Symmetri- it, and make CA equal to CA'. For R, acting along the central axis, substitute (by § 561) R at each end of AA'. Then, choosing this line AA' as the arm of the couple, and calling it cal case. Composition of parallel forces. G a a, we have at one extremity of it, two forces, perpendicular to the central axis, and R parallel to the central axis. ComG pounding these we get two forces, each equal to (+12+), through A and A' respectively, perpendicular to AA', and -1 561. A very simple, but important, case, is that of any number of parallel forces acting at different points of a rigid body. Here, for equilibrium, obviously it is necessary and sufficient. that the algebraic sum of the forces be nil; and that the sum of their moments about any two axes perpendicular to the common direction of the forces be also nil. This clearly implies (§ 553) that the sum of their moments about any axis whatever is nil. To express the condition in rectangular coordinates, let P1, P., &c. be the forces; (x, y,, ≈1), (X, Y, Z), &c. points in their lines of action; and l, m, n the direction cosines of a line parallel to them all. The general equations [§ 552 (1), (2)] of equilibrium of a rigid body become in this case, ΙΣΡ=0, ΣΡ=0, «ΣΡ=0; nΣPy - mΣPz = 0, lΣPz-nΣPx = 0, mΣPx − lΣPy = 0. These equations are equivalent to but three independent equations, which may be written as follows: If the given forces are not in equilibrium a single force may be found which shall be their resultant. To prove this let, if possible, a force R, in the direction (l, m, n), at a point Equation (2) determines R, and equations (3) are the equations of a straight line at any point of which a force equal to - R, applied in the direction (l, m, n), will balance the given system. Suppose now the direction (l, m, n) of the given forces to be varied while the magnitude P1, and one point (x,, y,,,) in the line of application, of each force is kept unchanged. We see by (3) that one point (x, y, z) given by the equations 2 The point (x, y, z) given by equations (4) is what is called the centre of the system of parallel forces P, at (x,, y1, 2), P2 at (x, y, z), &c.: and we have the proposition that a force in the line through this point parallel to the lines of the given forces, equal to their sum, is their resultant. This proposition is easily proved synthetically by taking the forces in any order and finding the resultant of the first two, then the resultant of this and the third, then of this second force, and so on. The line of the first subsidiary resultant, for all varied directions of the given forces, passes through one and the same point (that is the point dividing the line joining the points of application of the first two forces, into parts inversely as their magnitudes). Similarly we see that the second subsidiary resultant passes always through one determinate point: and so for the third, and so on for any number of forces. gravity. 562. It is obvious, from the formulas of § 230, that if masses Centre of proportional to the forces be placed at the several points of application of these forces, the centre of inertia of these masses will be the same point in the body as the centre of parallel Centre of gravity. Parallel forces whose algebraic forces. Hence the reactions of the different parts of a rigid body against acceleration in parallel lines are rigorously reducible to one force, acting at the centre of inertia. The same is true approximately of the action of gravity on a rigid body of small dimensions relatively to the earth, and hence the centre of inertia is sometimes (§ 230) called the Centre of Gravity. But, except on a centrobaric body (§ 534), gravity is in general reducible not to a single force but to a force and couple (§ 559 g); and the force does not pass through a point fixed relatively to the body in all the positions for which the couple vanishes. 563. In one case the proposition of § 561, that the system has a single resultant force, must be modified: that is the case sum is zero. in which the algebraic sum of the given forces vanishes. In this case the resultant is a couple whose plane is parallel to the common direction of the forces. A good example of this case is furnished by a magnetized mass of steel, of moderate dimensions, subject to the influence of the earth's magnetism. The amounts of the so-called north and south magnetisms in cach element of the mass are equal, and are therefore subject to equal and opposite forces, parallel in a rigorously uniform field of force. Thus a compass-needle experiences from the earth's magnetism sensibly a couple (or directive action), and is not sensibly attracted or repelled as a whole. Conditions of equili brium of three forces. Physical axiom. 564. If three forces, acting on a rigid body, produce equilibrium, their directions must lie in one plane; and must all meet in one point, or be parallel. For the proof we may introduce a consideration which will be very useful to us in investigations connected with the statics of flexible bodies and fluids. If any forces, acting on a solid, or fluid body, produce equilibrium, we may suppose any portions of the body to become fixed, or rigid, or rigid and fixed, without destroying the equilibrium. Applying this principle to the case above, suppose any two points of the body, respectively in the lines of action of two of the forces, to be fixed. The third force must have no moment axiom. about the ne joining these points; in other words, its direction Physical must pass through that line. As any two points in the lines of action may be taken, it follows that the three forces are coplanar. And three forces, in one plane, cannot equilibrate unless their directions are parallel, or pass through a point. um under of gravity. 565. It is easy, and useful, to consider various cases of Equilibriequilibrium when no forces act on a rigid body but gravity the action and the pressures, normal or tangential, between it and fixed supports. Thus if one given point only of the body be fixed, it is evident that the centre of inertia must be in the vertical line through this point. For stable equilibrium the centre of inertia need not be below the point of support (§ 566). stones. 566. An interesting case of equilibrium is suggested by Rocking what are called Rocking Stones, where, whether by natural or by artificial processes, the lower surface of a loose mass of rock is worn into a convex or concave, or anticlastic form, while the bed of rock on which it rests in equilibrium may be convex or concave, or of an anticlastic form. A loaded sphere resting on a spherical surface is a particular case. Let O, O' be the centres of curvature of the fixed, and rocking, bodies respectively, when in the position of equilibrium. Take any two infinitely small, equal arcs PQ, Pp; and at Q make the angle R G Now if ρ and be the radii of curvature OP, OP of the two surfaces, and @ the angle POp, the angle QO'R will be equal to ро σ ; and we have in the triangle QO'R (§ 112) If the lower surface be plane, p is infinite, and the condition becomes (as in § 291) PG <0. If the lower surface be concave the sign of p must be changed, and the condition becomes Equilibrium, about an axis, on a fixed surface. ρ-σ which cannot be negative, since p must be numerically greater than in this case. 567. If two points be fixed, the only motion of which the system is capable is one of rotation about a fixed axis. The centre of inertia must then be in the vertical plane passing through those points. For stability it is necessary (§ 566) that the centre of inertia be below the line joining them. 568. If a rigid body rest on a frictional fixed surface there will in general be only three points of contact; and the body will be in stable equilibrium if the vertical line drawn from its centre of inertia cuts the plane of these three points within the triangle of which they form the corners. For if one of these supports be removed, the body will obviously tend to fall towards that support. Hence each of the three prevents the body from rotating about the line joining the other two. Thus, for instance, a body stands stably on an inclined plane (if the friction be sufficient to prevent it from sliding down) when the vertical line drawn through its centre of inertia falls within the base, or area bounded by the shortest line which can be drawn round the portion in contact with the plane. Hence a body, which cannot stand on a horizontal plane, may stand on an inclined plane. |