pressure of these forces, with regard to N, the point of support, must be equal and contrary. a m fig. 36. 772 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; or, N a ma x Nm xm = m'a' x Nm' x m'. But 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 mxmn xoP 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 mis at rest, the rotatory pressure must be zero; hence in equilibrio, m × mn xoP = 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 Join OE, then CD = dx, and AB = dy. noE = AnD, because the triangle and parallelogram are on the same base and between the same parallels; also moE = AE: hence the triangle therefore m (xdy-ydx) = 0 is the rotatory pressure in the plane roy when m is in equilibrio. A similar equation must exist for each coordinate plane when m is in a state of equilibrium with regard to each axis, therefore also m (xdz - zdx) = 0, m (ydz - zdy) = 0. The same may be proved for every body in the system, consequently when the whole is in equilibrio on the point o Em(xdy - ydx) = 0 Em (xdz-zdx) = 0 133. This property may be expressed by means of virtual velocities, namely, that a system of bodies will be at rest, if the sum of the products of their momenta by the elements of their directions be zero, or by article 125 EmFds=0. Since the mutual distances of the parts of the system are invariable, if the whole system be supposed to be turned by an indefinitely small angle about the axis oz, all the co-ordinates z', z", &c., will be invariable. If d be any arbitrary variation, and if the mutual distance of the bodies m and m' whose co-ordinates are x, y, z; x', y', z', there will arise {(x-x) (y-y) δω - (y' - y) (x' – x) δω } = 0. So that the values assumed for δα, δу, δα', δy' are not incompatible with the invariability of the system. It is therefore a permissible assumption. Now if s be the direction of the force acting on m, its variation is since z is constant; and substituting the preceding values of δα, δу, the result is In the same manner with regard to the body m' and so on; and thus the equation EmFds = 0 becomes 0. 2F{-}=0 It follows, from the same reasoning, that 2F{-1}=0, In fact, if X, Y, Z be the components of the force F in the direction of the three axes, it is evident that But EmFy expresses the sum of the moments of the forces parallel to the axis of a to turn the system round that of z, and EmFx that of the forces parallel to the axis of y to do the same, бу δε but estimated in the contrary direction; and it is evident that the forces parallel to z have no effect to turn the system round 2. Therefore the equation EmF ( = 0, expresses that the sum of the moments of rotation of the whole system relative to the axis of z must vanish, that the equilibrium of the system may subsist. And the same being true for the other rectangular axes (whose positions are arbitrary), there results this general theorem, viz., that in order that a system of bodies may be in equilibro upon a point, the sum of the moments of rotation of all the forces that act on it must vanish when estimated parallel to any three rectangular co-ordinates. 134. These equations are sufficient to ensure the equilibrium of the system when o is a fixed point; but if o, the point about which it rotates, be not fixed, the system, as well as the origin o, may be carried forward in space by a motion of translation at the same time that the system rotates about o, like the earth, which revolves about the sun at same time that it turns on its axis. In this case it is not only necessary for the equilibrium of the system that its rotatory pressure should be zero, but also that the forces which cause the translation when resolved in the direction of the axis or, oy, oz, should be zero for each axis separately. On the Centre of Gravity. 135. If the bodies m, m', m", &c., be only acted on by gravity, its effect would be the same on all of them, and its direction may be considered the same also; hence are the same in this case for all the bodies, so that the equations of or, if X, Y, Z, be considered as the components of gravity in the three It is evident that these equations will be zero, whatever the direction of gravity in the three co-ordinates or, oy, oz, δε Σm; F. F. δε δε бу . Ση; are the forces which translate the system parallel to these axes. But |
