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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=f'; 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 gm. (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+f'), wf
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 bodies to be only three-m, m', and m". It is evident that m is attracted by the S force at S, and also urged by the reciprocal action of
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 f, f', ƒ", we shall have
E.mFds+2.pdƒ + Σ.pdƒ' ±, &c. = 0,
Σ being the sum of finite quantities; for it is evident that
sf=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 + Σ.pdƒ + &c. + Ròr + 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 df = 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)o,
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 dƒ= 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
Σ.mFds · Σ.ηυδη - 0.
The virtual velocities Sn, n', &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 2.mFds-2. mv2 = 0.
It has been shown that in all cases Σ.mFds = 0, if the virtual velocities be subject to the conditions of the system. Hence, also, Σ.mv2 = 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 billiard-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
pressure of these forces, with regard to N, the point of support, must
be equal and contrary.
Demonstration.-Let ma, m'a', fig.36, which are proportional to the velocities, represent the forces acting on m and m' during the indefinitely small time in which the bodies m and m' describe the indefinitely small spaces ma, m'a'. The distance of the direction of the forces ma, m'a' from the fixed point N, are Nm, Nm'; and the momentum of m into Nm, must be equal to the momentum of m' into Nm'; that is, the product of ma by Nm and the mass m, must be equal to the product of m'a' by Nm' and the mass m' when the lever is in equilibrio ;
ma x Nm x m = m'a' x Nm' x m'. But
max Nm is twice the triangle Nma, and
hence twice the triangle Nma into the mass m, is equal to twice the triangle Nm'a' into the mass m', and these are the rotatory pressures which cause the lever to rotate about the fulcrum; thus, in equilibrio, the rotatory pressures are equal and contrary, and the moments are inversely as the distances from the point of support.
Projection of Lines and Surfaces.
131. Surfaces and areas may be projected on the co-ordinate planes by letting fall perpendiculars from every point of them on these
be drawn parallel to oy, DC is the projection of mn on the axis or. In the same manner AB is the projection of the same line on oy.
Equilibrium of a System of Bodies invariably united.
132. A system of bodies invariably united will be in equilibrio upon a point, if the sum of the moments of rotation of all the forces that act upon it vanish, when estimated parallel to three rectangular co-ordinates.
Demonstration. Suppose a system of bodies invariably united, moving about a fixed point o in consequence of an impulse and a force of attraction; o being the origin of the attractive force and of the co-ordinates.
Let one body be considered at a time, and suppose it to describe the indefinitely small arc MN, fig. 37, in an indefinitely small time, and let mn be the projection of this arc on the plane roy. If m be the mass of the body, then m × mn is its momentum, estimated in the plane roy; and if oP be perpendicular to mn, it is evident that m x mn x op is its rotatory pressure. But mn x OP is twice the triangle mon; hence the rotatory pressure is equal to the mass m into twice the triangle mon that the body could describe in an element of time. But when m is at rest, the rotatory pressure must be zero; hence in equilibrio, m × mn × oP = 0.
Let omn, fig. 38, be the projected area, and complete the parallelogram oDEB; then if oD, oA, the co-ordinates of m, be represented by x and y, it is evident that y increases, while a diminishes; hence CD=- dr, and AB = dy. noEnD,
Join OE, then
because the triangle and parallelogram are on the same base and between the same parallels; also moEAE: hence the triangle
therefore m (xdy-ydx)= 0 is the rotatory pressure in the plane roy