| Fig. 22. |
The formal analytical reduction of a system of coplanar forces is as follows. Let (x1, y1), (x2, y2), ... be the rectangular co-ordinates of any points A1, A2, ... on the lines of action of the respective forces. The force at A1 may be replaced by its components X1, Y1, parallel to the co-ordinate axes; that at A2 by its components X2, Y2, and so on. Introducing at O two equal and opposite forces ±X1 in Ox, we see that X1 at A1 may be replaced by an equal and parallel force at O together with a couple −y1X1. Similarly the force Y1 at A1 may be replaced by a force Y1 at O together with a couple x1Y1. The forces X1, Y1, at O can thus be transferred to O provided we introduce a couple x1Y1 − y1X1. Treating the remaining forces in the same way we get a force X1 + X2 + ... or Σ(X) along Ox, a force Y1 + Y2 + ... or Σ(Y) along Oy, and a couple (x1Y1 − y1X1) + (x2Y2 − y2X2) + ... or Σ(xY − yX). The three conditions of equilibrium are therefore
Σ(X) = 0, Σ(Y) = 0, Σ(xY − yX) = 0.
(8)
If O′ be a point whose co-ordinates are (ξ, η), the moment of the couple when the forces are transferred to O′ as a new origin will be Σ{(x − ξ) Y − (y − η) X}. This vanishes, i.e. the system reduces to a single resultant through O′, provided
−ξ·Σ(Y) + η·Σ(X) + Σ(xY − yX) = 0.
(9)
If ξ, η be regarded as current co-ordinates, this is the equation of the line of action of the single resultant to which the system is in general reducible.
If the forces are all parallel, making say an angle θ with Ox, we may write X1 = P1 cos θ, Y1 = P1 sin θ, X2 = P2 cos θ, Y2 = P2 sin θ, .... The equation (9) then becomes
{Σ(xP) − ξ·Σ(P)} sin θ − {Σ(yP) − η·Σ(P)} cos θ = 0.