THE COMPOSITION OF FORCES

Experimental Mechanics by Robert S. Ball is part of the HackerNoon Books Series. You can jump to any chapter in this book here. THE COMPOSITION OF FORCES

LECTURE I. THE COMPOSITION OF FORCES.

Introduction.—The Definition of Force.—The Measurement of Force.—Equilibrium of Two Forces.—Equilibrium of Three Forces.—A Small Force can sometimes balance Two Larger Forces.

INTRODUCTION.

1. I shall endeavour in this course of lectures to illustrate the elementary laws of mechanics by means of experiments. In order to understand the subject treated in this manner, you need not possess any mathematical knowledge beyond an acquaintance with the rudiments of algebra and with a few geometrical terms and principles. But even to those who, having an acquaintance with mathematics, have by its means acquired a knowledge of mechanics, experimental illustrations may still be useful. By actually seeing the truth of results with which you are theoretically familiar, clearer conceptions may be produced, and perhaps new lines of thought opened up. Besides, many of the mechanical principles which lie rather beyond the scope of elementary works on the subject are very susceptible of being treated experimentally; and to the consideration of these some of the lectures of this course will be devoted.

Many of our illustrations will be designedly drawn from very commonplace sources: by this means I would try to impress upon you that mechanics is not a science that exists in books merely, but that it is a study of those principles which are constantly in action about us. Our own bodies, our houses, our vehicles, all the implements and tools which are in daily use—in fact all objects, natural and artificial, contain illustrations of mechanical principles. You should acquire the habit of carefully studying the various mechanical contrivances which may chance to come before your notice. Examine the action of a crane raising weights, of a canal boat descending through a lock. Notice the way a roof is made, or how it is that a bridge can sustain its load. Even a well-constructed farm-gate, with its posts and hinges, will give you admirable illustrations of the mechanical principles of framework. Take some opportunity of examining the parts of a clock, of a sewing-machine, and of a lock and key; visit a saw-mill, and ascertain the action of all the machines you see there; try to familiarize yourself with the principles of the tools which are to be found in any workshop. A vast deal of interesting and useful knowledge is to be acquired in this way.

THE DEFINITION OF FORCE.

2. It is necessary to know the answer to this question, What is a force? People who have not studied mechanics occasionally reply, A push is a force, a steam-engine is a force, a horse pulling a cart is a force, gravitation is a force, a movement is a force, &c., &c. The true definition of force is that which tends to produce or to destroy motion. You may probably not fully understand this until some further explanations and illustrations shall have been given; but, at all events, put any other notion of force out of your mind. Whenever I use the word Force, do you think of the words “something which tends to produce or to destroy motion,” and I trust before the close of the lecture you will understand how admirably the definition conveys what force really is.

3. When a string is attached to this small weight, I can, by pulling the string, move the weight along the table. In this case, there is something transmitted from my hand along the string to the weight in consequence of which the weight moves: that something is a force. I can also move the weight by pushing it with a stick, because force is transmitted along the stick, and makes itself known by producing motion. The archer who has bent his bow and holds the arrow between his finger and thumb feels the string pulling until the impatient arrow darts off. Here motion has been produced by the force of elasticity in the bent bow. Before he released the arrow there was no motion, yet still the bow was exerting force and tending to produce motion. Hence in defining force we must say “that which tends to produce motion,” whether motion shall actually result or not.

4. But forces may also be recognized by their capability or tendency to prevent or to destroy motion. Before I release the arrow I am conscious of exerting a force upon it in order to counteract the pull of the string. Here my force is merely manifested by destroying the motion that, if it were absent, the bow would produce. So when I hold a weight in my hand, the force exerted by my hand destroys the motion that the weight would acquire were I to let it fall; and if a weight greater than I could support were placed in my hand, my efforts to sustain it would still be properly called force, because they tended to destroy motion, though unsuccessfully. We see by these simple cases that a force may be recognized either by producing motion or by trying to produce it, by destroying motion or by tending to destroy it; and hence the propriety of the definition of force must be admitted.

THE MEASUREMENT OF FORCE.

5. As forces differ in magnitude, it becomes necessary to establish some convenient means of expressing their measurements. The pressure exerted by one pound weight at London is the standard with which we shall compare other forces. The piece of iron or other substance which is attracted to the earth with this force in London, is attracted to the earth with a greater force at the pole and a less force at the equator; hence, in order to define the standard force, we have to mention the locality in which the pressure of the weight is exerted.

It is easy to conceive how the magnitude of a pushing or a pulling force may be described as equivalent to so many pounds. The force which the muscles of a man’s arm can exert is measured by the weight which he can lift. If a weight be suspended from an india-rubber spring, it is evident the spring will stretch so that the weight pulls the spring and the spring pulls the weight; hence the number of pounds in the weight is the measure of the force the spring is exerting. In every case the magnitude of a force can be described by the number of pounds expressing the weight to which it is equivalent. There is another but much more difficult mode of measuring force occasionally used in the higher branches of mechanics (Art. 497), but the simpler method is preferable for our present purpose.

Fig. 1.

6. The straight line in which a force tends to move the body to which it is applied is called the direction of the force. Let us suppose, for example, that a force of 3 lbs. is applied at the point a, Fig. 1, tending to make a move in the direction ab. A standard line c of certain length is to be taken. It is supposed that a line of this length represents a force of 1 lb. The line ab is to be measured, equal to three times c in length, and an arrow-head is to be placed upon it to show the direction in which the force acts. Hence, by means of a line of certain length and direction, and having an arrow-head attached, we are able completely to represent a force.

EQUILIBRIUM OF TWO FORCES.

Fig. 2.

7. In Fig. 2 we have represented two equal weights to which strings are attached; these strings, after passing over pulleys, are fastened by a knot c. The knot is pulled by equal and opposite forces. I mark off parts cd, ce, to indicate the forces; and since there is no reason why c should move to one side more than the other, it remains at rest. Hence, we learn that two equal and directly opposed forces counteract each other, and each may be regarded as destroying the motion which the other is striving to produce. If I make the weights unequal by adding to one of them, the knot is no longer at rest; it instantly begins to move in the direction of the larger force.

8. When two equal and opposite forces act at a point, they are said to be in equilibrium. More generally this word is used with reference to any set of forces which counteract each other. When a force acts upon a body, at least one more force must be present in order that the body should remain at rest. If two forces acting on a point be not opposite, they will not be in equilibrium; this is easily shown by pulling the knot c in Fig. 2 downwards. When released, it flies back again. This proves that if two forces be in equilibrium their directions must be opposite, for otherwise they will produce motion. We have already seen that the two forces must be equal.

A book lying on the table is at rest. This book is acted upon by two forces which, being equal and opposite, destroy each other. One of these forces is the gravitation of the earth, which tends to draw the book downwards, and which would, in fact, make the book fall if it were not sustained by an opposite force. The pressure of the book on the table is often called the action, while the resistance offered by the table is the force of reaction. We here see an illustration of an important principle in nature, which says that action and reaction are equal and opposite.

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, 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.

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.

Fig. 6.

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.

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