Gravitational Field Strength

10 Key excerpts on "Gravitational Field Strength"

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  Foundations of Physics

  • Steve Adams(Author)
  • 2019(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  23

  Gravitational Field

  23.1    Gravitational Forces and Gravitational Field Strength

   

  Gravity is one of the four fundamental forces. It has infinite range and obeys a similar inverse-square law to electrostatics. All masses create gravitational fields, but, unlike the electrostatic forces between charges, which can be attractive or repulsive, gravitational forces are always attractive. The gravitational force acting on a mass close to the surface of the Earth is called weight.

  23.1.1   Newton’s Law of Gravitation

  Newton stated that two point masses would exert an attractive force on each other that is directly proportional to the product of the masses and inversely proportional to their separation. The minus sign indicates attraction.

  G is the universal constant of gravitation, G = 6.674 × 10−11 Nm2 kg−2 .

  Newton was also able to show that the force of attraction between spheres of uniform density is the same as the attraction between two point masses placed at their centers. This means that we can treat object like planets and stars as point masses when considering orbital motion. It is also important to note that, by Newton’s third law, the forces on each mass have the same magnitude, even if the masses are different. For example, the weight of an apple in the Earth’s gravitational field is the same as the weight of the Earth in the apple’s gravitational field. It is also the case that the gravitational force exerted on the Earth by the Moon is equal in magnitude to the gravitational force exerted on the Moon by the Earth.

  The resultant gravitational force on a body affected by the gravitational fields of several other objects (e.g., the Earth affected by the Sun, Moon, and other planets) is the vector sum of the gravitational forces from each of the other objects.

  23.1.2   Gravitational Field Strength

  The idea that gravitational forces arise from a gravitational field removes the difficulty of an action-at-a-distance explanation. The Moon is attracted to the earth because it experiences a force from the gravitational field where it is, i.e., a local force

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  Foundations of Physics

  • Steve Adams(Author)
  • 2023(Publication Date)
  • Mercury Learning and Information

    (Publisher)

  23

  THE GRAVITATIONAL FIELD

  23.1 GRAVITATIONAL FORCES AND Gravitational Field Strength

  Gravity is one of the four fundamental forces. It has an infinite range and obeys a similar inverse-square law to electrostatics. All masses create gravitational fields but unlike the electrostatic forces between charges, which can be attractive or repulsive, gravitational forces are always attractive. The gravitational force acting on a mass close to the surface of the Earth is called weight.

  23.1.1 Newton’s Law of Gravitation

  Newton stated that two-point masses would exert an attractive force on one another that is directly proportional to the product of the masses and inversely proportional to their separation. the minus sign indicates attraction.

  G is the universal constant of gravitation, G = 6.674 × 10− 11 Nm2 kg− 2 .

  Newton was also able to show that the force of attraction between spheres of uniform density is the same as the attraction between two point masses placed at their centers. This means that we can treat object like planets and stars as point masses when considering the orbital motion. It is also important to note that, by Newton’s third law, the forces on each mass have the same magnitude, even if the masses are different. For example, the weight of an apple in the Earth’s gravitational field is the same as the weight of the Earth in the apple’s gravitational field. It is also the case that the gravitational force exerted on the Earth by the Moon is equal in magnitude to the gravitational force exerted on the Moon by the Earth.

  The resultant gravitational force on a body affected by the gravitational fields of several other objects (e.g., the Earth affected by the Sun, Moon, and other planets) is the vector sum of the gravitational forces from each of the other objects.

  23.1.2 Gravitational Field Strength

  The idea that gravitational forces arise from a gravitational field removes the difficulty of an action-at-a-distance explanation. The Moon is attracted to the earth because it experiences a force from the gravitational field where it is

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  The Big Picture

  The Universe in Five S.T.E.P.S.

  • John Beaver(Author)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  But this is only because we happen to be relatively close to an enormous quantity of mass—the 5:972 ˆ 10 24 kg of mostly rock and metal that makes up Earth. For astronomers, it is gravity that holds it all together and has the greatest effect on the motions of large bodies in the universe, even though it is the intrinsically weakest of the physical forces, all else being equal. The reason is that all else is not equal; gravitation is the only one of the fundamental forces that is both long-range and only attractive. The much-stronger electrical force, for example, can either attract or repel, and so it tends to cancel out at large distances. 11.1. NEWTON’S GRAVITY 147 11.1.1 THE CLASSICAL GRAVITATIONAL FIELD Equation (11.1), taken at face value, asserts that one object exerts an instantaneous influence (a gravitational force) upon another, even if the two are separated by a vast expanse of the vacuum of space. But it is possible to look at Newton’s gravity in a different way; we replace the action at a distance with the local action of the gravitational field. Let us return to our example of two point masses, m 1 and m 2 , placed a distance d from each other. From the point of view of Newton, we say that m 1 exerts a gravitational force, at a distance, on m 2 . But from the perspective of the gravitational field, we say something subtly different: the presence of m 1 causes there to be a gravitational field, E g, that extends throughout space. We can think of this field as a property of space itself. And so we have the gravitational field, E g, due to the presence of m 1 . Our field theory must then tell us how to calculate this field. For this simple case of the gravitational field due to a single point mass, the answer is very simple. The field points everywhere toward m 1 , and it has this magnitude, for a point in space located a distance d from m 1 : g D G m 1 d 2 : (11.4) But this is only half the story; we must also say what the field does.

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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  The field extends throughout all space. It is not easy to grasp the field concept; it is particularly difficult to describe exactly what is filling all of space. We know that if we place a test mass at any point in the field it will feel a P(r) = field point r – r' r' r M = source point Figure 9.2 Particle M at the source point r  generates a gravitational field everywhere. The gravitational field at point P(r) (the “field point”) is given by Equation (9.3). 5 Sometimes, I like to change the definition of field slightly and state, “A field is a region of space in which some physical quantity is defined at every point.” This focuses the mind on the space rather than on the physical quantity. It is not, however, the standard definition of field. 9.2 The Gravitational Field 215 force, so we think of the mass M as producing something which exerts a force on any other mass in this space. The field is everywhere, but it exerts a force only when a material body is placed in the field. At the surface of the Earth the gravitational field is approximately given by g = − 9.8 m/s 2 ˆ k. Here ˆ k is a unit vector pointing upward at the surface. For points far above the surface, the field is approximately g(r) = −(GM E /r 2 ) ˆ r, where the origin is at the center of the Earth and M E is the mass of the Earth. (Do not confuse the field vector g(r) with the quantity g which is an abbreviation for the numerical value 9.8 m/s 2 .) The field concept is quite different from action at a distance. In action at a distance we consider the force one body exerts directly on another. When using the field concept, we think of the interaction of bodies M and m as a two-step process in which mass M generates a field and then mass m interacts with the field, rather than interacting with M directly. Thus the field approach decouples the sources from the test body used to determine the field.

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  Physics of Force and Friction (Concepts and Applications)

  • (Author)
  • 2014(Publication Date)
  • Academic Studio

    (Publisher)

  In this formula, quantities in bold represent vectors. where ________________________ WORLD TECHNOLOGIES ________________________ F 12 is the force applied on object 2 due to object 1, G is the gravitational constant, m 1 and m 2 are respectively the masses of objects 1 and 2, | r 12 | = | r 2 − r 1 | is the distance between objects 1 and 2, and is the unit vector from object 1 to 2. It can be seen that the vector form of the equation is the same as the scalar form given earlier, except that F is now a vector quantity, and the right hand side is multiplied by the appropriate unit vector. Also, it can be seen that F 12 = − F 21 . Gravitational field The gravitational field is a vector field that describes the gravitational force which would be applied on an object in any given point in space, per unit mass. It is actually equal to the gravitational acceleration at that point. It is a generalization of the vector form, which becomes particularly useful if more than 2 objects are involved (such as a rocket between the Earth and the Moon). For 2 objects (e.g. object 2 is a rocket, object 1 the Earth), we simply write r instead of r 12 and m instead of m 2 and define the gravitational field g ( r ) as: so that we can write: This formulation is dependent on the objects causing the field. The field has units of acceleration; in SI, this is m/s 2 . Gravitational fields are also conservative; that is, the work done by gravity from one position to another is path-independent. This has the consequence that there exists a gravitational potential field V ( r ) such that If m 1 is a point mass or the mass of a sphere with homogeneous mass distribution, the force field g ( r ) outside the sphere is isotropic, i.e., depends only on the distance r from the center of the sphere. In that case

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  Principles of the Gravitational Method

  • Alex Kaufman, Richard O. Hansen, Alex A. Kaufman(Authors)
  • 2007(Publication Date)
  • Elsevier Science

    (Publisher)

  Gravitational Field of the Earth 63 Now we demonstrate one interesting feature of the magnitude of the surface force as a function of the latitude. First, assuming that the fluid Earth is almost a sphere with a radius a , we can represent the distance r as r ¼ a cos l . Then, Equa-tion (2.18) has the form g s ¼ ð g a o 2 a cos 2 l Þ or g s ¼ ½ð g a o 2 a Þ þ o 2 a sin 2 l Š ð 2 : 22 Þ At points of the equator we have g e s ¼ g a o 2 a ð 2 : 23 Þ and, therefore, we obtain g s ð l Þ ¼ g e s ð 1 þ q sin 2 l Þ ð 2 : 24 Þ Here the parameter q ¼ o 2 a g e s ð 2 : 25 Þ is very small. In fact, assuming that g s ¼ 9 : 8 m = s 2 ; a ¼ 6 : 4 10 6 m ; and o ¼ 7 : 3 10 5 s 1 we have q ¼ 1 289 ð 2 : 26 Þ This parameter can be interpreted in the following way. At the pole Equation (2.24) gives g p s ¼ g e s ð 1 þ q Þ or q ¼ g p s g e s g e s ð 2 : 27 Þ Equations (2.24 and 2.27) look like as Clairaut’s formulas which will be derived later. However, this similarity is superficial, since the former do not contain the flattening of the earth. A variation of the field magnitude, g s with latitude, Equation (2.24), is caused by only a change of the centripetal acceleration on the spherical surface. 2.1.3. The gravitational field g In order to study the attraction of masses of the earth which moves around the axis of rotation, it seems appropriate to use the field g s , which depends on the distribution of masses and the angular velocity, as well as coordinates of the point. Besides, it has a physical meaning of the reaction force per unit mass. However, it has one very serious shortcoming, namely, unlike the attraction force it is directed outward. In other words, it differs strongly from the attraction field, in spite of the fact that the contribution of rotation is extremely small. To overcome this problem we introduce the gravitational field g which differs from the reaction field in direction only: g ð p Þ ¼ g s ð p Þ ð 2 : 28 Þ Methods in Geochemistry and Geophysics 64

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  Effects of Hypergravity and Microgravity on Biomedical Experiments, The

  • Thais Russomano, Gustavo Dalmarco, Felipe Prehn Falcao(Authors)
  • 2022(Publication Date)
  • Springer

    (Publisher)

  Gravitation is also responsible for keeping the Earth and the other planets in their orbits around the sun, the Moon in its orbit around the Earth, the formation of tides, and various other natural phenomena that are commonly observed (Figure 1.1). Gravity, however, is the gravitation related to Earth. It is then the gravitational force that occurs between the Earth and other bodies, the force acting to pull objects toward the Earth. It is expressed as 1G (capital “G,” as opposed to the acceleration g). Bodies with less mass than the Earth will have values lower than 1G (the Moon has 1/6G), and bodies with mass bigger than the Earth will have values higher than 1G (planet Jupiter has 3.5G). C H A P T E R 1 General Concepts in Physics— Definition of Physical Terms 2 EFFECTS OF HYPER- AND MICROGRAVITY ON BIOMEDICAL EXPERIMENTS Mass is a property of a physical object that quantifies the amount of matter and energy it is equivalent to and is expressed by the symbol m. Acceleration (expressed by the symbol a) is defined as the rate of change of velocity. It is thus a vector quantity with dimension length/time² (SI units m/s²), and the instantaneous accelera- tion of an objection is given by Equation 1.1. By being a vector, it must be described with both a direction and a magnitude. It can have positive and negative values — called acceleration (increasing speed) and deceleration (decreasing speed), respectively, as well as change in direction. a dv dt (1.1) where a is acceleration, v is velocity, t is time, and d is Leibniz’s notation for differentiation. Gravitation is one of the four fundamental interactions in nature, the other three being the electromagnetic force, the weak nuclear force, and the strong nuclear force. Compared to the other three fundamental interactions in nature, gravitation is the weakest one. It, however, acts over great distances and is always present. Gravitation is interpreted differently by classic mechanics, relativity, and quantum physics.

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  Relativistic Astrophysics

  • Marek Demiański, D. Ter Haar(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 1 GRAVITATIONAL FIELD 1.1. Newton's Theory of Gravitation Classical celestial mechanics and models of stars are based on the assumption that bodies attract each other according to Newton's theory of gravitation. The fundamental quantity describing the gravitational field in this theory is the gravitational potential φ. The distribution of mass is represented by the density of matter p as a function of position; the gravitational potential φ is then determined from the Poisson equation Αφ=4πβρ, (1.1) where (?=6.67·10~ 8 cm 3 g~ 1 s~ 2 is the gravitational constant. For the potential φ to be determined uniquely, boundary conditions must be given. For a bounded distribution of mass it is usually assumed that far from the sources, at infinity, the gravitational potential vanishes as 1/r. The Poisson equation can then be solved uniquely and φ can be written in the form *(r,0=-GJ^dV. (1.2) Far from the sources, for |f|=r>Ä, where R characterizes the size of the region filled with matter, : 7 . can be represented by the series |r-r'| r rj ta 2 rj tab The small Latin indices a,b,c, ... run over the values 1, 2, 3, and repeated indices denote summation, e.g. x a p a =x i p l +x 2 p 2 +x 3 P3· A comma denotes partial differentiation, e.g. dr r a =-— . Substituting this expansion into (1.2) gives • ox * ( r f 0 -~ + c ( i ) i x > ( ^ 0 d V -y G ( i ) J x ; x ; p ( * ' , 0 d V + ..., (1.4) where M = Jpd 3 jc. In the centre-of-mass system this reads *(r.O-— -J G ß -f i ) , (1.5) r 6 V r J.ab 1 (1.3) 2 GRAVITATIONAL FIELD where g<*= J p (jc', 0(3x'V 6 -x /c x;5 eb )dV (1.6) denotes the quadrupole moment of the mass distribution. The equations of motion for a test particle can be found from Newton's second law, which takes the form mx f l =-m^ t a . (1.7) Owing to the equality of gravitational and inertial mass, an experimentally confirmed fact, gravitational forces are locally indistinguishable from inertial forces.

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  Basic Concepts in Relativistic Astrophysics

  • L Z Fang, R Ruffini;;;(Authors)
  • 1983(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2 EFFECTS IN A WEAK GRAVITATIONAL FIELD 2-1 Gravitational Redshift 2 A system for which (2GM/Rc ) « 1, involves what is called a weak gravitational field. For such a system, Newtonian gravitation is sufficient and only very little correction need to be brought in using general relativity. However, for quite accurate measurements, such deviations are clearly observable. These weak field effects which will be discussed in this chapter, are of great importance in testing the validity of general relativity and its application to astrophysics. It is possible to discuss gravitational effects by just using the principle of equivalence without recourse to Einstein's gravitational redshift, that is, how the frequency of light changes when it propa-gates in a gravitational field. The energy of a photon of frequency v is hv, and the inertia! 2 mass of the photon is then hv/c . According to the principle of equi-valence, which is also obeyed by photons, the ratio of the inertia! mass to the gravitational mass is always the same for any material body. Suppose light propagates in the gravitational field near the Earth where the gravitational acceleration is g. Let us consider a 40 laboratory at height h in free-fall and assume that a light ray with frequency v moves from the floor through space to arrive and be detected at a later time on the ceiling. According to the principle of equivalence, there is no gravita-tional force in this frame, so that the propagation of light will be described by special relativity with the speed of light c. Therefore, the propagation will take time t = h/c from the floor to the ceiling, and its frequency arriving at the ceiling will still be v . This view is not shared by an observer at rest relative to the Earth, to which the laboratory is in accelerated motion. During the time t, the downward velocity increases by u = gt = gh/c.

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  Geophysical Field Theory and Method, Part A

  Gravitational, Electric, and Magnetic Fields

  • Alexander A. Kaufman(Author)
  • 1992(Publication Date)
  • Academic Press

    (Publisher)

  11.2 Determination of the Gravitational Field 165 caused by different masses within the body, and their contributions essen-tially depend on the location and distance of these masses from the observation point. In particular, masses located closer give a larger contri-bution, while remote parts of the body produce smaller effects. It is obvious that there are always such masses within the body that their contribution to the total field is so small that with the given accuracy of measuring it cannot be detected. For instance, we can imagine such changes of a shape, dimensions, location of the body, as well as the density of some of its parts, that the measured field would remain the same. In other words, due to errors in determination of the secondary field there can be an unlimited number of different distributions of masses that generate practically the same field at observation points. Inasmuch as the secondary field is caused by all masses within the body -that is, an integrated effect is measured-some changes of masses in relatively remote parts of the body can be significant; but their contribu-tion to the field would still remain small. At the same time similar changes in those parts of the body closer to observation points will result in much larger changes of the field. For this reason, in performing an interpreta-tion it is appropriate to distinguish at least two groups of parameters describing the distribution of masses, namely; 1. Parameters that have a sufficiently strong effect on the field; that is, relatively small changes of their values produce a change of the field that can be detected. 2. Parameters that have a noticeable influence on the field only if their values are significantly changed. It simply means that they cannot be defined from a field measured with some error.

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