Mathematics
Newton's Law of Gravitation
Newton's Law of Gravitation states that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. This law is fundamental in understanding the gravitational force between objects and is expressed mathematically as F = G * (m1 * m2) / r^2, where F is the force, G is the gravitational constant, m1 and m2 are the masses of the objects, and r is the distance between their centers.
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12 Key excerpts on "Newton's Law of Gravitation"
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
(When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him.) In modern language, the law states the following: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses: , where: • F is the magnitude of the gravitational force between the two point masses, • G is the gravitational constant, • m 1 is the mass of the first point mass, • m 2 is the mass of the second point mass, and • r is the distance between the two point masses. Assuming SI units, F is measured in newtons (N), m 1 and m 2 in kilograms (kg), r in meters (m), and the constant G is approximately equal to 6.674×10 −11 N m 2 kg −2 . The value of the constant G was first accurately determined from the results of the Cavendish experiment conducted by the British scientist Henry Cavendish in 1798, although Cavendish did not himself calculate a numerical value for G . This experiment was also the first test of Newton's theory of gravitation between masses in the laboratory. It took place 111 years after the publication of Newton's Principia and 71 years after Newton's death, so none of Newton's calculations could use the value of G ; instead he could only calculate a force relative to another force. Newton's Law of Gravitation resembles Coulomb's law of electrical forces, which is used to calculate the magnitude of electrical force between two charged bodies. Both are inverse-square laws, in which force is inversely proportional to the square of the distance between the bodies. Coulomb's Law has the product of two charges in place of the product of the masses, and the electrostatic constant in place of the gravitational constant.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Learning Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 1 Newton's Law of Universal Gravitation and Newton's Laws of Motion Newton's law of universal gravitation The mechanisms of Newton's law of universal gravitation; a point mass m 1 attracts another point mass m 2 by a force F 2 which is proportional to the product of the two masses and inversely proportional to the square of the distance ( r ) between them. Regardless of masses or distance, the magnitudes of | F 1 | and | F 2 | will always be equal. G is the gravitational constant. Newton's law of universal gravitation states that every massive particle in the universe attracts every other massive particle with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. (Separately it was shown that large spherically-symmetrical masses attract and are attracted as if all their mass were concentrated at their centers.) This is a general physical law derived from empirical observations by what Newton called induction. It is a part of classical mechanics and was formulated in Newton's work Philosophiae Naturalis Principia Mathematica (the Principia), first published on 5 July 1687. (When Newton's ________________________ WORLD TECHNOLOGIES ________________________ book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him.) In modern language, the law states the following: Every point mass attracts every single other point mass by a force pointing along the line intersecting both points.
eBook - PDF- John Matolyak, Ajawad Haija(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
As will be learned in later chapters, this idea does not hold for an extended object known as rigid body. 4.5 NEWTON’S GRAVITATIONAL LAW Newton’s gravitational law demonstrates another inclusive embracement of Newton’s third law. According to this law, masses attract each other. For two objects of masses m 1 and m 2 (Figure 4.2), there is a force of attraction F 2 on m 2 exerted by m 1 to which m 2 reacts exerting on m 1 a force F 1 , equal but opposite in direction to F 2 . And if there are no other forces acting on either of them, the two masses will start moving toward each other under equal forces, F 2 = F 1 , that is, m 1 a 1 = m 2 a 2 . m 1 F 1 m 2 F 2 FIGURE 4.2 Demonstration of Newton’s third law via depicting equal and opposite forces of attraction and reaction, that two masses m 1 and m 2 exert on each other. 70 Essential Physics © 2010 Taylor & Francis Group, LLC Since these masses are different, their accelerations will be different. The heavier mass will move more slowly than the lighter mass (see Newton’s second law). According to Newton’s gravitational law, the force of attraction F between two masses is (1) directly proportional to the product of the two masses and (2) is inversely proportional to the square of the separation between them. A proportionality may be written as an equation by inserting a multiplicative proportionality constant. The constant of proportionality is called the gravitational constant G. Applying this law to an object of mass m in the vicinity of the Earth, mass M E (Figure 4.3), the force on mass m takes the following form: F G m M r G = E 2 , where r is the distance between the center of mass of the object and the center of the Earth; the gravitational constant G = 6.67 × 10 − 11 N m 2 /kg 2 . As the force on any object according to Newton’s second law is F net = mass × acceleration, in the absence of any other forces on mass m, the above force is ma G mM r E 2 = , which leads to a G M r E 2 = .
- David Halliday, Jearl Walker, Patrick Keleher, Paul Lasky, John Long, Judith Dawes, Julius Orwa, Ajay Mahato, Peter Huf, Warren Stannard, Amanda Edgar, Liam Lyons, Dipesh Bhattarai(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
Although the gravitational force is still not fully understood, the starting point in our understanding of it lies in the law of gravitation of Isaac Newton. Newton’s law of gravitation Before we get to the equations, let’s just think for a moment about something that we take for granted. We are held to the ground just about right, not so strongly that we have to crawl to get to school (though an occasional exam may leave you crawling home) and not so lightly that we bump our heads on the ceiling when we take a step. It is also just about right so that we are held to the ground but not to each other (that would be awkward in any classroom) or to the objects around us (the phrase ‘catching a bus’ would then take on a new meaning). The attraction obviously depends on how much ‘stuff’ there is in ourselves and other objects: Earth has lots of ‘stuff’ and produces a big attraction, but another person has less ‘stuff’ and produces a smaller (even negligible) attraction. Moreover, this ‘stuff’ always attracts other ‘stuff’, never repelling it (or a hard sneeze could put us into orbit). In the past people obviously knew that they were being pulled downward (especially if they tripped and fell over), but they figured that the downward force was unique to Earth and unrelated to the apparent movement of astronomical bodies across the sky. But in 1665, 23‐year‐old Isaac Newton recognised that this force is responsible for holding the Moon in its orbit. Indeed, he showed that every body in the universe attracts every other body. This tendency of bodies to move towards one another is called gravitation, and the ‘stuff’ that is involved is the mass of each body. If the myth were true that a falling apple inspired Newton to his law of gravitation, then the attraction is between the mass of the apple and the mass of Pdf_Folio:241 CHAPTER 13 Gravitation 241 Earth. It is appreciable because the mass of Earth is so large, but even then it is only about 0.8 N.
Available until 25 Jan |Learn more- Tai L. Chow(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
In Chapter 6 on central force motion, we shall learn how Newton deduced the law of gravitation from Kepler’s three laws of planetary motion. He found that the gravitational force between two particles is inversely proportional to the square of the distance between them and proportional to the product of the masses of the two particles. As shown in Figure 2.2, if we denote the masses of the particles by m 1 and m 2 and the distance between them by r 12 , and arrowrightnosp F 12 is the force on m 1 by m 2 , then Newton’s law of gravity may be written as arrowrightnosp F Gm m r r 12 1 2 12 2 12 = -ˆ (2.6) in which the origin is set at m 2 , and ˆ ( / ) r r r 12 12 12 1 1 = is a unit vector pointing along the radius vector from the origin. The minus sign indicates that the force is always attractive. The quantity G is a uni-versal coefficient of proportionality independent of the nature of the interacting bodies. It is called the gravitational constant , and its value in the CGS system is 6.67 × 10 –8 cm 3 /g s 2 . The extremely small value of G shows that the force of gravitational attraction becomes con-siderable only for very large masses. Thus, gravitational force plays no great part in the mechan-ics of atoms and molecules, where the Coulomb forces are much larger in comparison. The static Coulomb force between two point charges q 1 and q 2 is similar in mathematical form to the gravita-tional force law: F 12 = q 1 q 2 / r 12 . (2.7) This force is attractive if the charges have opposite signs and is repulsive if the charges are of the same sign. The proportionality coefficient in Coulomb’s law has been made equal to unity by the appropriate choice of the unit of charge. A charge q moving in an electromagnetic field E and B experiences the Lorentz force: arrowrightnosp arrowrightnosp arrowrightnosp arrowrightnosp F q E v B c = + × ( / ) (2.8) where arrowrightnosp v is the velocity of the charge q .
- Daniel Fleisch, Julia Kregenow(Authors)
- 2013(Publication Date)
- Cambridge University Press(Publisher)
2 Gravity Even before taking an astronomy class, most people have a sense of how gravity works. No mathematics is needed to understand the idea that every mass attracts every other mass and that gravity is the force that causes apples to fall from trees. But what if you want to know how much you’d weigh on Saturn’s moon Titan, or why the Moon doesn’t come crashing down onto the Earth, or how it can possibly be true that you’re tugging on the Earth exactly as hard as the Earth is tugging on you? The best way to answer questions like that is to gain a practical understanding of Newton’s Law of Gravity and related principles. This chapter is designed to help you achieve that understanding. It begins with an overview of Newton’s Law of Gravity, in which you’ll find a detailed explanation of the meaning of each term. You’ll also find plenty of examples showing how to use this law – with or without a calculator. Later sections of this chapter deal with Newton’s Laws of Motion as well as Kepler’s Laws. And like every chapter in this book, this one is modular. So, if you’re solid on gravity but would like a review of Newton’s Third Law, you can skip to that section and dive right in. 2.1 Newton’s Law of Gravity The equation for Newton’s Law of Gravity may look a bit daunting at first but, like most equations, it becomes far less imposing when you take it apart and examine each term. To help with that process, we’ll write “expanded” versions of some of the important equations in this book, of which you can see an exam-ple in Figure 2.1 . As you can see, in an expanded equation, the meaning and units of each term are readily available in a text block with an arrow pointing to the relevant term. After the figure, you’ll find additional explanations of the 41
eBook - PDFSuperstrings and Other Things
A Guide to Physics, Second Edition
- Carlos Calle(Author)
- 2009(Publication Date)
- CRC Press(Publisher)
35 3 The Laws of Mechanics: Newton’s Laws of Motion THE CONCEPT OF FORCE Central to the laws of mechanics is the concept of force . Our idea of force is closely related to muscular activity. When we push or pull on an object, we exert a force on it (Figure 3.1). When we push a lawn mower across a yard, pull a hand truck loaded with boxes, push against the arms of a chair to get up from it, or when we turn the ignition key with our index finger and thumb to get the car started, we are applying a force. These forces associated with muscular activity are not the only ones that exist in nature. When you bring a small magnet near a nail, a magnetic force pulls the nail toward the magnet; and a gravitational force keeps the moon orbiting around the earth and the earth around the sun, and keeps us attached to the ground. The concept of force is directly involved in the formulation of the laws of motion. The discovery of these laws marks the birth of our modern understanding of the universe. THE ANCIENT IDEA OF MOTION We all know today that, neglecting the very small effect of air resistance, an object falling toward the ground experiences a constant acceleration caused by the gravita-tional attraction of the earth upon the object, and that all falling objects experience this acceleration. This was not known before the early 1600s. Until then, it was believed that heavier objects would fall toward the ground faster than lighter ones. This idea was based on the teachings of Aristotle, the greatest scientific authority of antiquity. Born in the Greek province of Macedonia in the year 384 BC, Aristotle was raised by a friend of the family, having lost both parents while still a child. At the age of 17 he went to Athens for his advanced education and later joined Plato’s Acad-emy, becoming “the intelligence of the school,” as Plato himself called him.
eBook - PDFGravity from the Ground Up
An Introductory Guide to Gravity and General Relativity
- Bernard Schutz(Author)
- 2003(Publication Date)
- Cambridge University Press(Publisher)
And then came Newton: gravity takes center stage 2 B orn in the same year, 1642, as Galileo died, Isaac Newton revolutionized the In this chapter: we learn about Newton’s postulate, that a single law of gravity, in which all bodies attract all others, could explain all the planetary motions known in Newton’s day. We also learn about Newton’s systematic explanation of the relationship between force and motion. When we couple this with Galileo’s equivalence principle, we learn how gravity makes time slow down. study of what we now call physics. Part of his importance comes from the wide range of subjects in which he made fundamental advances – mechanics (the study of motion), optics, astronomy, mathematics (he invented calculus), . . . – and part from his ability to put physical laws into mathematical form and, if neces- sary, to invent the mathematics he required. Although other brilliant thinkers made key contributions in his day – most notably the German scientist Gottfried Leibniz (1646–1716), who independently invented calculus – no physicist living between Galileo and Einstein rivals Newton’s impact on the study of the natural world. Nevertheless, it is hard to imagine that Newton could have made such progress in the study of motion and gravity if he had not had Galileo before him. Newton proposed three fundamental laws of motion. The first two are developed from ideas of Galileo that we have already looked at: Figure 2.1. Brilliant and demanding, Isaac Newton created theoretical physics. Besides devising the laws of gravity and mechanics, he invented calculus, still the central mathematical tool of physicists today. (Original engraving by unknown artist, courtesy aip Emilio Segr` e Visual Archives, Physics Today Collection.) The first law is that, once a body is set in motion, it will remain moving at constant speed in a straight line unless a force acts on it. This is just like the rubber ball dropped inside the airplane of Chapter 1.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
CHAPTER4 Forces and Newton's Laws of Motion PREVIEW In this chapter you will begin the study of dynamics, that branch of physics which explains why objects accelerate. You will be introduced to the concept of force, and study Newton's laws of motion, which apply to all forces that occur in nature. You will learn how to construct free-body diagrams, and use them to analyze systems subject to such forces as gravity and friction. You will also apply Newton's laws of motion to solve a number of different problems. The applications will include both equilibrium and non-equilibrium problems. QUICK REFERENCE Important Terms Force The push or pull required to change the state of motion of an object, as defined by Newton's second law. It is a vector quantity with units of newtons (N), dynes (dyn), or pounds (lb ). Inertia The natural tendency of an object to remain at rest or in uniform motion at a constant speed in a straight line. Mass A quantitative measure of inertia. Units are kilograms (kg), grams (g), or slugs (sl). Inertial Reference Frame A reference frame in which Newton's law of inertia is valid. Free-Body Diagram A vector diagram that represents all of the forces acting on an object. Gravitational Force The force of attraction that every particle of mass in the universe exerts on every other particle. Weight The gravitational force exerted by the earth (or some other large astronomical body) on an object. Normal Force One component of the force that a surface exerts on an object with which it is in contact. This component is directed normal, or perpendicular, to the surface. Friction The force that an object encounters when it moves or attempts to move along a surface. It is always directed parallel to the surface in question. Tension The tendency of a rope (or similar object) to be pulled apart due to the forces that are applied at either end. Equilibrium The state an object is in if it has zero acceleration.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
The distance from the center of the earth to the telescope is r. Equation 4.4 is Newton’s law of universal gravitation, and Equation 4.5 is Newton’s sec- ond law (net force equals mass times acceleration) incorporating the acceleration g due to gravity. These expressions make the distinction between mass and weight stand out. The weight of an object whose mass is m depends on the values for the universal gravitational constant G, the mass M E of the earth, and the distance r. These three parameters together determine the acceleration g due to gravity. The specific value of g 5 9.80 m/s 2 applies only when r equals the radius R E of the earth. For larger values of r, as would be the case on top of a mountain, the effective value of g is less than 9.80 m/s 2 . The fact that g decreases as the distance r increases means that the weight likewise decreases. The mass of the object, however, does not depend on these effects and does not change. Conceptual Example 7 further explores the difference between mass and weight. CONCEPTUAL EXAMPLE 7 | Mass Versus Weight A vehicle designed for exploring the moon’s surface is being tested on earth, where it weighs roughly six times more than it will on the moon. The acceleration of the vehicle along the ground is measured. To achieve the same acceleration on the moon, will the required net force be (a) the same as, (b) greater than, or (c) less than that on earth? Reasoning Do not be misled by the fact that the vehicle weighs more on earth. The greater weight occurs only because the mass and radius of the earth are different from the mass and radius of the moon. In any event, in Newton’s second law the net force is proportional to the vehicle’s mass, not its weight. Answers (b) and (c) are incorrect. According to Newton’s second law, for a given acceleration, the net force depends only on the mass.
eBook - PDFTheoretical Concepts in Physics
An Alternative View of Theoretical Reasoning in Physics
- Malcolm S. Longair(Author)
- 2020(Publication Date)
- Cambridge University Press(Publisher)
Newton’s calculations did not give quite such good agreement, but were sufficiently close to persuade him that the same force which holds the Moon in its orbit about the Earth and the planets in their orbits about the Sun is exactly the same force which causes the acceleration due to gravity on Earth. From this result, the general formula for the gravitational attraction between any two bodies of masses M 1 and M 2 follows immediately: f = GM 1 M 2 r 2 , (4.5) where G is the gravitational constant and r is the distance between the bodies. The force acts along the line joining the two bodies and is always an attractive force. This work was not published, because there were a number of steps in the calculation which needed further elaboration. (1) Kepler had shown that the orbits of the planets are ellipses and not circles – how did that affect the calculation? (2) Newton was uncertain about the influence of the other bodies in the Solar System upon each others’ orbits. (3) He was unable to explain the details of the Moon’s motion about the Earth, which we now know is influenced by the fact that the Earth is not spherical. (4) Probably most important of all, there is a key assumption made in the calculation that all the mass of the Earth can be located at its centre in order to work out the acceleration due to gravity at its surface and its influence upon the Moon. The same assumption was 55 4.5 Cambridge 1667–96 made for all the bodies in the Solar System. In his calculations of 1665–66, he regarded this step as an approximation. He was uncertain about its validity for objects close to the surface of the Earth. Newton laid this work aside until 1679. 4.5 Cambridge 1667–96 The University reopened in 1667 and Newton returned to Trinity College in the summer of that year. He became a fellow of the college in the autumn of 1667 and two years later, at the age of 26, he was elected Lucasian Professor of Mathematics, a position which he held for the next 32 years.
eBook - PDFApplied Mechanics
Made Simple
- George E. Drabble(Author)
- 2013(Publication Date)
- Made Simple(Publisher)
The first law gives us a definition of force: force is something which, by itself, produces an acceleration. This is the definition of force, and is clearly more satisfactory than the 'push or pull' we have had to accept so far. It only gives us a qualitative definition: it does not tell us how to measure force, but we shall find the answer to this problem in the second law. Let us now see how general the first law is in its application, and how it must have appeared to cut right across contemporary beliefs. It is natural to assume that a body which is not acted upon by a force would be in a state of rest: it is not so obvious to assume a possible state of motion in a straight line. All terrestrial experience of Newton's day must have pointed to other con-clusions. All bodies on the Earth, if set moving, came 'naturally' to rest. On the other hand, bodies apparently free from earthly interference (planets, for instance) were known to move in approximately circular paths. From the time of the Greeks, one school of thought accepted the 'natural' motion of bodies as circular. We now know that earth-bound bodies are subjected to fric-tional force, and that planets are subjected to gravitational force: both of 40 Applied Mechanics Made Simple these forces cause departure from straight-line uniform motion. In fact, perhaps the most exasperating aspect of Newton's theories is that no body ever observed has a 'natural' motion unaffected by force. There is an intri-guing and amusing dialogue on this topic between Newton and the artist, Kneller in G. B. Shaw's play, 'In Good King Charles's Golden Days'. It is perhaps typical of Newton's genius that he was able to perceive, without the benefit of direct observation, what the unforced motion of a body would be. Present-day students have less difficulty with this idea than those of an earlier age.
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