Newton's Law of Gravitation

11 Key excerpts on "Newton's Law of Gravitation"

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  eBook - ePub

  Understanding Physics

  • Michael M. Mansfield, Colm O'Sullivan(Authors)
  • 2020(Publication Date)
  • Wiley

    (Publisher)

  5 Force fields

  AIMS

  • to introduce the fundamental law which governs the attractive force between masses (Newton's Law of Gravitation)
  • to describe how the concept of a force field may used to describe the behaviour of bodies under the influence of a variety of forces
  • to extend the definition of mechanical work, energy, and power to two and three dimensions and to demonstrate the value of these quantities in the analysis of complicated mechanical systems.
  • to introduce the concept of angular momentum and to see how it is applied in the description of the rotational movement of bodies about a fixed centre such as the orbits of planets about the Sun

  5.1 Newton's law of universal gravitation

  The development of mankind's understanding of the universe and, in particular, the role of gravity, from the early ideas of philosophers in ancient Greece to the proposal of the law of gravitation by Newton in the seventeenth century, is one of the most extraordinary stories in the history of human knowledge. No brief treatment here could do justice to the excitement and heroism of the tale; readers are encouraged to study any one of the many books on the subject (for example, Koestler, A. (1968) The Sleepwalkers; London: Hutchison).

  What Newton proposed was that, between any two point masses, there is a force of attraction which acts along the line joining them and which is directly proportional to each of the masses involved and inversely proportional to the square of the distance between them. Thus, if a point mass m2 has a displacement r from a point mass m1 (Figure 5.1 ) the force on m2 due to m1 is given by

  (5.1)

  where is a unit vector (Section 4.2 ) in the r‐direction. In Equation (5.1) the magnitude of F depends on so that the law of gravitation is described as an inverse square law. The constant of proportionality G in Equation (5.1) is called the gravitational constant and its value must be determined experimentally. The minus sign indicates that the force is attractive (the force on m2 is directed towards m1 ). Because the gravitational force between laboratory masses is so small, very sensitive apparatus is required to obtain an accurate value of G. A laboratory technique developed in 1798 by Henry Cavendish (1731–1810) can be used for this purpose, the basic principles of which will be discussed in Chapter 7 (Section 7.5 ). The value of the gravitational constant in SI units (to three significant figures) is measured to be

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  Violent Phenomena in the Universe

  • Jayant V. Narlikar(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  gravitational constant.

  It is often stated that Newton arrived at this law of gravitation from his encounter with the falling apple while he was sitting in an orchard in his native village of Woolsthorpe in Lincolnshire in the year 1666, when he left Cambridge to avoid the ravages of the great plague. This legend, which first seems to have appeared in the writings of Voltaire, while highlighting Newton’s originality and speculative power, hardly does justice to his mathematical and deductive ability. In any case the first complete description of the law as given by Newton appeared some 21 years later in his Principia. Considerable discussion has gone on in literature on the history of science about why Newton waited so long before publishing the law of gravitation and whether the genesis of the law rests solely with him. For instance, there are indications that Robert Hooke (1635–1703) had also arrived at the concept of the inverse square law of attraction from the data on planetary motion. We will not go any further into the controversial questions about the genesis of this law. This much is clear: the law would not have been discovered without the input from astronomy, from the data available on planetary motion.

  Celestial Mechanics

  Astronomy continued to provide evidence in support of the law of gravitation. Many of the developments came after Newton’s lifetime. One of these was the appearance of a comet in 1758 as predicted by Newton’s contemporary Edmond Halley in 1682.

  Halley had noticed that a particular comet was seen in the vicinity of the Sun in the year 1682. Although comets come and go every year Halley discovered that on previous occasions comets of somewhat striking appearance (similar to the comet of 1682) were seen in the years 1456, 1531, and 1607. The comets come near the Sun, go round it, and then recede from the solar neighbourhood. Could the comets seen in the years 1456, 1531, and 1607 be one and the same comet appearing periodically after every 75–6 years? Halley thought so and in terms of Newton’s laws of motion and gravitation he could give a reason for his supposition. The comet was moving in an elliptical orbit around the Sun just as any planet would do. The difference between the elliptical orbits of a comet and a planet lies in the circumstance that the latter is nearly circular while the former is highly elongated. In a highly elongated ellipse one end of the major axis lies very close to the focal point where the Sun is situated, while the other end of the major axis lies far away from the Sun. The comet is seen only when it is in the part of the orbit close to the Sun and appears to fade away as it recedes towards the far end of the major axis. The situation is illustrated in Fig. 2.4

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  Reasoning About Luck

  • Vinay Ambegaokar(Author)
  • 2017(Publication Date)
  • Dover Publications

    (Publisher)

  The ubiquity of frictional forces opposing motions in our everyday world makes it easy to understand why astronomical observations played such an important role in the insights of Newton and his contemporaries. On the other hand, although the original impetus for the development of mechanics may have been the explanation of celestial motions, one of Newton’s great triumphs was the unification of celestial and terrestrial dynamics. We are very close to being able to understand this synthesis. What follows is a small digression in that direction. Thereafter, I shall return to the main path of extracting those concepts from mechanics which allow one to understand the random motions which we experience as heat.

  Newton’s law of gravitation

  Newton discovered that he could explain the elliptical trajectories of the planets around the sun, as well as other properties of these orbits systematized by Johannes Kepler [1571–1630], by assuming (i) that the sun attracted each planet with a force proportional to the inverse square of the distance between them, and (ii) that planets followed the dynamical laws we discussed in the last section. He then made the astonishingly bold assumption that every particle of matter in the universe attracts every other particle in the same way. This ‘universal’ law of gravitation can be written in the form

  In this equation F is the magnitude of the gravitation force, m 1 and m 2 are the masses of the two particles, r 12 is the distance between the two particles, and G is the so called gravitational constant. The direction of the force is along the line joining the two particles: particle 1 attracts particle 2 along this line, and vice versa.

  Various motions are possible for two masses moving under the influence of their mutual gravitational attraction. Two cases that are well within our ability to understand completely are circular motion and a head-on collision. To keep everything as simple as possible, let us assume that one of the objects is very much more massive than the other. It is then a good first approximation to assume that the more massive object is not accelerating at all – because the accelerations produced by the equal and opposite forces are inversely proportional to the masses – and to take the center of this object as the origin of the position vector of the other mass. As an example of circular motion, we may think of the earth-moon system, because the moon is, in fact, in an almost circular orbit around the earth. For a head-on collision we may think of an apple falling to the earth. If these are two manifestations of the same basic laws of nature, then there must be connections between them. What are they?

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  Mathematics for the Nonmathematician

  • Morris Kline(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  2 . Thus Newton’s law of gravitation met its first test, for it yielded as a special case a well established fact.

  EXERCISES

  1.  Suppose that the gravitational force varies with the distance between two definite masses according to the formula F = 6/r 2 . Show graphically how F varies with r .

  2.  Knowing that the acceleration of objects near the surface of the earth is 32 ft/sec2 , use formula (5) to calculate the acceleration which the earth exerts on objects 1000 mi above the surface of the earth.

  3.  Suppose that an object falls to earth from a point 1000 mi above the surface. May we use the formula d = 16t 2 to compute the time it takes to fall this distance?

  4.  What is the mass of an object which weighs 150 lb? (One pound of weight is 32 poundals.)

  5.  How much force is required to give an automobile weighing 3000 lb an acceleration of 12 ft/sec2 ?

  15–9  FURTHER DISCUSSION OF MASS AND WEIGHT

  With some support for the law of gravitation we can now, following Newton’s example, adopt it as an axiom of physics and see what conclusions we may draw from this axiom and the other axioms of physics and mathematics. The law itself states that the force of gravitation F between any two masses m and M is given by the formula

  where r is the distance between the masses. Formula (6) leads immediately to a better understanding of the relationship between weight and mass and to an extension of the concept of weight. Let M be the mass of the earth and let m be the mass of some other object. Since F is the force with which the earth attracts this object, we can regard F as the weight of the object, for this attractive force is what we have meant by weight. However, we now see that the force or weight depends upon the distance r between the two masses. Hence, the weight of an object is not really a fixed number but varies with the distance of the object from the earth or, more precisely, from the center of the earth (see Section 15–8 ). If an object of mass m

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  The Handy Physics Answer Book

  • Charles Liu(Author)
  • 2020(Publication Date)
  • Visible Ink Press

    (Publisher)

  G is the symbol for the universal gravitational constant. It is called universal because it is the same for objects made of any material and of any mass—from an apple to a galaxy.

  If the sun suddenly disappeared, what would happen to the sun’s gravity?

  W hat would happen to Earth if the sun suddenly disappeared? How soon would Earth recognize that the sun’s gravitational field was gone? It couldn’t happen instantaneously because according to Einstein’s Special Theory of Relativity, no information can travel faster than the speed of light. It would take about eight minutes before Earth would both experience the lack of sunlight and the lack of gravitational force at the same time.

  The equation that describes the gravitational field at a specific location is

  g = −GM/r2

  Here, M is the mass of the attracting object, and r is the distance from the center of this mass to the location in space. The gravitational field is a vector quantity, and its direction is toward the center of the attracting object.

  What is the difference between a gravitational field and gravitational acceleration?

  When Isaac Newton developed his Universal Law of Gravitation, nobody had yet understood the concept of a gravitational field. Instead, Newton found g = −GM/r2 to be the acceleration of an object by another object due to the force of gravity. About 200 years after Newton’s work, Albert Einstein developed the General Theory of Relativity; one part of that theory is called the Principle of Equivalence, which explains that you cannot tell the difference between being accelerated by a pulling force and being acted upon by gravity.

  How does a gravitational field relate to force?

  According to Newton’s Second Law of Motion, the force exerted on an object is equal to the object’s mass times its acceleration. Because of the Principle of Equivalence, it is appropriate to say that the force of gravity on an object is equal to the object’s mass times the gravitational field strength or, written as an equation, F = mg = −GMm/r2

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  eBook - ePub

  Subtlety in Relativity

  • Sanjay Moreshwar Wagh(Author)
  • 2017(Publication Date)
  • Jenny Stanford Publishing

    (Publisher)

  From Kepler’s laws again, Newton also concluded that the assumed pull varies inversely with the square of the distance between the sun and the planet. This is Newton’s inverse-square gravitation.

  Newton’s inverse-square gravitation is only an assumption about the nature of the pull that is causing the deviation of a planetary body from the state of uniform rectilinear motion. This assumption says that a body like the sun acts on the planetary body situated at a distance and this force is therefore known as the action-at-a-distance force.

  As the action at a distance is an assumption, no explanation can be found within the Newtonian framework of concepts for why bodies exert such a pull on other bodies. Interestingly, Newton himself felt the need to replace the assumption of the action-at-a-distance nature of the aforementioned pull.

  If we change the position of a body, then the effects of this change will have to be instantaneously felt by all the other bodies in the universe when the pull is of the action-at-a-distance type. This is quite contrary to what we observe in nature, for the effect of moving one body reaches another body situated at a distant location only after the lapse of some time.

  After his general critique of Newtonian mechanics, Einstein admiringly writes: Newton, forgive me; you found the only way which, in your age, was just about possible for a man of highest thought and creative power. The concepts, which you created, are even today still guiding our thinking in physics, although we now know that they will have to be replaced by others farther removed from the sphere of immediate experience, if we aim at a profounder understanding of relationships.

  Both Newton and Galileo had no reason to consider bodies of vanishing inertia, for they imagined nonzero inertia to be an intrinsic physical property of every natural body. But if a body has zero inertia, then Newton’s laws of motion are not usable for it. Thus, interactions of inertialess bodies among themselves and with bodies having inertia cannot be analyzed using Newton’s laws of motion. That is to say, light does not obey Newton’s laws of motion if a light body has zero mass.

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  Newtonian Dynamics

  An Introduction

  • Richard Fitzpatrick(Author)
  • 2021(Publication Date)
  • CRC Press

    (Publisher)

  a , on itself, without the aid of any external agency. It will, therefore, accelerate forever under its own steam. We know, from experience, that this sort of behavior does not occur in real life. For instance, a person cannot grab hold of their own shoelaces and, thereby, pick themselves up off the ground. In other words, the person in question cannot self-generate a force that will spontaneously lift them into the air; they need to exert forces on other objects around them in order to achieve this goal. Thus, Newton’s third law essentially acts as a guarantee against the absurdity of self-generated forces.

  4.6 Mass, Weight, and Reaction

  The terms mass and weight are often confused with one another. However, in physics, their meanings are quite distinct.

  A body’s mass is a measure of its inertia; that is, its reluctance to deviate from uniform straight-line motion under the influence of external forces. According to Newton’s second law, Equation (4.4), if two objects of differing masses are acted upon by forces of the same magnitude then the resulting acceleration of the larger mass is less than that of the smaller mass. In other words, it is more difficult to force the larger mass to deviate from its preferred state of uniform motion in a straight-line. Incidentally, the mass of a body is an intrinsic property of that body, and, therefore, does not change if the body is moved to a different location.

  4.6.1 Block Resting on Earth’s Surface

  Imagine a block of granite resting on the surface of the Earth. See Figure 4.4 . The block experiences a downward force, f

  g

  , due to the gravitational attraction of the Earth. This force is of magnitude

  m   g

  , where m is the mass of the block, and

  g = 9.81  

  m/s

  − 2

  is the acceleration due to gravity at the surface of the Earth. The block transmits this force to the ground below it, which is supporting it, and, thereby, preventing it from accelerating downward. In other words, the block exerts a downward force, f

  W

  , of magnitude

  m   g

  , on the ground immediately beneath it. We usually refer to this force (or the magnitude of this force) as the weight of the block. According to Newton’s third law, the ground below the block exerts an upward reaction force, f

  R

  , on the block. This force is also of magnitude

  m   g

  . Thus, the net force acting on the block is

  f g

  +

  f R

  = 0

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  Physics for Students of Science and Engineering

  • A. L. Stanford, J. M. Tanner(Authors)
  • 2014(Publication Date)
  • Academic Press

    (Publisher)

  Chapter 3 . For example, we will no longer restrict ourselves to frictionless surfaces but will introduce a more realistic and practical representation of surfaces. Then we will consider how Newton’s second law may be applied to particles in circular motion. Next, another of Newton’s discoveries, the law of universal gravitation, will be discussed and applied to appropriate situations. Finally, in this chapter we will look at the special case of Newton’s second law in which physical bodies are in static equilibrium. In this case of bodies at rest, we will consider physical bodies that have extension in space, that is, bodies that are not treated merely as mathematical points. These considerations provide the occasion to introduce the concept of torque and the conditions that must be satisfied in order that a body be in static equilibrium. This chapter, then, introduces several new concepts while extending the usefulness of Newton’s laws of motion.

  4.1 Friction

  When two bodies are in contact, there are interactions at the molecular level within the region of contact. Although an adequate theoretical treatment that explains the contact forces from fundamental principles is not now feasible, we can make a number of experimental observations that permit us to analyze the macroscopic effects of contact between physical bodies. Let us look at some experimental situations using a book on a desk.

  Suppose first that a book is resting on a horizontal surface, as illustrated in Figure 4.1(a) . Only two forces act on the book: Its weight W is a downward force on the book, and the table exerts a force that, because the book is at rest, must be oppositely directed from the weight and equal in magnitude. The force that the table exerts upward on the book is perpendicular (or normal) to the horizontal contact surface and is, therefore, known as the normal force N . Now suppose that we push on the book with a horizontal force P 1 directed toward the right, as in Figure 4.1(b) and that P 1 is a small force. We observe that the book does not move, even though a horizontal force has been applied. The acceleration of the book is still zero, which means the resultant force on the book must still be zero. The normal force N still balances the weight W in the vertical direction, so we must conclude that there is a horizontal force to the left that balances the push P 1 to the right. Because only the table is present to produce additional contact forces, we must further conclude that the table exerts a force on the book parallel to the surface of contact. This force that is exerted by the table on the book in a direction parallel to the contact surface is known as the force of static friction fs

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  Classical Mechanics

  • Hiqmet Kamberaj(Author)
  • 2021(Publication Date)
  • De Gruyter

    (Publisher)

  These examples indicate that the acceleration of an object is directly proportional to the resultant force acting on it. The acceleration of an object should also depend on its mass. For example, consider the following experiment: If you apply a force F to a block of ice moving on a horizontal frictionless surface, then the block undergoes some acceleration, a. If you double the mass of the block, then the same applied force produces an acceleration of a / 2. If the mass is tripled, then the same applied force delivers acceleration of a / 3, and so on. This observation indicates that the magnitude of the acceleration of an object is inversely proportional to its mass. These observations lead to Newton’s second law. Newton’s second law The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. We can relate mass and force using the following expression, which represents a mathematical statement of the second law of the Newton: (5.7) ∑ i F i = m a. Since this expression is a vectorial equation, we can project along the three coordinate axes as follows: (5.8) ∑ i F x i = m a x (5.9) ∑ i F y i = m a y (5.10) ∑ i F z i = m a z. In the SI unit system, the force has the units of newton (N), which is defined as the force acting on a mass of 1 kg gaining an acceleration. of 1 m/s 2. From the definition of Newton’s second law, we see that the newton can be expressed in terms of the units of mass, length, and time as follows: (5.11) 1 N ≡ 1 kg · m s 2. In the engineering system or British system, the unit of force is pound, which is defined as the force acting on a mass of 1 slug 1 to produce an acceleration of 1 ft/s 2 : (5.12) 1 lb ≡ 1 slug · ft/s 2. An approximation is (5.13) 1 lb ≈ 1 4 N. 5.5 Newton’s third law When we press against a corner of a textbook with the fingertip, the book pushes back and makes a small dent in our skin

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  Explaining Science's Success

  Understanding How Scientific Knowledge Works

  • John Wright(Author)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  Further on in his first corollary to proposition V, Newton observes that it follows from his third law of motion (“to every action there is always opposed an equal reaction”) that since the planets gravitate towards the Sun, the Sun must also gravitate towards the planets, and that since the Moons of a planet gravitate toward that planet, so the planet must also gravitate towards its moons. It also follows, he notes in the third corollary, that “all the planets do mutually gravitate toward one another”. These claims follow deductively from conclusions already established, together with his third law of motion.

  Finally, in the “Scholium” to proposition V, Newton argues that the centripetal force acting on all orbiting bodies must be gravity. He has already argued that gravity is the force keeping the Moon in its orbit. But since the orbits exhibited by the other planets are phenomena of the same sort as the orbit of the Moon, it follows that, by his first and second rules of reasoning, they are to be explained in the same way as the orbiting of the Moon, that is as due to gravity. It is plain that this hypothesis explains the orbiting behaviour of all bodies in a way that maximizes the independence of theory from data: to explain the orbit of the Moon in one way and the orbits of other bodies in another would plainly increase the ratio of DECs to components of data explained.

  Proposition VI asserts: “That all bodies gravitate towards every planet, and that the weights of bodies towards any one at equal distances from the centre of the planet are proportional to the quantities of matter which they severally contain.”

  We will initially consider the argument Newton presents for the second part of proposition VI. First, let us remind ourselves of the conceptual distinction between the weight of a body and the quantity of matter

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  Doing Physics with Scientific Notebook

  A Problem Solving Approach

  • Joseph Gallant(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Chapter 5

  Newton’s Laws of Motion

  In the previous chapters on kinematics, we described motion in a quantitative sense. Once we know an object’s acceleration, we can describe its motion in terms of position, velocity, and time. Knowing how its motion changes lets us calculate how far, how fast, and how long the object moves.

  Now we move to a branch of physics known as dynamics. Dynamics is the study of the effects of forces on an object’s motion. Dynamics explains changes in motion by relating the cause of the changes (forces) to the effect (acceleration). Newton’s Second Law provides the rule relating the acceleration to the net force.

  There are three ingredients to Newton’s Second Law: the object’s acceleration, mass, and the net force acting on it. As we discussed in Chapter 2, the object’s acceleration is the rate of change in its velocity. Mass is a property of the object that determines how much change the net force produces. An object’s mass tells you how difficult it is to change its velocity and how much matter it has.

  A force is a push or a pull that can cause changes in motion. Forces are vectors, so they have magnitude and direction. This is consistent with your experience. When you exert a force on something, two things matter: how hard you push or pull and which way. Often objects have more than one force acting on them. The net force acting on an object is the vector sum of all the forces acting on it.

  Newton’s First Law

  Newton’s First Law tells us what happens when there is no net force acting on an object.

  Newton’s 1st Law: An object will remain in a state of rest or continue in motion at a constant velocity unless compelled to change by a non-zero net force.

  When there is no net force acting on the object, there is no change in the object’s velocity. If it is at rest, it remains at rest. If it is moving, it keeps moving at constant velocity. Since velocity is a vector, constant velocity means no change in both speed and direction. Motion at constant velocity is motion in a straight line at a constant speed.

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