14. This remarkable geometrical figure is called the parallelogram of forces. Stated in its general form, the property we have discovered asserts that two forces acting at a point have a resultant, and that this resultant is represented both in magnitude and in direction by the diagonal of the parallelogram, of which two adjacent sides are the lines which represent the forces.
Fig. 5.
15. The parallelogram of forces may be illustrated in various ways by means of the apparatus of [Fig. 3]. Attach, for example, to the middle hook g 1·5 lb., and place 1 lb. on each of the remaining hooks e, f. Here the three weights are not equal, and symmetry will not enable us, as it did in the previous case, to foresee the condition which the cords will assume; but they will be observed to settle in a definite position, to which they will invariably return if withdrawn from it.
Let o p, o q ([Fig. 5]) be the directions of the cords; o p and o q being each of the length which corresponds to 1 lb., while o s corresponds to 1·5 lb. Here, as before, o p and o q together may be considered to counteract o s. But o s could have been counteracted by an equal and opposite force o r. Hence o r may be regarded as the single force equivalent to o p and o q, that is, as their resultant; and thus it is proved experimentally that these forces have a resultant. We can further verify that the resultant is the diagonal of the parallelogram of which the equal forces are the sides. Construct a parallelogram on a piece of cardboard having its four sides equal, and one of the diagonals half as long again as one of the sides. This may be done very easily by first drawing one of the two triangles into which the diagonal divides the parallelogram. The diagonal is to be produced beyond the parallelogram in the direction o s. When the cardboard is placed close against the cords, the two cords will lie in the directions o p, o q, while the produced diagonal will be in the vertical o s. Thus the application of the parallelogram of force is verified.
Fig. 6.
16. The same experiment shows that two unequal forces may be compounded into one resultant. For in [Fig. 5] the two forces o p and o s may be considered to be counterbalanced by the force o q; in other words, o q must be equal and opposite to a force which is the resultant of o p and o s.
17. Let us place on the central hook g a weight of 5 lbs., and weights of 3 lbs. on the hook e and 4 lbs. on f. This is actually the case shown in [Fig. 3]. The weights being unequal, we cannot immediately infer anything with reference to the position of the cords, but still we find, as before, that the cords assume a definite position, to which they return when temporarily displaced. Let [Fig. 6] represent the positions of the cords. No two of the angles are in this case equal. Still each of the forces is counterbalanced by the other two. Each is therefore equal and opposite to the resultant of the other two. Construct the parallelogram on cardboard, as can be easily done by forming the triangle o p r, whose sides are 3, 4, and 5, and then drawing o q and r q parallel to r p and o p. Produce the diagonal o r to s. This parallelogram being placed behind the cords, you see that the directions of the cords coincide with its sides and diagonal, thus verifying the parallelogram of forces in a case where all the forces are of different magnitudes.
18. It is easy, by the application of a set square, to prove that in this case the cords attached to the 3 lb. and 4 lb. weights are at right angles to each other. We could have inferred, from the parallelogram of force, that this must be the case, for the sides of the triangle o p r are 3, 4, and 5 respectively, and since the square of 5 is 25, and the squares of 3 and of 4 are 9 and 16 respectively, it follows that the square of one side of this triangle is equal to the sum of the squares of the two opposite sides, and therefore this is a right-angled triangle (Euclid, i. 48). Hence, since p r is parallel to o q, the angle p o q must also be a right angle.