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Equilibrium of a Particle

attraction, force, forces, centre, law and distance

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EQUILIBRIUM OF A PARTICLE that orbit is a parabola; if v V it is a hyperbola. We see that V is the velocity which must be possessed by the particle in order that its orbit may go to an infinite dis tance from the centre of attraction ; it is termed the critical velocity corresponding to the distance r.

Newton's Law of Gravitation.-36. We have assumed in the preceding paragraph that the attractive force on the particle varies inversely as the square of its distance from the centre of attraction. When the force is due to the attraction of a second particle, this is equivalent to assuming that the attraction varies inversely as the square of the distance between the two particles.

Newton's "law of universal gravitation" asserts that a mutual attraction, satisfying this relation, is exerted between every pair of particles in the material universe. If two particles have masses m, m', it asserts that the force of their mutual attraction, when their distance apart is r, will be where y is a universal constant, called the constant of gravitation.

That branch of mechanics known as the

"theory of attractions" is concerned with the consequences of this law in regard to the attractions of bodies of finite extent; i.e., bodies composed of a large number of particles. One result may be stated here : a spher ical body (either solid or hollow), of which the density is the same for all points at the same distance from its centre, exerts the same attraction on any particle or body outside it as would be exerted by a particle of the same total mass, situated at its centre. Thus in calculating, e.g., the motion of a planet under the attrac tion of the sun, we may replace each body by a particle of equal mass. The investigation of § 35 relates directly to this problem, if we assume (as is very nearly true) that the sun's mass is so large, in relation to that of the planet, that the centre of the sun can be identified with the mass-centre of the two, i.e., with the centre of attraction. More elaborate investigations, by Newton

and his successors, have taken account of the attractions of the planets on the sun and on one another : they have abundantly confirmed the accuracy of the inverse square law, by showing that it is able to explain the actual motions of the planets in minute detail.

Statics treats of forces at rest and therefore in equilibrium. The second law of motion is : "Change of motion is proportional to the moving force impressed and takes place in the direction in which the force acts." By change of motion, Newton meant change in Polygon of Forces.-38. We start by considering a particle, i.e., a body of infinitesimal size. Suppose first that two forces are acting, represented in direction and magnitude by lines 0P1, OP2 (fig. 12) passing through 0, the particle. According to the vector law, their resultant will be repre sented by the diagonal of the parallelogram of which 0P1, are the sides : it follows that the point could have been found by drawing to represent the first force and to represent the second.

If a third force acts in addition, represented by the result ant of the three forces is the resultant of and By the same argument, it will be represented by where is drawn (parallel and equal to to represent the third force. The process can be repeated for any number of forces: we obtain points such as . . . etc., and the successive resultants will be represented by OR2, OR3 . . . etc. It is however a con dition of equilibrium that the resultant of all the forces shall be zero; so we see that the last of the points obtained in this way must coincide with 0. That is to say, if ()PI, P2R1, R1R2, . . . etc. are drawn to represent all the forces which act on a particle in equilibrium, these must be sides of a closed polygon, which is called the polygon of forces. The order in which the forces are taken is evidently immaterial, and the polygon is not necessarily confined to one plane.

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