MECHANICS 729 irallel rces. condition above shows that they must be equal and oppo site, and the second that they must act in the same line, if they are to maintain equilibrium. When only three forces act, the first condition shows that their directions must lie in one plane, the second that their lines of action must be parallel or must meet in one point, if they are to maintain equilibrium. If their directions meet in one point we have again the problem of the equilibrium of a single particle under three forces ; for there can be no moment about this point. When the directions are parallel, one of the forces must obviously be equal to the sum of the other two, and must act in the opposite direction. Also its line of action must lie between those of the other two, for their moments about any point in it must be equal and opposite. Hence it is impossible that any single force should balance a couple, unless we adopt the mathematical fiction of an infinitely small force acting in a line everywhere at an infinite distance ; so that its moment may be finite, and equal and opposite to that of the couple. 222. When any number of parallel forces act simul taneously on a rigid body, their resultant is a single force equal to their algebraic sum, with a couple whose plane is obviously parallel to the common direction of the forces. The forces of this couple may be made, by lengthening or shortening the arm, equal to the resultant force. One of them will neutralize it, and the other remains the final resultant, which passes through a definite point called the entre." " centre of parallel forces." Thus any set of parallel forces necessarily has a single force as a resultant, excepting in the special case when their algebraic sum is zero. avity. 223. Excellent examples are furnished by heavy bodies of moderate dimensions, where the weights of their parts are forces practically in parallel lines. The single re sultant force, in such cases, is the whole weight of the body. Its direction always passes through the centre of inertia ( 109) because weight (in any one locality) is proportional to mass. For this reason all heavy bodies of mtreof moderate dimensions are said to have a "centre of gravity," avity. which coincides with the centre of inertia. But it must be noticed that the two ideas are radically different, and that, while every piece of matter has a true centre of inertia, it is, in general, only approximately that we can predicate of it that it has a centre of gravity. In fact a body has a true centre of gravity only when it attracts, and is attracted by, all other gravitating matter as if its whole mass were con- centrited in that point. See POTENTIAL. When there is a centre of gravity in a body, it is necessarily coincident with the centre of inertia. In gravitation cases, where bodies of moderate size are concerned, the resultant is, at least approximately, a single force. But, when we deal with large non-barycentric bodies like the earth, we find that the resultant of the sun s attraction is a force (deter mining the orbit) and a couple (producing precession, &c.). When a mass is laid on a three-legged table, we find the pressure which each leg supports by simply taking moments about the line joining the upper ends of the other two. The leg is thus seen to support a fraction of the weight of the mass, whose nu merator is the distance of the centre of gravity of the mass from this line, and its denominator the distance of the leg from the same line. Thus we have a physical proof of the geomet rical proposition that if any point, P, be taken in the plane of a triangle ABC (fig. 60), and perpendiculars be drawn from it and from the angles, we have Fi
a /"i Lc If the mass of the table is to be reckoned, Pmust be taken as the centre of gravity of the system of table and load together. If the table be of uniform material, triangular, and supported by legs at its corners, similar reasoning shows that when it is unloaded (or loaded at its centre of gravity) each leg supports one third of the weight. 224. Examples in which the resultant is a single Resultant couple are found in rigidly magnetized bodies placed in a cou P le - a uniform magnetic field. As the amounts of N. and S. magnetism in a body are always equal, there is no force of translation in a uniform field. The resultant couple depends for its magnitude on the orientation of the body, and the positions of equilibrium are those for which its moment vanishes. 225. Let P at x, y, z, be one of a system of parallel forces, their direction cosines being A, /j., v . Let Q be the resultant force, and R, with direction cosines A , //, j/ , the axis of the resultant couple. Then our conditions become Q-3KP), The last three equations give the following conditions determin ing R, A , p, v : A R +At2(Pa)-v2(Py)-0, - A2( Pz) + /R + 1/2( Pa-) = , + -S(Y > y)- f j.2(rx) + v ll =0. From these AVC have the equation of condition AA + nn + vv = 0, showing that the axis of the couple is at right angles to the common direction of the parallel forces. We have also R a = (2(Poj)) + (2(P?/)) 2 + (2(P~)} 2 - (A2(Px) + M 2(Py) + 2(P~~)) 2 . This expression is of the same form as that in 77, and we there fore conclude that, if A", // , v" be the direction cosines of aline in the body such that A" /j." v" sTi^r^pz/r^ the magnitude of the resultant couple is directly as the sine of the angle between this line and the common direction of the paralh 1 forces. In fact the mere form of the three equations above proves this result. In the case of a body of moderate dimensions, acted on by gravity, P is the weight of the element at x, y, z, and therefore proportional to its mass, so that if the centre of inertia be taken as the origin we have and there is no couple. The whole effect is therefore the same as if the mass were condensed at the centre of iuerlia. In the case of a magnet, 2(P)-0, and there is no translatory force. The couple, as we have seen, depends upon the orientation of the body as regards the direction of the line of the earth s magnetic force. 226. We have seen that any system of forces acting Reduction on a rigid body may be reduced to a force and a couple ; also to two that when the force is in the plane of the couple the forces< resultant can always be put in the form of a single force acting in a definite line in the body. When the force is not in the plane of the couple, we may resolve the couple into two components, the plane of one being parallel, of the other perpendicular, to the force. The first, when compounded with the force, merely shifts the line in which it acts. Thus any system of forces may be reduced to a single force, acting in a definite line called the " central Central axis," and a couple in a plane perpendicular to it. One of axis - the forces of the couple may now be compounded with the single force, and thus we obtain, as the resultant of any system of forces, a pair of forces in non-intersecting lines not perpendicular to one another. This is only one of an infinite number of ways in which fancy, or convenience, may lead us to represent the equivalent of a group of forces. Many very curious theorems have been met with in investigations on this subject. For instance, by compounding one of the forces of the resultant couple with the resultant force (not
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