Newton’s law of attraction

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... 

Newton s law of attraction states that the force of interaction of particles is inversely proportional to the square of the distance between them. However, in a general case of arbitrary bodies the behavior of the force as a function of a distance can be completely different. 

Thus, we have derived a generalization of Newton s law of attraction. 

In the previous section we established two fundamental features of the attraction field, and both of them follow from Newton s law of attraction and the principle of... 

As a summary, it is useful to illustrate a transition from Newton s law of attraction to the system of equations in the form of diagram. The arrows show that all equations are derived from Newton s law of attraction, and they do not contain more information than this law. At the same time, the system (1.38) allows us to... 

Then, in accordance with the Newton s law of attraction this potential is... 

For what follows one can accept this equation as a fundamental natural law of the mechanics of electrons, the correctness of which has been tested many times by experiment, just as Newton s law of attraction is a fundamental law of macroscopic mechanics. De Broglie s relation is then a special result of this equation. 

Ozorio goes on to consider Newton s law of attraction as applied to the formation of chemical compounds as well as the approximation of chemistry and astronomy and the formation of atoms according to A. DuponcheF° before ending the chapter by presenting Mendeleev s opinions on, for instance, ammoniac with a planetary system, sodium chloride as a double star of sodium and chloride, and so on. 

Newton s law of gravitation states that if two particles are a distance r apart, the mutual attraction force between them can be expressed as follows... 

Inasmuch as determination of the field of attraction is an important element of interpretation of gravitational data, let us derive some equations allowing us to simplify the calculation of the useful signal. As follows from the Newton s law of... 

The molecular dynamics method is based on the time evolution of the path (p (t), for each particle to feel the attractions and repulsions from all other particles, following Newton s law of motion. The simplest case is a dilute gas following the hard sphere force field, where there is no interaction between molecules except during brief moments of collision. The particles move in straight lines at constant velocities, until collisions take place. For a more advanced model, the force fields between two particles may follow the Lennard-Jones 6-12 potential, or any other potential, which exerts forces between molecules even between collisions. 

The relation between the speed v of the electron in. a circular orbit about the nucleus and the radius r of the orbit can be derived by use of Newton s laws of motion. A geometrical construction shows that the acceleration of the electron toward the center of the orbit is v2/r, and hence the force required to produce this acceleration is mv2/r. This force is the force of attraction Ze2/r2 of the electron and the nucleus hence we write the equation... 

In the above calculation the system has been treated as though the nucleus were stationary and the electron moved in a circular orbit about the nucleus. The correct application of Newton s laws of motion to the problem of two particles with inverse-square force of attraction leads to the result that both particles move about their center of mass. The center of mass is the point on the line between the centers of the two particles such that the two radii are inversely proportional to the masses of the two particles. The equations for the Bohr orbits with consideration of motion of the nucleus are the same as those given above, except that the mass of the electron, m, is to be replaced by the reduced mass of the two particles, /, defined by the expression 1/m = 1/m + 1/M, where M is the mass of the nucleus. 

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 ... 

Equation (1.2) is Newton s law of gravitation, giving the force of attraction between two particles of masses m and mi, a distance r apart. 

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. 

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. 

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. 

The hydrogen atom consists of an electron and a proton. The interaction of their electric charges, —e and +e. respectively, corresponds to inverse-square attraction, in the same way that the gravitational interaction of the earth and the sun corresponds to inverse-square attraction. If Newton s laws of motion were applicable to the hydrogen atom we should accordingly expect that the electron, whose mass is small compared with that of the nucleus, would revolve about the nucleus in an elliptical orbit, in the same way that the earth revolves about the sun. The simplest orbit for the electron about the nucleus would be a circle, and Newton s laws of motion would permit the circle to be of any size, as determined by the energy of the system. 

We are about to provide a microscopic definition of the collision cross-section. Two ingredients come in. One is the definition of a collision. On the basis of Newton s laws of motion we took it that a collision occurs whenever two molecules exercise a force on one another. The outcome of the collision can be that a chemical reaction took place, or only that the two molecules deflected from flieir unperturbed straight-line motion, or anything in between. Whatever the outcome, when a force due to the potential acted, a collision is said to have taken place. Nor need this force be repulsive, and indeed the long-range part of the force is in general weakly attractive. Section 2.1.8. The other point is that the cross-section is that area, drawn in a plane perpendicular to the initial velocity, that the relative motion of the molecules needs to cross if a coUision is to take place. 

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