Experimental Mechanics / A Course of Lectures Delivered at the Royal College of Science for Ireland

by Robert S. Ball

SAILING.

30. The action of the wind upon the sails of a vessel affords a very instructive and useful example of the decomposition of forces. By the parallelogram of forces we are able to explain how it is that a vessel is able even to sail against the wind. A force is that which tends to produce motion, and motion generally takes place in the line of the force. In the case of the action of wind on a vessel through the medium of the sails, we have motion produced which is not necessarily in the direction of the wind, and which may be to a certain extent opposed to it. This apparent paradox requires some elucidation.

Fig. 13.

31. Let us first suppose the wind to be blowing in a direction shown by the arrows of [Fig. 13], perpendicular to the line ab in which the ship’s course lies.

In what direction must the sail be set? It is clear that the sail must not be placed along the line ab, for then the only effect of the wind would be to blow the vessel sideways; nor could the sail be placed with its edge to the wind, that is, along the line o w, for then the wind would merely glide along the sail without producing a propelling force. Let, then, the sail be placed between the two positions, as in the direction p q. The line o w represents the magnitude of the force of the wind pressing on the sail.

We shall suppose for simplicity that the sail extends on both sides of o. Through o draw o r perpendicular to p q, and from w let fall the perpendicular w x on p q, and w r on o r. By the principle of the parallelogram of forces, the force o w may be decomposed into the two forces o x and o r, since these are the sides of the parallelogram of which o w, the force of the wind, is the diagonal. We may then leave o w out of consideration, and imagine the force of the wind to be replaced by the pair of forces o x and o r; but the force o x cannot produce an effect, it merely represents a force which glides along the surface of the sail, not one which pushes against it; so far as this component goes, the sail has its edge towards it, and therefore the force produces no effect. On the other hand, the sail is perpendicular to the force o r, and this is therefore the efficient component.

The force of the wind is thus measured by o r, both in magnitude and direction: this force represents the actual pressure on the mast produced by the sail, and from the mast communicated to the ship. Still o r is not in the direction in which the ship is sailing: we must again decompose the force in order to find its useful effect. This is done by drawing through r the lines r l and r m parallel to o a and o w, thus forming the parallelogram o m r l. Hence, by the parallelogram of forces, the force o r is equivalent to the two forces o l and o m.

The effect of o l upon the vessel is to propel it in a direction perpendicular to that in which it is sailing. We must, therefore, endeavour to counteract this force as far as possible. This is accomplished by the keel, and the form of the ship is so designed as to present the greatest possible resistance to being pushed sideways through the water: the deeper the keel the more completely is the effect of o l annulled. Still o l would in all cases produce some leeway were it not for the rudder, which, by turning the head of the vessel a little towards the wind, makes her sail in a direction sufficiently to windward to counteract the small effect of o l in driving her to leeward.

Thus o l is disposed of, and the only force remaining is o m, which acts directly to push the vessel in the required direction. Here, then, we see how the wind, aided by the resistance of the water, is able to make the vessel move in a direction perpendicular to that in which the wind blows. We have seen that the sail must be set somewhere between the direction of the wind and that of the ship’s motion. It can be proved that when the direction of the sail supposed to be flat and vertical, is such as to bisect the angle w o b, the magnitude of the force o m is greater than when the sail has any other position.

32. The same principles show how a vessel is able to sail against the wind: she cannot, of course, sail straight against it, but she can sail within half a right angle of it, or perhaps even less. This can be seen from [Fig. 14].

The small arrows represent the wind, as before. Let o w be the line parallel to them, which measures the force of the wind, and let the sail be placed along the line p q; o w is decomposed into o x and o y, o x merely glides along the sail, and o y is the effective force. This is decomposed into o l and o m; o l is counteracted, as already explained, and o m is the force that propels the vessel onwards. Hence we see that there is a force acting to push the vessel onwards, even though the movement be partly against the wind.

Fig. 14.

It will be noticed in this case that the force o l acting to leewards exceeds o m pushing onwards. Hence it is that vessels with a very deep keel, and therefore opposing very great resistance to moving leewards, can sail more closely to the wind than others not so constructed; a vessel should be formed so that she shall move as freely as possible in the direction of her length, for which reason she is sharpened at the bow, and otherwise shaped for gliding through the water easily; this is in order that o m may have to overcome as little resistance as possible. If the sail were flat and vertical it should bisect the angle a ow for the wind to act in the most efficient manner. Since, then, a vessel can sail towards the wind, it follows that, by taking a zigzag course, she can proceed from one port to another, even though the wind be blowing from the place to which she would go towards the place from which she comes. This well known manœuvre is called “tacking.” You will understand that in a sailing-vessel the rudder has a more important part to play than in a steamer: in the latter it is only useful for changing the direction of the vessel’s motion, while in the former it is not only necessary for changing the direction, but must also be used to keep the vessel to her course by counteracting the effect of leeway.

ONE FORCE RESOLVED INTO THREE FORCES
NOT IN THE SAME PLANE.

Fig. 15.

33. Up to the present we have only been considering forces which lie in the same plane, but in nature we meet with forces acting in all directions, and therefore we must not be satisfied with confining our inquiries to the simpler case. We proceed to show, in two different ways, how a force can be decomposed into three forces not in the same plane, though passing through the same point. The first mode of doing so is as follows. To three points a, b, c ([Fig. 15]) three spring balances are attached; a, b, c are not in the same straight line, though they are at the same vertical height: to the spring balances cords are attached, which unite in a point o, from which a weight w is suspended. This weight is supported by the three cords, and the strains along these cords are indicated by the spring balances. The greatest strain is on the shortest cord and the least strain on the longest. Here the force w lbs. produces three forces which, taken together, exceed its own amount. If I add an equal weight w, I find, as we might have anticipated, that the strains indicated by the scales are precisely double what they were before. Thus we see that the proportion of the force to each of the components into which it is decomposed does not depend on the actual magnitude of the force, but on the relative direction of the force and its components.

Fig. 16.

34. Another mode of showing the decomposition of one force into three forces not in the same plane is represented in [Fig. 16]. The tripod is formed of three strips of pine, 4' × 0"·5 × 0"·5, secured by a piece of wire running through each at the top; one end of this wire hangs down, and carries a hook to which is attached a weight of 28 lbs. This weight is supported by the wire, but the strain on the wire must be borne by the three wooden rods: hence there is a force acting downwards through the wooden rods. We cannot render this manifest by a contrivance like the spring scales, because it is a push instead of a pull. However, by raising one of the legs I at once become aware that there is a force acting downwards through it. The weight is, then, decomposed into three forces, which act downwards through the legs; these three forces are not in a plane, and the three forces taken together are larger than the weight.

35. The tripod is often used for supporting weights; it is convenient on account of its portability, and it is very steady. You may judge of its strength by the model represented in the figure, for though the legs are very slight, yet they support very securely a considerable weight. The pulleys by means of which gigantic weights are raised are often supported by colossal tripods. They possess stability and steadiness in addition to great strength.

36. An important point may be brought out by contrasting the arrangements of [Figs. 15] and [16]. In the one case three cords are used, and in the other three rods. Three rods would have answered for both, but three cords would not have done for the tripod. In one the cords are strained, and the tendency of the strain is to break the cords, but in the other the nature of the force down the rods is entirely different; it does not tend to pull the rod asunder, it is trying to crush the rod, and had the weight been large enough the rods would bend and break. I hold one end of a pencil in each hand and then try to pull the pencil asunder; the pencil is in the condition of the cords of [Fig. 15]; but if instead of pulling I push my hands together, the pencil is like the rods in [Fig. 16].

37. This distinction is of great importance in mechanics. A rod or cord in a state of tension is called a “tie”; while a rod in a state of compression is called a “strut.” Since a rod can resist both tension and compression it can serve either as a tie or as a strut, but a cord or chain can only act as a tie. A pillar is always a strut, as the superincumbent load makes it to be in a state of compression. These distinctions will be very frequently used during this course of lectures, and it is necessary that they be thoroughly understood.