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meter, or as 1.57079 to 2. Thus the straight line AB, though the shortest that can be drawn between the points B and A, is not the line of quickest descent.
Curve of quickest Descent.
113. In order to find the curve in which a heavy body will descend from one given point to another in the shortest time possible, let CP = z, PM = y, and CM = s, fig. 33. The velocity of a body moving in the
must be a minimum, or, by the method of variations,
The values of z and z' are the same for any curves that can be drawn between the points M and m': hence Ɛdz = 0 Ɛdz' = 0. Besides, whatever the curves may be, the ordinate om' is the same for all; hence dydy' is constant, therefore (dy + dy') = 0: whence Sdy-ddy'; and d.
0, from these considerations,
= 0. Now it is evident, that the
second term of this equation is only the first term in which each variable quantity is augmented by its increment, so that
But is the sine of the angle that the tangent to the curve
ds makes with the line of the abscissæ, and at the point where the tan
gent is horizontal this angle is a right angle, so that = 1: hence
the equation to the cycloid, which is the curve of quickest descent.
ON THE EQUILIBRIUM OF A SYSTEM OF BODIES.
Definitions and Axioms.
114. ANY number of bodies which can in any way mutually affect each other's motion or rest, is a system of bodies.
115. Momentum is the product of the mass and the velocity of a body.
116. Force is proportional to velocity, and momentum is proportional to the product of the velocity and the mass; hence the only difference between the equilibrium of a particle and that of a solid body is, that a particle is balanced by equal and contrary forces, whereas a body is balanced by equal and contrary momenta.
117. For the same reason, the motion of a solid body differs from the motion of a particle by the mass alone, and thus the equation of the equilibrium or motion of a particle will determine the equilibrium or motion of a solid body, if they be multiplied by its mass.
118. A moving force is proportional to the quantity of momentum generated by it.
Reaction equal and contrary to Action.
119. The law of reaction being equal and contrary to action, is a general induction from observations made on the motions of bodies when placed within certain distances of one another; the law is, that the sum of the momenta generated and estimated in a given direction is zero. It is found by experiment, that if two spheres A and B of the same dimensions and of homogeneous matter, as of gold, be suspended by two threads so as to touch one another when at rest, then if they be drawn aside from the perpendicular to equal heights and let fall at the same instant, they will strike one another centrically, and will destroy each other's motion, so as to remain at rest in the perpendicular. The experiment being repeated with spheres of homogeneous matter, but of different dimensions, if the velocities be inversely as the quantities of matter, the bodies
after impinging will remain at rest. It is evident, that in this case, the smaller sphere must descend through a greater space than the larger, in order to acquire the necessary velocity. If the spheres move in the same or in opposite directions, with different momenta, and one strike the other, the body that impinges will lose exactly the quantity of momentum that the other acquires. Thus, in all cases, it is known by experience that reaction is equal and contrary to action, or that equal momenta in opposite directions destroy one another. Daily experience shows that one body cannot acquire motion by the action of another, without depriving the latter body of the same quantity of motion. Iron attracts the magnet with the same force that it is attracted by it; the same thing is seen in electrical attractions and repulsions, and also in animal forces; for whatever may be the moving principle of man and animals, it is found they receive by the reaction of matter, a force equal and contrary to that which they communicate, and in this respect they are subject to the same laws as inanimate beings.
Mass proportional to Weight.
120. In order to show that the mass of bodies is proportional to their weight, a mode of defining their mass without weighing them must be employed; the experiments that have been described afford the means of doing so, for having arrived at the preceding results, with spheres formed of matter of the same kind, it is found that one of the bodies may be replaced by matter of another kind, but of different dimensions from that replaced. That which produces the same effects as the mass replaced, is considered as containing the same mass or quantity of matter. Thus the mass is defined independent of weight, and as in any one point of the earth's surface every particle of matter tends to move with the same velocity by the action of gravitation, the sum of their tendencies constitutes the weight of a body; hence the mass of a body is proportional to its weight, at one and the same place.
121. Suppose two masses of different kinds of matter, A, of hammered gold, and B of cast copper. If A in motion will destroy the
motion of a third mass of matter C, and twice B is required to produce the same effect, then the density of A is said to be double the density of B...
Mass proportional to the Volume into the Density.
122. The masses of bodies are proportional to their volumes multiplied by their densities; for if the quantity of matter in a given cubical magnitude of a given kind of matter, as water, be arbitrarily assumed as the unit, the quantity of matter in another body of the same magnitude of the density p, will be represented by p; and if the magnitude of the second body to that of the first be as m to 1, the quantity of matter in the second body will be represented by mxp.
123. The densities of bodies of equal volumes are in the ratio of their weights, since the weights are proportional to their masses; therefore, by assuming for the unit of density the maximum density of distilled water at a constant temperature, the density of a body will be the ratio of its weight to that of a like volume of water reduced to this maximum.
This ratio is the specific gravity of a body.
Equilibrium of two Bodies.
124. If two heavy bodies be attached to the extremities of an inflexible line without mass, which may turn freely on one of its points; when in equilibrio, their masses are reciprocally as their distances from the point of motion.
Demonstration.-For, let two heavy bodies, m and m', fig. 34, be attached to the extremities of an inflexible line, free to turn round one of its points n, and suppose the
line to be bent in n, but so little, that m'nm only dif
fers from two right angles by an indefinitely small angle amn, which may be represented by w. If g be the force of gravitation, gm, gm' will be the gravitation of the two bodies. But the gravitation gm acting in the direction na may be resolved into two forces, one in the