CHAPTER XIII GRAVITATION 103. Attraction and Repulsion.-In the case of two portions of matter between which a stress exists, and in which we are unable to trace any material connecting link, it is usual, if the stress tends to make the bodies move towards one another, to say that the bodies attract one another. If, on the other hand, the stress is such as to tend to make the bodies separate, then we say they repel one another. For convenience, the force in the case where the bodies repel one another is generally regarded as positive, and that in the case where they attract one another as negative. 104. The Law of Inverse Squares.—In general, when the distance between two particles which attract or repel one another changes, the stress between the particles alters. The only case which we shall examine, since it is by far the most important in physics, is that in which the stress varies inversely as the square of the distance between the particles, and takes place in the direction of the line joining them. As a particular case of this general law, which is called the law of the inverse square, we may take the case of the attraction between two particles of mass m and m'. The stress between the particles depends directly on the product of the masses and inversely on the square of the distance (a) between them. Hence if F is the force, which either particle exerts on the other, If, instead of having only two particles, there are a number, then experiment shows that the force exerted on each particle is the resultant of all the forces which would be exerted by each of the other particles separately. 105*. Work done by Attraction or Repulsion.-If the distance between two particles which attract one another is increased, work will have to be done on the system. If the particles approach one another, however, they are capable of doing work. The maximum of work will be done by or on the system when the particles are brought from an infinite Kepler's laws are as follows: P 1. The areas swept out by the straight line joining a planet to the sun are proportional to the time. Thus in Fig. 79, if s is the position of the sun, and PP2P3... P is the orbit of a planet, and in a given interval the planet moves from P to P, or from P2 to P3, or from Pa to P, &c., then the areas P1SP PSP3, and PSP, &c., are all equal. 2. The orbit of a planet is an ellipse, the sun being at one of the foci. 3. The squares of the time taken to describe its orbit by different planets are proportional to the cubes of the mean distances of the planets S FIG. 79. from the sun. Thus if 7, and 71⁄2 are the times taken by two planets to describe their orbits, and D, and D2 are their distances from the sun, then Kepler's third law states that 2 108. Newton's Law of Gravitation.—Although Kepler's laws give us a description of the motion of the planets, they do not tell us anything about the forces which serve to determine these motions. Newton, however, discovered the dynamical interpretation of Kepler's laws, and ' showed that if we suppose that a stress is set up between each of the planets and the sun directly proportional to the mass of the planet and inversely proportional to the square of the distance of the planet from the sun, then the motions of the planets will be just such as would satisfy Kepler's laws. Although it had been previously suggested that the sun as a whole attracted each planet as a whole, and the law of the inverse square had even been enunciated, it is to Newton that we owe the law of gravitation in the form in which it remains to this day, viz. every portion of matter attracts every other portion of matter, and the stress between them is proportional to the product of their masses divided by the square of their distance apart. As a test of the truth of his law, Newton showed that it correctly accounted for the force necessary to retain the moon in her orbit. If we assume that the orbit of the moon (with reference to the earth) is a circle of radius R, then by § 42 the acceleration of the moon towards the earth necessary to keep it moving in this orbit will be where is the linear velocity with which the moon is moving in the kmm' 2 I (d1 = dd)2+(d, − 2òd)2] ôd, and so on. The total work done will be the sum of a number of such terms taken for the whole path. If d is the distance AB' or AE, then the work done between B and E, and therefore also between B and B', can be shown1 to be kmm' If the point B is at an infinite distance from A, then the work done by the body in the case of attraction, or against the body in the case of repulsion, when moved up from an infinite distance to a distance d from A, is kmm' If the body which is moved is of unit mass, i.e. if m'= 1, then the work done is km da In the case of two bodies which repel one another, and are at a distance do, they possess a certain potential energy due to their mutual repulsion. The amount of this potential energy is kmm' since this expression gives the maximum amount of work which could be done by the mutual repulsion of the bodies, for there is no force exerted between the bodies when they are at an infinite distance, so that their potential energy, as far as their mutual repulsion is concerned, is zero. 106*. Potential.-It has been shown, in the last section, that the work which has to be done to remove a unit mass from a given point in the neighbourhood of another mass, to a place where there ceases to be any attractive or repulsive force between the masses, is a fixed quantity depending simply on the position of the point with reference to the attracting mass. This quantity of work may be looked upon as an attribute of the given point, the attracting mass of course being supposed to remain fixed in position, and it is then called the potential of the given point. Thus the potential at a point at a distance d from a mass m is equal to km 107. Kepler's Laws.-Astronomical observations having shown that the earth and the other planets move round the sun in approximately circular (really elliptical) orbits, it follows that there must be attraction between each planet and the sun, for otherwise the planets would travel in straight lines. Kepler, by a careful study of the observations on the motion of the planets made by Tycho Brahe, deduced three laws which now bear his P 3 P Pa Ps Ро 1. The areas swept out by the straight line joining a planet to the sun are proportional to the time. Thus in Fig. 79, if s is the position of the sun, and P1P2P3... P is the orbit of a planet, and in a given interval the planet moves from P to P, or from P2 to P, or from Ps to P4, &c., then the areas P1SP, PSP, and PSP4, &c., are all equal. 2. The orbit of a planet is an ellipse, the sun being at one of the foci. 3. The squares of the time taken to describe its orbit by different planets are proportional to the cubes of the mean distances of the planets P S FIG. 79. from the sun. Thus if 7, and T2 are the times taken by two planets to describe their orbits, and D, and D2 are their distances from the sun, then Kepler's third law states that 108. Newton's Law of Gravitation.-Although Kepler's laws give us a description of the motion of the planets, they do not tell us anything about the forces which serve to determine these motions. Newton, however, discovered the dynamical interpretation of Kepler's laws, and showed that if we suppose that a stress is set up between each of the planets and the sun directly proportional to the mass of the planet and inversely proportional to the square of the distance of the planet from the sun, then the motions of the planets will be just such as would satisfy Kepler's laws. Although it had been previously suggested that the sun as a whole attracted each planet as a whole, and the law of the inverse square had even been enunciated, it is to Newton that we owe the law of gravitation in the form in which it remains to this day, viz. every portion of matter attracts every other portion of matter, and the stress between them is proportional to the product of their masses divided by the square of their distance apart. As a test of the truth of his law, Newton showed that it correctly accounted for the force necessary to retain the moon in her orbit. If we assume that the orbit of the moon (with reference to the earth) is a circle of radius R, then by § 42 the acceleration of the moon towards the earth necessary to keep it moving in this orbit will be where is the linear velocity with which the moon is moving in the circular orbit. If 7 is the time the moon takes to complete the orbi (a sidereal month), then Since the force exerted by the attraction of the earth on a given mass is proportional to the acceleration produced in the mass (§ 61), it follows that if Newton's law is true, i.e. if the force decreases as the square of the distance, then if is the radius of the earth and g the acceleration produced by gravity at the surface of the earth, the acceleration (a') produced by the gravitational attraction of the earth at the distance of the moon will be given by In order to calculate the values of a and a1 we may take Since g is expressed in feet per second per second, we must reduce R and to feet and 7 to seconds, then substituting we get The agreement is as good as the approximate values we have assumed for the various quantities will allow. 109. The Cavendish Experiment.-The calculation given in the last section shows that the moon is attracted by the earth with a force which follows the same law as the attraction exerted by the earth on bodies on its surface. We now proceed to show that two bodies of such a size that we can handle them attract one another. The experimental difficulties of carrying out this investigation are, however, very great, since the mass of the largest body which we can employ is so excessively small as compared with the mass of the earth, and hence the attraction between any two bodies we can use is only a small fraction of the weight of either. The first apparatus for measuring the gravitational attraction of two bodies was designed by the Rev. John Michell, but he died before he |