121.* (iv) CURVILINEAR MOTION ABOUT A CENTER OF FORCE.-Here the direction of the acceleration will always pass through a fixed point, and therefore the motion is that investigated in Arts. 61-85; the general properties of this kind of motion will be found in Arts. 62-66. If we suppose the force to be attractive and to vary as the distance or as (distance), the acceleration will be towards the center, and will vary according to the same law. 122. If the acceleration =λ (distance), the motion is that discussed in Arts. 68-75; the path being an ellipse whose center is the center of force: the period of revolution is 2π 仄 (Art. 75); therefore all bodies revolving about a common center of force varying as the distance describe their elliptic orbits in the same time, of whatever dimensions these orbits may be. 123. If the acceleration = (distance), the motion is that discussed in Arts. 76-85: the path is a conic section (generally an ellipse), and the center of force is in the focus. In the case of the ellipse the period of revolution is 2π 仄 a (Art. 84); whence if a number of bodies describe ellipses about a common center of force varying as the (distance), the squares of their periods of revolution vary as the cubes of the major axes of their orbits, i. e. as the cubes of their mean distances from the center of force. 124. In all these cases, when the acceleration is known, the force acting on the body in motion at any instant can be ascertained, by multiplying the numerical measure of the acceleration by the numerical measure of the mass of the body (Art. 107). 125. KEPLER'S LAWS.-The motion of the planets in their orbits about the Sun were discovered by Kepler to be subject to the three following laws; *If the reader be unacquainted with the Differential Calculus, he will find the results of the following Articles established in Newton, sect. ij. iij. ↑ (j) That each planet describes an ellipse about the Sun, which is situated in the focus thereof. (ij) That the sectorial areas swept out by the line joining the planet and the Sun in all equal intervals of time are equal. (iij) That the squares of the periods of any of the planets about the Sun vary as the cubes of their mean distances from it. 126. From the second of these laws we have the sectorial area swept out in any time o time of describing it; therefore if the Sun be taken for origin of polar co-ordinates, and A be the sectorial area described in time t, A= Ct, where C is a constant; whence, by Art. 34, there is no acceleration perpendicular to the radius vector, or the whole acceleration is along the radius vector; wherefore the whole force on any one planet is constantly directed towards the Sun. (NEWTON, Prop. 2.) 127. From the first law, compared with Art. 87, it will appear that the acceleration on the motion of any one planet varies as the (distance) from the Sun, or that the force on any planet varies as the (distance). -2 128. From the third law, if we take any two planets whose orbits have a, a' for their semi-major axes, the absolute accelerations* on them being λ, λ', at at :: period of first planet: period of second *By the absolute acceleration is meant the acceleration at distance unity. = whence λ, or the absolute accelerations on all the planets are equal; therefore, if they were all placed at the same distance from the Sun, the accelerations on them all would be the same. 129. It appears then that on the motion of the planets the force is central, varying as (distance), and producing an acceleration independent of the particular planet under discussion: we most reasonably conclude that this force resides in the Sun, i. e. that the planets move under the Sun's attraction only. This at first sight militates against the law of universal gravitation, but more delicate observations have shewn that these laws are not rigidly true, but very nearly so: and the fact that the mass of any of the planets is very small, compared with that of the Sun, would lead us to expect that this would be the case if the law of universal gravitation were true. CHAPTER VIII. OF CONSTRAINED MOTION OF PARTICLES. 130. IN these cases certain geometrical conditions are required to be satisfied, and unknown forces are involved arising from the constraint; these produce unknown accelerations; and the method pursued to determine the motion is deduced from the consideration, that if external forces be applied continually equal to the forces arising from the constraint, the cause of constraint may be withdrawn, and the motion is unaltered: this then becomes the case of free motion under some unknown forces but with some known conditions, i. e. the motion of a point under some unknown accelerations but with some known geometrical properties. 131. To determine the motion of a heavy particle down a smooth inclined plane. Let W be the weight of the particle; R the normal pressure on the plane, which by the Third Law of Motion = the reaction of the plane on the particle; g, E, the accelerations due to these respectively; the inclination of the plane to the horizon. W Then we have by the Second Law of Motion (Art. 105 (4)). And if a force continually equal to R be made to act constantly on the particle, in the direction from the plane to it, the plane may be removed, and the motion is free and the same as the constrained motion under consideration. Then the force in the direction down the plane = Wsin ; and the force perpendicular to the plane Wcos - R: therefore the acceleration down the plane = g sin ʊ, and the acceleration perpendicular to the plane = g cos i - §. But since there is no motion perpendicular to the plane, we must have And the acceleration down the plane, being g sin, is constant, and therefore the motion is that determined in Arts. 38-41, where a is to be replaced by g sin. giving the pressure on the plane, which we see is of constant intensity throughout the motion. If the particle had initially any velocity not directly up or down the plane, the motion would be parabolic, as in Arts. 42-47, wherein a will=g sin. 132. In the motion directly down an inclined plane, we have This is the same formula as that for a heavy body falling freely through a space AC, which is the difference of the heights of the two positions of the body in the case in question. The above reasoning will also apply if the motion be up the plane. This is generally cited thus: that the change of velocity is due to the vertical space through which the body has descended or ascended, irrespective of the inclination of the plane. L. L |