CASE 6. Forces applied to different points, but not pa rallet—If they are in one plane, you may produce them till they intersect, and then compound them two and two. If they arc not in one plane, but converge toward one point, you may consider them all as applied at that point, and then find their equivalent, as in Case 2. If they do not converge toward the same point, the case requires the reduction of forces, and will be explained under Art. 4. of next Section.
Resolution of Forces.
T1115 consists in resolving a single force into other to which it is equivalent, and which are to have given directions.
1. If the given directions are all in the same plant; with that of the given force, we can always perform the resolution into any number of forces, by inverting the process of Case I. last Section.
Thus, if two, we will make a parallelogram, of which the diagonal represents the given force, and the sides are parallel to the given directions. If three, we may first resolve a part of the given force into two of the given directions, and then the remainder into one of these directions, and the third direction.
2. When the given directions are not in the same plane with that of the given force, we cannot resolve it into two, but we can always resolve it into three given directions, provided two of them are in one plane. We can first resolve it into two forces, one of which is pa rallel to one of the given directions, and the other paral lel to the plane of the other two ; and then resolve this other into the other two given directions.
3. One important application of the resolution of for ces, is to find the effect of a force in a given direction, or in other words, to reduce it to a given direction.
It is evident that, in order to do this, we have only to resolve the force into two, one of which is parallel to the given direction, and the other at right angles to it. The latter can have no effect in the given direction, and therefore the other will express the whole effect.
4. Another important application of this subject is to the composition of forces.
The last case of composition cannot be managed un less by resolution. You may fix on three directions, two of them in one plane; resolve each force into three, parallel to these directions, find the equivalent of all those parallel to one line, then of those parallel to ano ther, Sze. then compound the three equivalents, if they pass through the same point; if not, the problem is im possible, that is, there is no single force equivalent to them all.
The most convenient directions to fix on will be three straight lines passing through the same point at right angles to one another, and the most convenient way of resolving, will be to calculate on the principle mention ed in Case 2d of last Section, that the three adjacent edges of a parallelopiped compose a force equal to the diagonal. Hence, since in this case the parallelopiped is rectangular, letf denote any force.
a the angle it makes with the parallel to one of the assumed lines.
Then R : cos. a : :f the force estimated in that direction.
Let b, c, d, denote the three equivalents found ; then, if they pass through the same point, they will be three edges of a rectangular parallelopiped, of which the dia gonal is the equivalent sought, which call m; then +c2. The same method may be applied with advantage to Case 2d of composition, when the forces are not all in one plane; and even when the for ces are all in one plane, a similar method will some times be convenient, only in that case it will be suffi cient to resolve each force in the direction of two per pendicular liAes. It is evident that in Case 2d the composition can always be effected, because the three equivalents will always pass through the point to which all the given forces are supposed to be applied.