acting in assigned directions. Thus a force can be uniquely
resolved into two components acting in two assigned directions
in the same plane with it by an inversion of the parallelogram
construction of fig. 1. If, as is usually most convenient, the
two assigned directions are at right angles, the two components
of a force P will be P cos θ, P sin θ, where θ is the inclination
of P to the direction of the
former component. This leads
to formulae for the analytical
reduction of a system of coplanar
forces acting on a
particle. Adopting rectangular
axes Ox, Oy, in the plane of
the forces, and distinguishing
the various forces of the system
by suffixes, we can replace the
system by two forces X, Y, in the direction of co-ordinate axes;
viz.—
Y = P1 sin θ1 + P2 sin θ2 + . . . = Σ (P sin θ).
These two forces X, Y, may be combined into a single resultant
R making an angle φ with Ox, provided
whence
For equilibrium we must have R = 0, which requires X = 0, Y = 0; in words, the sum of the components of the system must be zero for each of two perpendicular directions in the plane.
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| Fig. 5. |
A similar procedure applies to a three-dimensional system. Thus if, O being the origin, OH→ represent any force P of the system, the planes drawn through H parallel to the co-ordinate planes will enclose with the latter a parallelepiped, and it is evident that OH→ is the geometric sum of OA→, AN→, NH→, or OA→, OB→, OC→, in the figure. Hence P is equivalent to three forces Pl, Pm, Pn acting along Ox, Oy, Oz, respectively, where l, m, n, are the “direction-ratios” of OH→. The whole system can be reduced in this way to three forces
X = Σ (Pl), Y = Σ (Pm), Z = Σ (Pn),
acting along the co-ordinate axes. These can again be combined into a single resultant R acting in the direction (λ, μ, ν), provided
If the axes are rectangular, the direction-ratios become direction-cosines, so that λ2 + μ2 + ν2 = 1, whence
The conditions of equilibrium are X = 0, Y = 0, Z = 0.
§ 2. Statics of a System of Particles.—We assume that the mutual forces between the pairs of particles, whatever their nature, are subject to the “Law of Action and Reaction” (Newton’s Third Law); i.e. the force exerted by a particle A on a particle B, and the force exerted by B on A, are equal and opposite in the line AB. The problem of determining the possible configurations of equilibrium of a system of particles subject to extraneous forces which are known functions of the positions of the particles, and to internal forces which are known functions of the distances of the pairs of particles between which they act, is in general determinate. For if n be the number of particles, the 3n conditions of equilibrium (three for each particle) are equal in number to the 3n Cartesian (or other) co-ordinates of the particles, which are to be found. If the system be subject to frictionless constraints, e.g. if some of the particles be constrained to lie on smooth surfaces, or if pairs of particles be connected by inextensible strings, then for each geometrical relation thus introduced we have an unknown reaction (e.g. the pressure of the smooth surface, or the tension of the string), so that the problem is still determinate.
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| Fig. 6. |
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| Fig. 7. |
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| Fig. 8. |
The case of the funicular polygon will be of use to us later. A number of particles attached at various points of a string are acted on by given extraneous forces P1, P2, P3 . . . respectively. The relation between the three forces acting on any particle, viz. the extraneous force and the tensions in the two adjacent portions of the string can be exhibited by means of a triangle of forces; and if the successive triangles be drawn to the same scale they can be fitted together so as to constitute a single force-diagram, as shown in fig. 6. This diagram consists of a polygon whose successive sides represent the given forces P1, P2, P3 . . ., and of a series of lines connecting the vertices with a point O. These latter lines measure the tensions in the successive portions of string. As a special, but very important case, the forces P1, P2, P3 . . . may be parallel, e.g. they may be the weights of the several particles. The polygon of forces is then made up of segments of a vertical line. We note that the tensions have now the same horizontal projection (represented by the dotted line in fig. 7). It is further of interest to note that if the weights be all equal, and at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical. To prove this statement, let A, B, C, D . . . be successive vertices, and let H, K . . . be the middle points of AC, BD . . .; then BH, CK . . . will be vertical by the hypothesis, and since the geometric sum of BA→, BC→ is represented by 2BH→, the tension in BA: tension in BC: weight at B
The tensions in the successive portions of the string are therefore proportional to the respective lengths, and the lines BH, CK . . . are all equal. Hence AD, BC are parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through A, B, C, D. The same holds for the four points B, C, D, E and so on; but since a parabola is uniquely determined by the direction of its axis and by three points on the curve, the successive parabolas ABCD, BCDE, CDEF . . . must be coincident.
§ 3. Plane Kinematics of a Rigid Body.—The ideal rigid body is one in which the distance between any two points is invariable. For the present we confine ourselves to the consideration of displacements in two dimensions, so that the body is adequately represented by a thin lamina or plate.
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| Fig. 9. |
The position of a lamina movable in its own plane is determinate when we know the positions of any two points A, B of it. Since the four co-ordinates (Cartesian or other) of these two points are connected by the relation which expresses the invariability of the length AB, it is plain that virtually three independent elements are required and suffice to specify the position of the lamina. For instance, the lamina may in general be fixed by connecting any three points of it by rigid links to three fixed points in its plane. The three independent elements may be chosen in a variety of ways (e.g. they may be the lengths
