EQUILIBRIUM OF THREE FORCES.
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.
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.
A SMALL FORCE SOMETIMES BALANCES
TWO LARGER FORCES.
19. Cases might be multiplied indefinitely by placing various amounts of weight on the hooks, constructing the parallelogram on cardboard, and comparing it with the cords as before. We shall, however, confine ourselves to one more illustration, which is capable of very remarkable applications. Attach 1 lb. to each of the hooks e and f; the cord joining them remains straight until drawn down by placing a weight on the centre hook. A very small weight will suffice to do this. Let us put on half-a-pound; the position the cords then assume is indicated in [Fig. 7]. As before, each force is equal and opposite to the resultant of the other two. Hence a force of half-a-pound is the resultant of two forces each of 1 lb. The apparent paradox is explained by noticing that the forces of 1 lb. are very nearly opposite, and therefore to a large extent counteract each other. Constructing the cardboard parallelogram we may easily verify that the principle of the parallelogram of forces holds in this case also.
Fig. 7.
20. No matter how small be the weight we suspend from the middle of a horizontal cord, you see that the cord is deflected: and no matter how great a tension were applied, it would be impossible to straighten the cord. The cord could break, but it could not again become horizontal. Look at a telegraph wire; it is never in a straight line between two consecutive poles, and its curved form is more evident the greater be the distance between the poles. But in putting up a telegraph wire great straining force is used, by means of special machines for the purpose; yet the wires cannot be straightened: because the weight of the heavy wire itself acts as a force pulling it downwards. Just as the cord in our experiments cannot be straight when any force, however small, is pulling it downwards at the centre, so it is impossible by any exertion of force to straighten the long wire. Some further illustrations of this principle will be given in our next lecture, and with one application of it the present will be concluded.
21. One of the most important practical problems in mechanics is to make a small force overcome a greater. There are a number of ways in which this may be accomplished for different purposes, and to the consideration of them several lectures of this course will be devoted. Perhaps, however, there is no arrangement more simple than that which is furnished by the principles we have been considering. We shall employ it to raise a 28 lb. weight by means of a 2 lb. weight. I do not say that this particular application is of much practical use. I show it to you rather as a remarkable deduction from the parallelogram of forces than as a useful machine.
Fig. 8.
A rope is attached at one end of an upright, a ([Fig. 8]), and passes over a pulley B at the same vertical height about 16' distant. A weight of 28 lbs. is fastened to the free end of the rope, and the supports must be heavily weighted or otherwise secured from moving. The rope ab is apparently straight and horizontal, in consequence of its weight being inappreciable in comparison with the strain (28 lbs.) to which it is subjected; this position is indicated in the figure by the dotted line ab. We now suspend from c at the middle of the rope a weight of 2 lbs. Instantly the rope moves to the position represented in the figure. But this it cannot do without at the same moment raising slightly the 28 lbs., for, since two sides of a triangle, cb, ca, are greater than the third side, ab, more of the rope must lie between the supports when it is bent down by the 2 lb. weight than when it was straight. But this can only have taken place by shortening the rope between the pulley b and the 28 lb. weight, for the rope is firmly secured at the other end. The effect on the heavy weight is so small that it is hardly visible to you from a distance. We can, however, easily show by an electrical arrangement that the big weight has been raised by the little one.
22. When an electric current passes through this alarum you hear the bell ring, and the moment I stop the current the bell stops. I have fastened one piece of brass to the 28 lb. weight, and another to the support close above it, but unless the weight be raised a little the two will not be in contact; the electricity is intended to pass from one of these pieces of brass to the other, but it cannot pass unless they are touching. When the rope is straight the two pieces of brass are separated, the current does not pass, and our alarum is dumb; but the moment I hang on the 2 lb. weight to the middle of the rope it raises the weight a little, brings the pieces of brass in contact, and now you all hear the alarum. On removing the 2 lbs. the current is interrupted and the noise ceases.
23. I am sure you must all have noticed that the 2 lb. weight descended through a distance of many inches, easily visible to all the room; that is to say, the small weight moved through a very considerable distance, while in so doing it only raised the larger one a very small distance. This is a point of the very greatest importance; I therefore take the first opportunity of calling your attention to it.
LECTURE II.
THE RESOLUTION OF FORCES.
Introduction.—One Force resolved into Two Forces.—Experimental Illustrations.—Sailing.—One Force resolved into Three Forces not in the same Plane.—The Jib and Tie-rod.