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the dark spaces represent the condensed parts, and the light spaces the rarified parts of the waves; from a to b is a wave length. It would appear from this that we should be able to tell the length of the waves that produce any particular note. And so we can. Suppose the fork in Fig. 5 performs 435 vibrations per second, and that the velocity of the sound is 1,120 ft. per second. Now, when the fork is made to vibrate, in one second the front of the first wave sent forth will be 1,120 ft. ahead; but during this one second the fork will have sent forth 435 waves; so that this space of 1,120 ft. is made up of 435 waves, therefore the length of each wave is 1,120 ft. divided by 435, that is, rather more than 2A ft. Thus the length of the waves corresponding to any note is found by dividing the velocity of the sound per second by the number of vibrations per second due to the note. It follows that the higher the pitch of a note, the shorter the waves: if one note be an octave higher than another, the waves producing the higher note are just half the length of those producing the lower.
In a large class of musical instruments the sound is produced by the vibration of highly stretched strings, the intensity of the sound being increased by the vibrations being communicated to suitable sound-boards, as in the
violin, piano, harp, &c From experiments made with the simple instrument (Fig. 6) called a monochord, in which, by means of the movable bridge, C, any length of the string may be vibrated, it is shown that the vibrations of a string depend on the following laws:
1. The shorter the string the more rapid the vibration, and consequently the higher the pitch of the note: with half the length of the string the vibrations are doubled, and produce the octave of the whole string; with one-third the length of the string the vibrations are trebled, and produce a note a fifth above the octave of the whole string. In general terms, the number of vibrations is inversely proportional to the length of the string.
2. The more tightly the string is stretched, the more rapidly it vibrates, and it is found that the number of vibrations is proportional to the square root of the stretching weight.
3. The vibrations depend also on the thickness of the string; the length and stretching weight continuing the same, we find that the number of vibrations varies inversely as the thickness.
4. All other things remaining the same, the number of vibrations is inversely proportional to the square root of the density of the string. The two last laws may be expressed in one, thus: the length and stretching weight continuing the same, the number of vibrations is inversely proportional to the square root of the weight of the string. In the violin and other stringed instruments, we avail ourselves of thickness instead of length to obtain the grave notes.
We can make any length of string vibrate without using the bridge C, Fig. 6. If we remove the bridge, and touch the string in the middle, while a fiddle-bow is drawn across one of its halves, the two halves will vibrate separately, as represented in Fig. 7, in which only the string of the monochord is shown. The note produced is an octave of that produced by the string vibrating as a whole, as shown in Fig. 8.
The point, C, between the vibrating segments remains at rest, and is called a node. By touching the string in the proper place and then drawing the bow across it, we can divide it into three, four, or five, or any number of vibrating segments, separated by the corresponding number of nodes. The position of these segments and nodes is readily observed by putting riders of paper along the string; those situated at the vibrating segments will be unhorsed, while
those at the nodes remain on, as in Fig. 9. The same thing takes place when plates of glass are made to vibrate, and the position of the nodal lines in these is beautifully shown by scattering fine sand over the plates; the sand assumes a curious rapid motion, and finally settles along the nodal lines. If a square piece of glass be held at the centre by a suitable clamp, and fine sand scattered over it, when a violin bow is drawn across its edge near one of the corners,
while the middle point of one of its sides is touched with the finger, the sand is tossed about and settles along the nodal lines (Fig. 10) which divide the plate into vibrating segments. The note produced by this division is the lowest noteof the plate.
If one of the corners be touched, and the bow drawn across the middle of one of the sides, the sand arranges itself as in Fig. 11: the note produced this time is a fifth above the last. If the plate be held near one of the corners, and the bow drawn as in the last case, the sand assumes the form shown in Fig. 12, and we obtain a note higher than either of the former. By thus agitating and touching the plates in different parts, Chladni was able to obtain an immense number of beautiful figures.
In organ-pipes the sound is produced not by the vibration of the material of which the pipes are made, but by the vibration of the air in the pipes. The lowest or fundamental note of a pipe, open at both ends, is that whose semiwave equals the length of the pipe; and if the pipe be closed at one end, its lowest note is that whose semi-wave is half the length of the pipe. From this \t follows that if two pipes be the same length, and one be open and the other closed, the fundamental note of the former is an octave lower than that of the latter. Since the longer the wave the deeper the note, the longer the pipe the deeper is its fundamental note. Higher notes may be got from a pipe by increasing the velocity of the current of air passing over its mouth; for then the air in the pipe divides itself into vibrating segments. The rates of vibration of the notes that can be got from an open pipe are as the numbers 1, 2, 3, 4, 5, &c, and from a pipe closed at one end, as the numbers 1, 3, 5, 7, 9, &c Thus with an open pipe, the next note that can be produced above the fundamental note is its octave; but with a closed pipe, the first note above the fundamental is a fifth above its octave.
In Mechanies we treat of the effects of forces on bodies; a force being anything which produces or tends to produce motion in a body, or which changes or tends to change the motion of a body. If one force alone acts on a body, only one effect can take place—the body must move in a straight line; for whenever a body at rest appears to be acted on by a single force, a little consideration will show that one or more others are in play. And when a body is subjected to the action of two or more forces, some one of two effects must take place: cither the body must continue at rest, or it must begin to move in some definite direction and at some definite speed. If the body continue at rest, the forces that act upon it must be so related as to their intensity and direction, that they neutralize each other's effects. They are then said to be in equilibrium, and the science that treats of the relations that must exist between two or more forces in order that they may be in equilibrium is called Staties; while that which treats of the effects of forces not in equilibrium— that is, of the motion of bodies—is called Dynamies.
In staties, the magnitude or intensity of a force is measured by the weight which it would be able to support. A force that would support a weight of 1 lb. is called a force of 1 lb.; that which would support a weight of 2 lbs. is called a force of 2 lbs.; or, speaking of the two with reference to each other, the former may be called a force 1, and the latter a force 2.
Since the only things necessary to describe a force arc its magnitude and its direction (that is, the direction in which it would tend to move any body to which it is applied), it is plain we can represent forces by lines, simply by drawing the lines in the directions in which the forces act, and of lengths proportional to their magnitudes. Thus if we want to represent, by means of lines, two forces of 3 lbs. and 5 lbs. respectively, acting on a body, O, at right
angles with each other, we draw trom the point O two lines, O A and o B, at right angles to each other, as in Fig. 1. From O A we measure off O P=3 in., and from o B we measure off O Q=5 in.; then O P and O Q represent the said forces. Of course we might have taken three half-inches and five half-inches, making a line of \ in. represent a force of I lb., or any other convenient lengths, taking care always to make the lines O P and O Q, which represent the forces, in the proportion of three to five.
Here is a body at O, Fig. 2. If it be acted on by a force which tends to move it in the direction o A, and at the same time be acted on by another force which tends to move it in the direction O B, it will obey neither, but will move off in a direction between them. From this it is evident that a single force might be found that would have the same effect in moving the body as the other two forces combined. That single force is called the resultant of the other two, and how to find it we shall now show. Let the force tending to send the body o in the direction o A be 3 lbs., and the force tending to send it in the direction O B be 4 lbs.; from O A measure off o P=3 in., and from O b measure off 0 Q=4 in. Now the lines O P and O Q represent the said forces. Complete the parallelogram, as in the figure, and draw the diagonal OR: OR represents the resultant. Let us measure it on the same scale as we measured O P and o Q on; we find it contains 5 in., and therefore represents a force of 5 lbs. Thus a single force of 5 lbs. acting in the direction O R would have the same effect in moving the body, O, as the other two forces combined have. The reader may, perhaps, now ask, What is the use of finding the resultant of two forces? Well, it is this: it enables us to tell both the magnitude and direction of a single force that would balance two given forces; and of the great and practical advantage of this no one will doubt. For instance, in the example that we have just taken, if a force of 5 lbs. be applied to the body at O, and be made to act in the opposite direction to O R, it will balance the other two forces, namely, the force of 3 lbs. acting in the direction O A, and the force of 4 lbs. acting in the direction O B. Two forces are called, in relation to their resultant, components; and the method of finding the resultant is called the composition of forces.
Having done this example, we shall now give in words the general rule for finding the resultant of any two forces acting on a body at the same point, but in different directions. Represent the forces by lines; complete the parallelogram of which these lines will be the adjacent sides, and draw the diagonal from the point at which the forces act. This diagonal represents the resultant. This rule is founded on the principle called the parallelogram of forces, which is in these words: if the two forces acting on a body be represented by the sides of a parallelogram, then the diagonal of this parallelogram, drawn in the direction of the forces, will represent the resultant.
We shall show by an experiment that this principle is true. Knot three strings together, and hang two of them over fixed pulleys, M and N; let the third hang from the point O; attach weights, P, Q, and R—R being less than the sum of P and Q: the whole, when left to itself, will balance in some cbfuvte position, as in the figure. The tension at every point of the string O M
is the weight P, and at every point of 0 N the tension is the weight Q. Now, since R, acting vertically downwards through O, balances P and Q, acting in the directions O M and O N respectively, the resultant of P, acting in the direction o M, and Q, acting in the direction O N, must be a force equal to R, and acting vertically upwards through o.
Let us now find the resultant of P and Q, according to the method of the parallelogram of forces, and it will be found to agree with that which we have just obtained by experiment. To do it, from o M measure off O A, containing as many inches as there are ounces in P, and from O N measure off O B, containing as many inches as there are ounces in Q; complete the parallelogram as in the figure, and draw the diagonal O C. O C will be found on trial to contain as many inches as R does ounces, and to be drawn vertically upwards through o; it therefore represents the resultant found by experiment. This proves the truth of the principle; for, however the position of the pulleys may be varied, the resultant found by the parallelogram of forces will be found always to agree with that found by experiment.
If we take five rods jointed together, and form four of them into a parallelogram, making the fifth rod, which should be longer than the others, a diagonal, as shown in Fig. 4, and let the rods, O P and O Q, represent two forces acting