Fig. 3.
9. We now come to the important case where three forces act on a point: this is to be studied by the apparatus represented in [Fig. 3]. It consists essentially of two pulleys h, h, each about 2" diameter,[1] which are capable of turning very freely on their axles; the distance between these pulleys is about 5', and they are supported at a height of 6' by a frame, which will easily be understood from the figure. Over these pulleys passes a fine cord, 9' or 10' long, having a light hook at each of the ends e, f. To the centre of this cord d a short piece is attached, which at its free end g is also furnished with a hook. A number of iron weights, 0·5 lb., 1 lb., 2 lbs., &c., with rings at the top, are used; one or more of these can easily be suspended from the hooks as occasion may require.
10. We commence by placing one pound on each of the hooks. The cords are first seen to make a few oscillations and then to settle into a definite position. If we disturb the cords and try to move them into some new position they will not remain there; when released they will return to the places they originally occupied. We now concentrate our attention on the central point d, at which the three forces act. Let this be represented by o in [Fig. 4], and the lines op, oq, and os will be the directions of the three cords.
On examining these positions we find that the three angles p o s, q o s, p o q, are all equal. This may very easily be proved by holding behind the cords a piece of cardboard on which three lines meeting at a point and making equal angles have been drawn; it will then be seen that the cords coincide with the three lines on the cardboard.
Fig. 4.
11. A little reflection would have led us to anticipate this result. For the three cords being each stretched by a tension of a pound, it is obvious that the three forces pulling at o are all equal. As O is at rest, it seems obvious that the three forces must make the angles equal, for suppose that one of the angles, p o q for instance, was less than either of the others, experiment shows that the forces o p and oq would be too strong to be counteracted by o s. The three angles must therefore be equal, and then the forces are arranged symmetrically.
12. The forces being each 1 lb., mark off along the three lines in [Fig. 4] (which represent their directions) three equal parts o p, o q, o s, and place the arrowheads to show the direction in which each force is acting; the forces are then completely represented both in position and in magnitude.
Since these forces make equilibrium, each of them may be considered to be counteracted by the other two. For example, o s is annulled by o q and o p. But o s could be balanced by a force o r equal and opposite to it. Hence or is capable of producing by itself the same effect as the forces o p and oq taken together. Therefore o r is equivalent to o p and oq. Here we learn the important truth that two forces not in the same direction can be replaced by a single force. The process is called the composition of forces, and the single force is called the resultant of the two forces. o r is only one pound, yet it is equivalent to the forces o p and o q together, each of which is also one pound. This is because the forces o p and o q partly counteract each other.
13. Draw the lines p r and q r; then the angles p o r and q o r are equal, because they are the supplements of the equal angles p o s and q o s; and since the angles p o r and q o r together make up one-third of four right angles, it follows that each of them is two-thirds of one right angle, and therefore equal to the angle of an equilateral triangle. Also o p being equal to o q and o r common, the triangles o p r and o q r must be equilateral. Therefore the angle p r o is equal to the angle r o q; thus p r is parallel to o q; similarly q r is parallel to o p; that is, o p r q is a parallelogram. Here we first perceive the great law that the resultant of two forces acting at a point is the diagonal of a parallelogram, of which they are the two sides.