaxiality of the celestial bodies; an example of this is the motion of artificial Earth satellites. In recent time, many perturbing forces like oblateness, triaxiality, variation of masses for the primaries and of the infinitesimal mass etc, have been included in the study of the existence and linear stability of the equilibrium points in the Robe’s circular restricted three-body problem.
To this end, we modify the model by considering the effect of small perturbations in the Coriolis and centrifugal forces on the Robe’s circular restricted three-body when the first primary is an oblate spheroid. Taking into account the full buoyancy force and other forces acting on the infinitesimal body when the density parameter is not zero (?.?.,?1≠?3). The Robe’s model under consideration can be applied in space mission design; it also has enormous applications in various astronomical problems. Examples include small oscillations of the earth’s core in the gravitational field of Earth-Moon system.
1.4 Research Methodology
We adopt a rotating coordinate system to derive the equations of motion of the infinitesimal body with respect to the primaries as given in chapter three.
1.5 Aim and Objectives of the Study
The research work is aimed at investigating the motion of an infinitesimal mass in the perturbed Robe’s circular restricted three body problem under an oblate spheroid. In view of the problem stated above, we therefore set the following objectives:
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(a) To derive the equations of motion of the infinitesimal mass
(b) To determine the locations of the axial equilibrium points
(c) To examine the linear stability of the axial equilibrium points
(d) To investigate the model when the density parameter is zero (?.?.,?1=?3)
1.6 Preliminary Ideas
1.6.1 Circular restricted three-body problem
The circular restricted three-body problem (CR3BP) describes the motion of a body of infinitesimal mass under the gravitational influence of two massive bodies, called the primaries, which orbit about their common centre of mass in circular orbits on account of their mutual attraction. The usefulness of studying this problem was verified by the discovery of the Trojan asteroids at the stable Lagrange points of the Sun-Jupiter orbital system.
The classical restricted three-body problem possesses five equilibrium points: three collinear points ?1,2,3 and two triangular points ?4,5, where the gravitational and centrifugal forces just balance each other. The collinear points are unstable, while the triangular points are stable for the mass ratio ?<0.03852… (Szebehely, 1967a). Their stability occurs in spite that the potential energy has a maximum rather than a minimum at the latter points. The stability is actually achieved through the influence of the Coriolis force, because the coordinate system is rotating (Wintner, 1941; Contopoulus, 2012). These equilibrium points are of great astronomical importance. The Solar and Heliospheric observatory (SOHO) and WMAP launched by NASA are in operation at the Sun-Earth ?1and ?2 respectively. The Laser Interferometer Space Antenna (LISA) pathfinder is planned to go to ?1.
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1.6.2 Equations of motion of two-body problem
In the circular restricted three-body problem, the motion of the primaries is governed by the equations of motion of two-body problem with constant masses. Let ?1 ??? ?2 be the masses of the two primaries, 0 be a fixed point in the space of the motion. Let ? 1 ??? ? 2 be their respective vectors from 0 to the respective masses and ? be the radius vector between the bodies. Then, the force of attraction on the particle of mass ?1 due to another particle of mass ?2 by Newton’s law of gravitation as in Singh and Sandah (2012) is given as ? 1=??1?2?2? ? (1.1)
and the force acting on the particle of mass ?2 due to the particle of mass ?1 is ? 2=−??1?2?2? ? (1.2)
Where G is the gravitational constant and ? is the distance between the primaries. By Newton’s second law ? 1=?2? 1??2?1, ? 2=?2? 2??2?2. (1.3) Putting the equations of system (1.3) in Eq. (1.1) and Eq. (1.2), we get ?2? 1??2?1=??1?2?2? ?, ?2? 2??2?2=−??1?2?2? ? or ?1 =??2?2? ?, ?2 =−??1?2? ? ????? ? =? 2−? 1 (1.4)
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Equations of system (1.4) are the vectorial equations of motion of the bodies of masses ?1 ??? ?2.
1.6.3 Equations of relative motion
We assume the velocity of the bodies ?1 ??? ?2 to be ? 1 ??? ? 2 respectively, and ? R be the velocity vector of the body ?2 ???? ??????? ?? ?1.
Then, ? R= ? 2 −? (1.5) Now the vectorial equations of system (1.4) take the form ?? 1??=??2?2? ? , (1.6) ?? 2??=−??1?2? ?. (1.7) Subtracting equation (1.6) from equation (1.7), we get ?? 2??− ?? 1??=−??1?2? ?−??2?2? ?=−? ?1+ ?2 ?2? ? or ???(? 2 –? 1) =−? ?1+ ?2 ?2? ? Using equation (1.5), we have ?? ???=−? ?1+ ?2 ?2? ? (1.8) ??. 1.8 ??????? ?? ???=−??2? ? , ????? ?=? ?1+ ?2 And removing the suffix, we have
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?? ??=−??2? ? (1.9)
Equation (1.9) is the relative motion of the body of mass ?2 with respect to the body of mass ?1, where r is the distance between the bodies.
1.7 Theoretical Framework
The theoretical framework upon which the derivations and finding of this research work is built on, is given in this section
1.7.1 The buoyancy force
The force exerted on an object that is completely or partly immersed in a fluid is known as buoyancy force. It acts upward, generally opposite to the direction of the frame of reference acceleration and its magnitude is equal to the weight of the fluid displaced by the submerged object. This force is caused by the difference between the pressure at the top of the object, which pushes downward and the pressure at the bottom of the object which pushes upward; every submerged object feels upward buoyancy force, because the pressure at the bottom of the object is always greater than the pressure at the top.
Consider a small particle of density d and volume V, falling with a velocity ? in a fluid of density ?1, the force acting on this particle as in Singh and Sandah ( 2012) are given as
(i) The gravitational force ??
(ii) The buoyancy force, given by
??=?1?g (1.10) where, g is the gravity of the fluid.
The pressure distribution on a submerged particle is statistically equivalent to a single point force ?? then acts at the location called the center of pressure (center of the mass of
8
the fluid displaced). Equation (1.10) is derived by relating hydrostatic distribution to an equivalent force – the buoyancy force (??). Now, consider a completely submerged cylinder of length l and radius R oriented vertically in the fluid basin with depth L (see figure 1.1 below)
Figure 1.1 A cylinder of height h and radius R fully immersed in a fluid
The distance between the top surface of the cylinder and the below surface fluid is ?1, the thrust at the top and bottom of the cylinder are ? T and ? B respectively acting in opposite direction. Now, from the relation ????????=????????? (1.11) The force acting on the top of the surface due to hydrostatics is given by ? ??? =− ? ???? ?? (1.12) where AT is the area of the top of the cylinder.
9
Now, the equation of pressure field in hydrostatics is given by ????=−? (1.13) here, p is a constant for an incompressible fluid, by integrating the last equation above with respect to z, we have
?=−???+?
At ?=?, ?=?0 ?? ???? ?=?0+??? ??? ? ? =?0+?? ?−? . Thus ???1=− ?????1 (1.14)
Evaluating equation (1.14) at ?=?−?1, we immediately have
?=?0+??(?−?+?1)
Or ?=?0+???1 (1.15) Substituting equation (1.15) in (1.12), we have ? ??? =− ? ? ?0+???1 ?? ?? or ? ??? =−? ? ?0+???1 ?? ??
????? ?? ?? is actually the cross sectional area of the cylinder, therefore the force acting at the top of the solid is
10
? ??? =−? ???2 ?0+???1 (1.16) Similarly, the force acting at the bottom of the cylinder is ? ??? =− ? ???? ??=−? ???2 ?0+?? ?1+? (1.17)
while we have substituted ?=?−(?1+?) in equation (1.12).
Now, since ? ?=? ?=?, where ? is a unit vector along positive z direction and perpendicular to the ?−????, consequently, equation (1.17) may be written as ? ??? =? ???2 ?0+?? ?1+? (1.18) Hence, the net vertical force (i.e., force acting downwards) on the cylinder can be obtained using equations (1.16) and (1.18) as ? ???=? ???+? ???=−? ???2 ?0+???1 +? ???2 ?0+?? ?1+?
=? ???2 −?0−???1+?0+???1+??? ? ???=? ???2 ??? or ? ???=? ????2? (1.19)
Since the volume of cylinder is ??2?, where ? is radius and ? is the height. The net force on the cylinder then takes the form ? ???=? ??? (1.20) therefore the buoyancy force is ? ?=??? ? ??? (1.21)
11
?cyl here refers to the volume of the cylinder, ? is the density of the fluid and ? is the gravity of the fluid. Thus, the buoyancy force acting upward (positive z – direction) is equal to the weight of the displaced fluid.
Consider an arbitrary shape body and extend the above analysis using the projected area theorem. Let the submerged body in a fluid be separated into two parts, with the upper surface denoted by ?2 and the lower surface by ?1. Therefore, the force acting on the submerged body is given by ? =− ?2 ? ?2 ??− ?2 ?1 ? ?1 ?? ?1 (1.22)
where ? ?1 ??? ? ?2 are the pressure acting on the lower and upper surface of the body.
The component of this force in the ? direction gives the buoyancy force. Taking the component of ? ?? ??? ? direction, the buoyancy force is then ??=? .? =− ? 2.? ? ?2 ??− ?2 ? 1.? ? ?1 ?? ?1 (1.23) From the projection theorem, we have ? 2.? ??=???, ? 1.? =−??? (1.24)
where ??? is the projection of ?? on the ??−?????. The use of equation (1.24) in equation (1.23), we have the magnitude of the buoyancy force (the component of the force acting in the vertical direction)
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??=− ? ?2 ???− ?2 ? ?1 ??? ?1 (1.25) Now, since the hydrostatic pressure at any point on z in the fluid is given by equation (1.14), then ??= ?? ?2−?1 ??? ?2 or ??=?? ?2−?1 ??? (1.26) ?2 From the principal surface of the integral, the equation (1.26) is just the volume of the solid, that is ?2−?1 ???=?? ?2 Therefore, the buoyancy force is ??=???? (1.27)
where ?? is the volume of the solid, this shows that the body is buoyed up by the weight of the displaced fluid.
Consider a solid spherical body of mass ?3 with center of mass ?3, density ?3 and radius b, so that ?3=?343??3 (1.28) or ?3=3?34??3 (1.29)
13
Using equation (1.29) in (1.27) to get the buoyancy force (in magnitude) in a fluid of ?1 ??=?1?43?×3?34??3 or ??=?1??3?3 this implies that ? ?=−?3?1?3? (1.30) The weight of the buoyed body is ? ?=??? ?? (1.31)
where ? ? ??? ?? are the density and volume of the solid body respectively, while ? is the gravity of the fluid ?1 acting at ?3. Equation (1.31) may be expressed as ? ?=??? ??=??1?3?1?3 ?133 (1.32)
Where ?1 is the center of the spherical shell having radius R filled with fluid ?1 and ?3?1 =?31=?13 ??? ?3=???? Putting equation (1.28) in equation (1.32), we have ??1?3? 13?133=?3?43??3 (1.33) using equation 1.29 ,we get ??1?3?1?3 ?133=?343?×3?34??3 which implies ? = ??1?1?3 ?133. ????? ?1=43??3?1 (1.34) therefore,
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? = 43???3?1?3?1 ?133 ??? ?=?13=?31 ?3?1 , the above takes the value ? = 43???3?1?3?1 ?133 or ? = 43???1?3?1 (1.35) Putting equation (1.35) in equation (1.30), we have the buoyant force as ? ?=−4?3?123?3???3?1 (1.36)
1.7.2 Robe’s restricted three-body problem
Robe (1977) introduced a new kind of restricted three-body problem in which he regarded one of the primaries of mass ?1 as a rigid spherical shell filled with homogenous, incompressible fluid of density ?1; the second one is a point mass located outside the shell and moving around the mass ?1 in a Keplerian orbit; the infinitesimal mass ?3 is a small solid sphere of density ?3 moving inside the shell and is subject to the attraction of ?2 and the buoyancy force due to the fluid of the first primary. He discussed the linear stability of an equilibrium point obtained in two cases; in the first case, the orbit of ?2 around ?1 is circular and in the second case, the orbit is elliptic, while the shell is empty (without fluid) or the densities of ?1 and ?3 are equal.
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Figure 1.2 Robe’s restricted three-body problem
Since we modified Robe’s model in our work .The derivation of the equations of motion of Robe’s problem (1977) is given in this section
Let the orbital plane of ?2 around ?1∗ (the shell with fluid ?1) be taken as ??−?????, and also let the origin of the coordinate system be at the center of mass, 0, of the two finite masses. Then the forces acting on the third body are;
1) The attraction of ?2
2) The gravitational force ? ? exerted by the fluid of density ?1 is given by
? ?=??1?3? 13?133
Substituting the mass of the fluid ?1 in the above gives ? ?=−?4??3?1?3? 133?133 (1.37)
Since the body of mass ?3 maintains a spherical symmetry about ?1 (center of ?1) due to the pressure of the fluid inside ?1, in this case, the radius R becomes ?=?13. Therefore, the gravitational force equation (1.37) is recast to the form
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? ?=− 43 ???12?3?1?3 (1.38)
3) The buoyancy force ? ?, is obtained from equation (1.36) as
? ?=−43???12?3?1?3 ?3 (1.39)
The equation of motion of the infinitesimal mass ?3 under the listed forces above, can be written as, Robe (1977).
? – 2? =??,
? + 2? = ?? , (1.40)
? = ??, with ?=12 ?2+?2+?2 +? 1−?−?2 2+?2+?2−?2 1−?−?2 2+?2+?2 , ?=?2?1+?2≤1 ??? ?=43??1 1−?1?3 . System (1.40) is equations of motion for the Robe’s CR3BP.
1.7.3 Perturbation
In astronomy, perturbations is the deviation in the motion of a celestial object caused either by the gravitational force of a passing object or by a collision with it. For example, predicting the Earth’s orbit around the Sun would be rather straightforward if not for the slight perturbations in its orbital motion caused by the gravitational influence of the other planets.
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1.7.4 Coriolis and centrifugal forces
1.7.4.1 Coriolis force: In classical mechanics, the Coriolis force is an inertial force described by the 19th-century French engineer-mathematician Gustave-Gaspard Coriolis in 1835. He described this force in terms of its effect. The effect of the Coriolis force is an apparent deflection of the path of an object that moves within a rotating coordinate system. The object does not actually deviate from its path, but it appears to do so because of the motion of the coordinate system.
The Coriolis effect has great significance in astrophysics and stellar dynamics, in which it is a controlling factor in the directions of rotation of sunspots. It is also an important consideration in ballistics, particularly in the launching and orbiting of space vehicles. (Encyclopedia Britainica Ultimate , 2014)
1.7.4.2 Centrifugal force: A fictitious force, peculiar to a particle moving on a circular path, that has the same magnitude and dimensions as the force that keeps the particle on its circular path (the centripetal force) but points in the opposite direction.
Although it is not a real force according to Newton’s laws, the centrifugal-force concept is a useful one. For example, when analyzing the behaviour of the fluid in a cream separator or a centrifuge, it is convenient to study the fluid’s behaviour relative to the rotating container rather than relative to the Earth; and, in order that Newton’s laws be applicable in such a rotating frame of reference, an inertial force, or a fictitious force (the centrifugal force), equal and opposite to the centripetal force, must be included in the equations of motion. (Encyclopedia Britainica Ultimate 2014)

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