Newton’s law of universal gravitation

The force of gravity dominates our macroscopic world. Gravity can be described as the universal attraction between all objects. Even though gravity is the weakest of the four fundamental forces, it is ultimately responsible for perhaps the most violent of all objects in the universe, black holes. Newton s Law of Universal Gravitation gives us the mathematical description of the attractive gravitational force between two point objects of mass mx and m2 ... 

Dalton s first atomic theory was a physical one. From his 1801 presentation we see his depiction of the four atmospheric gases (water, oxygen, nitrogen, and carbonic acid). Separately, each gas repels like atoms (top of Figure 226), but mixed atoms of different gases do not repel or attract (bottom of Figure 226). Dalton, modest Quaker that he was, nonetheless compared his theory to Newton s law of universal gravitation. This comparison was not immodest. A few years later, Dalton would realize that his theory explained chemistry as well as physics. 

Figure 1.14 It does not matter how many times skydivers leap from a plane Newton s law of universal gravitation applies every time. 

Gravitational force is defined as the force of attraction between all masses in the universe. Every object exerts gravitational force on every other object. This force depends on the masses of the objects and the distance between them. The gravitational force between any two masses is given by Newton s law of universal gravitation, which states that the force is inversely proportional to the square of the distance between the masses. 

The sign conventions for force In Figure 4.1a the force is attractive when F is positive. This is the usual convention in materials science (and in Newton s law of universal gravitation). The force is attractive if A > 0 and negative if A < 0. Beware in electrostatics, the convention is that a negative force is attractive. 

The phenomenon of attraction of masses is one of the most amazing features of nature, and it plays a fundamental role in the gravitational method. Everything that we are going to derive is based on the fact that each body attracts other. Clearly this indicates that a body generates a force, and this attraction is observed for extremely small particles, as well as very large ones, like planets. It is a universal phenomenon. At the same time, the Newtonian theory of attraction does not attempt to explain the mechanism of transmission of a force from one body to another. In the 17th century Newton discovered this phenomenon, and, moreover, he was able to describe the role of masses and distance between them that allows us to calculate the force of interaction of two particles. To formulate this law of attraction we suppose that particles occupy elementary volumes AF( ) and AF(p), and their position is characterized by points q and p, respectively, see Fig. 1.1a. It is important to emphasize that dimensions of these volumes are much smaller than the distance Lgp between points q and p. This is the most essential feature of elementary volumes or particles, and it explains why the points q and p can be chosen anywhere inside these bodies. Then, in accordance with Newton s law of attraction the particle around point q acts on the particle around point p with the force d ip) equal to... 

Berzelius s accomplishment was a great step forward. However, in the mid-nineteenth century, confusion continued to reign. It appeared that the more that chemists knew, the more puzzles that confronted them. First, there was the question of why there were so many different chemical elements. When Dalton died in 1844, about 50 were known. Was the universe really made of 50 different building blocks The physicists had discovered that fundamental physical laws could be based on simple assumptions. Newton s law of gravitation, for example, could be written using just a few mathematical symbols. Why then should the world of chemistry be so complicated ... 

Scientific iaw Sometimes, many scientists come over and over again to the same conclusion about certain relationships in nature. They find no exceptions. For example, you know that no matter how many times skydivers leap from a plane, they always wind up back on Earth s surface. Sir Isaac Newton was so certain that an attractive force exists between all objects that he proposed his law of universal gravitation. 

Most philosophers discussions of issues relating to "laws of nature" and "scientific theories" have concentrated heavily on examples from classical physics. Newton s laws of motion and of gravitation and the various conservation laws are often discussed. This area of science provides very clear examples of the type of universal generalization that constitutes the widely accepted view of what a law of nature or a scientific theory "ought to be."... 

The first issue, universal quantification, is a requirement that there be no exceptions to a law. It must be "in all cases" or "in no case." Even "in the vast majority of cases" will not do. The problem with this clause is an implicit ceteris paribus requirement for most laws (Cartwright, 1983). Newton s law of gravitation, for example, is a perfectly respectable law of nature in the epistemological and historical senses. It purports to quantify the force of interaction between a pair of massive objects. It has been empirically verified by measurement of the force between a pair of suspended metal spheres in a laboratory. But the law does not quantify the force, even as an approximation, if we take the trouble to provide the spheres with an electrostatic charge. 

Suppose an astronomer wishes to predict when the next lunar eclipse will occur. As we know, the data accumulated after centuries of speculation and observation led, in the last quarter of the 17th century, to a theory that perfectly explains non-relativistic astronomical phenomena Newtonian mechanics. From Newton s laws, it is possible to deduce the behavior of heavenly bodies as a logical consequence of their gravitational interactions. This is an example of a mechanistic model with it we can predict trajectories of planets and stars because we know what causes their movements, that is, we know the mechanism governing their behavior. An astronomer only has to apply Newtonian mechanics to his data and draw the necessary conclusions. Moreover, he need not restrict his calculations to our own solar system Newton s laws apply universally. In other words, Newtonian mechanics is also a global model. 

Newton s law of gravitation There is a force of attraction between any two massive particles in the universe. For any two point masses m, and m2, separated by a distance d, the force of attraction f is given by f= mijnfilcP, where G is the gravitational constant. Real bodies having spherical symmetry act as point masses positioned at their centres of mass. 

In 1687, Newton summarized his discoveries in terrestrial and celestial mechanics in his Philosophiae naturalis principia mathematica (Mathematical Principles of Natural Philosophy), one of the greatest milestones in the history of science. In this work he showed how his (45) principle of universal gravitation provided an explanation both of falling bodies on the earth and of the motions of planets, comets, and other bodies in the heavens. The first part of the Principia, devoted to dynamics, includes Newton s three laws of motion the second part to fluid motion and other topics and the third part to the system of the (50) world, in which, among other things, he provides an explanation of Kepler s laws of planetary motion. 

A short glance back into the past gives an interesting perspective on this question. Aristotle (350 BC) maintained that an iron ball falls faster than a feather, implying that the interaction of these two bodies with the Earth is different. Galileo (1604) has experimentally shown that all bodies fall with the same acceleration, a property known as the principle of universality of free fall. This principle can be derived from the Newton s gravitational law (1687) which, when combined with the Newton s equation of motion shows that the gravitational acceleration of all bodies is the same. 

Geocentric model based on religious beliefs, but explains observed phenomena. Careful observations (Brahe, 9 planets eventually Kepler) point to discovered. Discovery of Heliocentric Model first Neptune confirms Newton s suggested by Copernicus. theory of universal Telescope confirms model. gravitation. Anomaly in orbit of Mercury resists solution with Newton s laws. Precession of Mercury s orbit is solved by Einstein s Theory of General Relativity. Theory and observations agree. Pluto is demoted to non-planet status (4). 

Legend has it that a falling apple inspired Newton s theory of gravitation. More likely the theory was the culmination of much thinking and several observations, of which the last perhaps involved an apple. Once his theory was tested in various situations and found satisfactory, it became known as a universal law. Newton s encounter with an apple may or may not have happened, but nevertheless the story conveys the most common method of discovery. This method, in which a few particular observations are extended to a single broad generality, is called induction. Tlie method is summarized schematically on the left side of Figure 0.1. (For more on the role of induction in scientific discovery, see Polya [1].)... 

Boyle was also a close colleague of another natural philosopher, who would come to have even greater distinction than he, Isaac Newton (1642—1727). Newton s crowning achievement was the elucidation of the law of gravitation and its application to celestial and terrestrial phenomena. He was a professor of mathematics at Cambridge University. He was also for many years president of the Royal Society of London and, more briefly, a member of Parliament and Master of the Mint. He was, in short, the very model of a modern major scientist and statesman of science. Until recently, historians have accepted that strict model and been reluctant to recognize that he was also a serious student and practitioner of alchemy. It is arguable that alchemy was as important to him as mathematical physics and astronomy. Newton and the age in which he lived were clearly more complex than the old historical model perceived. 

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