CASE 1. Two forces applied at the same point cf a body.
If they act in the same direction, the equivalent will be in that direction, and equal to their sum; if in oppo site directions, the equivalent will be the direction of the greater, and equal to this difference ; tf at an the equivalent will be in the same plane, and represeht ed by the diagonal of a parallelogram of which the two sides represent the simple forces.
This follows immediately from the second law of mo tion ; the only case which needs any demonstration is that when the forces act at an angle.
If the one force alone would make the body move along A13, (Plate CCXLI. Fig. 1.) and the other along AC in the same time, the two by their joint action will make it move along the diagonal AD in the same time.
For by the second law, its motion parallel to AB will he equal to AB, and its motion parallel to AC will be equal to AC ; hence at the end of the time it must be at D. In like manner, suppose that by the separate action of the forces, the body would at any intermediate point of time be at E and I'', then completing the para lelogram FE, their joint action will bring it to G at that time. But it is easy to sec that AB : AE : : AC : AF. For both motions being uniform, (first law of motion,) the spaces past over in both cam.. s are proportional to the times, and hence proportional to one another. Hence the two parallelograms are similar, and consequently about the saute diagonal, (Euc. xxvi. 6 ) If the forces are pressures, the effects of thc momen tart' action will be extremely small, but the same de monstration still applies. Though equilibrium is not the direct object of this article, yet the following remarks may not be improper, as tending to illustrate the subject. If no motion ensues, still the diagonal will express the tendency to move, as well as the direction of that tenden cy. It will express the force applied in the contrary' di rection, which has destroyed the motion, whether that force be an active one, or merely the reaction of a fixed obstacle. In the case of pressures, it will express the
pressure that arises against the obstacle; a conclusion which we are justified in drawing from the analogy es tablished in the first Section betwixt pressures and the motions which they generate, and which, besides, is con firmed by experience. The case of impulses is illus trated in some treatises of natural philosophy, by a com plex piece of machinery. The case of pressures may be illustrated by the following simple apparatus.
Let the three weights hung around the two fixed pul leys, as in the Figure, (Fig. 2.) be in a state of equili let All, AC, be taken proportional to the weights N, i‘1, respectively, which act in the directions of All. AC; then the parallelogram being constructed, the dia gonal AD, which is the equivalent according to our pro position, will be found to be vertical, that is directly op posite to P, and also proportional to P.
Colt. I. The equivalent is represented by the third side of a triangle, whose other two sides are drawn in succession parallel and equal to the single forces. This is evident by inspecting the triangle ABD (Fig. I.) It is evident also, that the included angle of this triangle is the supplement of that at which the forces act.
The sides of any triangle perpendicular respectively to those of ABD, will also express the magnitudes of the forces, and of their equivalent ; for such a triangle will be similar to ABD.
COB. 2. Given two forces and the angle which their directions make, the equivalent may be found either by constructing the paralellogram, or the triangle, or by trigonometrical calculation. The angle BAD being found, will express the direction of the equivalent.
It is easy to slim by help of Eucl. 12, 13. 2. that if a, b denote the two forces, c the angle which they make. and d their equivalent, Then 2a bx cos. , the radius being unity ; also tato sine of the angle which d makes with a. is ax sine, c d It is easy to see from 47.1 Eucl. that if the angle is right, then (1=V .