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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 m×p.
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
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
direction mn, which is destroyed by the fixed point n, and another acting on m' in the direction m'm. Let mn=f, m'n=ƒ'; then m'mf+f' very nearly. Hence the whole force gm is to the part acting on m' :: na: mm', and the action of m on m', is gm (f+f'); but m'n : na :: 1 : w, for the arc is so small that it
may be taken for its sine. Hence na w.f', and the action of m on m' is 5m. (ƒ +ƒ')
In the same manner it may be shown that the action of m' on m gm' (f+f'); but when the bodies are in equilibrio, these forces
must be equal: therefore gm (ƒ +ƒ')
gm.f=gm'.f', or gm: gm' :: f'f, which is the law of equilibrium in the lever, and shows the reciprocal action of parallel forces.
Equilibrium of a System of Bodies.
125. The equilibrium of a system of bodies may be found, when the system is acted on by any forces whatever, and when the bodies also mutually act on, or attract each other.
Demonstration.-Let m, m', m', &c., be a system of bodies attracted by a force whose origin is in S, fig. 35; and suppose each body to act on all the other bodies, and also to be itself subject to the action of each, the action of all these forces on the bodies m, m', m", &c., are as the masses of these bodies and the intensities of the forces conjointly.
Let the action of the forces on one body, as m, be first considered; and, for simplicity,
suppose the number of bo-
the bodies m' and m".
Suppose m' and m" to remain fixed, and that m is arbitrarily moved to n: then mn is the virtual velocity of m; and if the per
pendiculars na, nb, nc be drawn, the lines ma, mb, mc, are the virtual velocities of m resolved in the direction of the forces which act on m. Hence, by the principle of virtual velocities, if the action of the force at S on m be multiplied by ma, the mutual action of m and m' by mb, and the mutual action of m and m" by mc, the sum of these products must be zero when the point m is in equilibrio; or, m being the mass, if the action of S on m be F.m, and the reciprocal actions of m on m' and m" be p, p', then
mF × ma + p × mb + p' × mc = 0.
Now, if m and m" remain fixed, and that m' is moved to n', then m'F' x m'a' + p × m'b' + p'' × m'c' = 0.
And a similar equation may be found for each body in the system. Hence the sum of all these equations must be zero when the system is in equilibrio. If, then, the distances Sm, Sm', Sm", be represented by s, s', s", and the distances mm', mm", m'm", by ƒ, f', ƒ", we shall have
E.mFds+2.pdf + Σ.pdf' ±, &c. = 0,
Σ being the sum of finite quantities; for it is evident that
df = mb + m'b', df' = mc + m''c", and so on.
If the bodies move on surfaces, it is only necessary to add the terms Ror, R'dr', &c., in which R and R' are the pressures or resistances of the surfaces, and dr dr' the elements of their directions or the variations of the normals. Hence in equilibrio
Σ.mFds +2.pdf + &c. + Rdr + R'dr', &c. = 0.
Now, the variation of the normal is zero; consequently the pressures vanish from this equation: and if the bodies be united at fixed distances from each other, the lines mm', m'm", &c., or f, f', &c., are constant:-consequently f= 0, f' 0, &c.
The distance f of two points m and m' in space is
ƒ = √(x − x)2 + (y' − y)2 + (z' — z)3,
x, y, z, being the co-ordinates of m, and x', y', z', those of m'; so that the variations may be expressed in terms of these quantities: and if they be taken such that df = 0, dƒ' = 0, &c., the mutual action of the bodies will also vanish from the equation, which is reduced to
126. Thus in every case the sum of the products of the forces into the elementary variations of their directions is zero when the system is in equilibrio, provided the conditions of the connexion of the
system be observed in their variations or virtual velocities, which are the only indications of the mutual dependence of the different parts of the system on each other.
127. The converse of this law is also true-that when the principle of virtual velocities exists, the system is held in equilibrio by the forces at S alone.
Demonstration.-For if it be not, each of the bodies would acquire a velocity v, v', &c., in consequence of the forces mF, m'F', &c. If dn, dn', &c., be the elements of their direction, then
The virtual velocities dn, dn', &c., being arbitrary, may be assumed equal to vdt, v'dt, &c., the elements of the space moved over by the bodies; or to v, v', &c., if the element of the time be unity. Hence Σ. mv 0.
It has been shown that in all cases 2.mFds 0, if the virtual velocities be subject to the conditions of the system. Hence, also, Σ.mv3 = 0; but as all squares are positive, the sum of these squares can only be zero if v = = 0, v' = 0, &c. Therefore the system must remain at rest, in consequence of the forces Fm, &c., alone.
128. Rotation is the motion of a body, or system of bodies, about a line or point. Thus the earth revolves about its axis, and bil
liard-ball about its centre.
129. A rotatory pressure or moment is a force that causes a system of bodies, or a solid body, to rotate about any point or line. It is expressed by the intensity of the motive force or momentum, multiplied by the distance of its direction from the point or line about which the system or solid body rotates.
On the Lever.
130. The lever first gave the idea of rotatory pressure or moments, for it revolves about the point of support or fulcrum.
When the lever mm', fig. 36, is in equilibrio, in consequence of forces applied to two heavy bodies at its extremities, the rotatory