Читать книгу Experimental Mechanics - Robert S. Ball - Страница 9

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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.