[We may see the great force of friction in the brake, by which a large weight running down a long inclined plane has its motion moderated and stopped; in the windlass, where a few coils of the rope round a cylinder sustain the stress and weight of a large iron anchor; in the mode of raising large blocks of granite by an iron rod driven into a hole in the stone. Probably no greater forces are exercised in any processes in the arts than the force of friction; and it is always employed to produce rest, stability, moderate motion. Being always ready and never wearied, always at hand, and augmenting with the exigency, it regulates, controls, subdues all motions; counteracts all other agents; and, finally, gains the mastery over all other terrestrial agencies, however violent, frequent, or long-continued. The perpetual action of all other terrestrial forces appears, on a large scale, only as so many interruptions of the constant and stationary rule of friction. "Astronomy and General Physics," by Professor Whewell.]

It has been proved by experiment that friction increases as the weight or pressure increases, and as the surfaces in contact are more extensive, and as the roughness is greater. With surfaces of the same material it is nearly proportional to the pressure. The time which the surfaces have been in contact appears to have a considerable influence, though this differs much with surfaces of different kinds. As a general rule, similar substances give rise to greater friction than dissimilar ones.

On the contrary, friction diminishes as the pressure is less, as the polish of the moving surfaces is more perfect, and as the surfaces in contact are smaller. It may also be diminished by anointing the surfaces with some suitable unguent or greasy material. Among such substances as are commonly used are the different fats, tar, and black-lead. By such means, friction may be reduced to one-fourth.

The

Of the friction produced by sliding and rolling motions, the latter, under similar circumstances, is far the least. This partly arises from the fact that the surfaces in contact constitute a mere line, and partly because the asperities are not abraded or pushed aside before motion can ensue. nature of this distinction may be clearly understood by observing what takes place when two brushes with stiff bristles are moved over one another, and when a round brush is rolled over a flat one. In this instance, the rolling motion lifts the resisting surfaces from one another; in the former, they require to be forcibly pushed apart.

Though, in many instances, friction acts as a resisting agency, and diminishes the power we apply to machines, in some cases its effects are of the utmost value. Thus, when nails or screws are driven into bodies, with a view of holding them together, it is friction alone which maintains them in their places. The case is precisely the same as in the action of a wedge.

RESISTANCE OF MEDIA.

A great many results in natural philosophy illustrate the resistance which media offer to the passage of bodies through them. The experiment known under the name of the guinea and feather experiment establishes this for atmospheric air. In a very tall air-pump receiver there are suspended a piece of coin and a feather in such a way that, by turning a button, the piece on which they rest drops, and permits them to fall to the pump-plate. Now, if the receiver be full of atmospheric air, on letting the objects fall, it will be found that, while the coin descends with rapidity, and reaches in an instant the pump-plate, the feather comes down leisurely, being buoyed up

by the air, and the speed of its motion resisted. But if the air is first extracted by the pump, and the objects allowed to fall in vacuo, both precipitate themselves simultaneously with equal velocity, and accomplish their fall in equal times.

In the vibrations of a pendulum, the final stoppage is due partly to friction, and partly to this cause. And in the case of motions taking place in water, we should, of course, expect to find a greater resistance, arising from the greater density of that liquid.

The resisting force of a medium depends upon its density, upon the surface which the moving body presents, and on the velocity with which it moves. Water, which is 800 times more dense than air, will offer a resistance 800 times greater to a given motion. Of the two mills represented in Fig. 35 (page 25), that which goes with its edge first runs far longer than that which moves with its plane first. We are not, however, to understand that the effect of the medium on a body moving through it increases directly as the transverse section of the body; for a great deal depends upon its figure. A wedge, going with its edge first, will pass through water more easily than if impelled with its back first, though, in both instances, the area of the transverse section is of course the same. It is stated that spherical balls encounter one-fourth less resistance from the air than would cylinders of equal diameter; and it is upon this principle that the bodies of fishes and birds are shaped, to enable them to move with as little resistance as may be through the media they inhabit.

The resistance of a medium increases with the velocity with which a body moves through it, being as the square of the velocity, so long as the motion is not too rapid; but when a high velocity is reached, other causes come into operation, and disturb the result.

As with friction, so with the resistance of media, a great many results depend on this impediment to motion; among such may be mentioned the swimming of fish through water, and the flight of birds through the air. It is the resistance of the air which makes the parachute descend with moderate velocity downward, and causes the rocket to rise swiftly upward.

RIGIDITY OF CORDAGE.

In the action of pulleys, in machinery in which the use of cordage is involved, the rigidity of that cordage is an impediment to motion. When a cord acts round a pulley, in consequence of imperfect flexibility, it obtains a leverage on the pulley, as may be understood from Fig. 183, in which let

Fig. 183.

CK D be the pulley working on a pivot at 0: let A and B be weights suspended by the rope, A CKD B. From what has been said respecting the theory of the pulley, the action of the machine may be regarded as that of a lever, C O D, with equal arms, ČO, O D. Now, if the cord were perfectly inflexible, on making the weight, A, descend by the addition of a small weight to it, it would take the position at A', the rope being a tangent to the pulley at C'; at the same time B, ascending, would take the position B', its cord being a tangent at D'. From the new positions, A' B', which the inflexible cord is thus supposed to have assumed, draw the perpendiculars, A/ E, B F, then will O E, O F,

represent the arms of the lever on which they act-a diminished leverage on the side of the descending, and an increased leverage on the side of the ascending weight is the result.

In practice the result does not entirely conform to the foregoing imaginary case, because cords are, to a certain extent, flexible. As their pliability diminishes, the disturbing effect is greater. The degree of inflexibility depends on many casual circumstances, such as dampness or dryness, or the nature of the substance of which they are made. Inflexibility increases with the diameter of a cord, and with the smallness of the pulley over which it

runs.

SECTION VI.-UNDULATORY MOTIONS.

CHAPTER XXXI.

OF UNDULATIONS.

Origin of Undulations-Progressive and Stationary Undulation— Course of a Progressive Wave--Nodal Points-Three different kinds of Vibration -Transverse Vibration of a Cord-Vibrations of Rods-Vibrations of Elastic Planes-Vibrations of Liquids-Waves on Water-Law of the Reflection of Undulations-Applied in the case of a Plane, a Circle, an Ellipse, a Parabola-Case of a Circular Wave on a Plane-Interference of Waves-Inflection of Waves-Intensity of Waves-Method of Combining Systems of Waves.

WHEN an elastic body is disturbed at any point, its particles gradually return to a position of rest, after executing a series of vibratory movements. Thus, when a glass tumbler is struck by a hard body, a tremulous motion is communicated to its mass, which gradually declines in force until the movement finally ceases.

In the same manner a stretched cord, which is drawn aside at one point, and then suffered to go, is thrown into a vibratory or undulatory movement; and, according as circumstances differ, two different kinds of undulation may be established: 1st, progressive undulations; 2nd, stationary undulations. In progressive undulations the vibrating particles of a body communicate their motion to the adjacent particles; a successive propagation of movement, therefore, ensues. Thus, if a cord is fastened at one end, and the other is moved up and down, a wave or undulation, m D n E o, Fig. 184, is produced. The part, m D n, is the elevation of the wave, D being the summit, n E o is the depression, E being the lowest point, D p is the height, q E the depth, and m o the length of the wave.

But, under the circumstances here considered, the moment this wave has formed, it passes onward, and successively assumes the positions indicated at I, II, III.~ When it has arrived at the other end of the cord, it at once returns with an inverted motion, as shown at IV and V. This, therefore, is a progressive undulation.

Again, instead of the cord receiving one impulse, let it be agitated equally at equal intervals of time; it will then divide itself, as shown in Fig. 185, into equal elevations and depressions with intervening points, m n, which

These are stationary undulations, and the points are called

The agents by which undulatory

- movements are established are chiefly elasticity and

m

Fig. 185.

H

館

Fig. 184.

which enables it to transmit the vibratory motions which constitute sound, and, for the same reason, steel rods and plates of glass may be thrown into musical vibrations. In the case

of threads and wires, a sufficient degree of elasticity may be given by forcibly stretching them. Waves on

the surface of liquids are produced by the agency of gravity. There are three different kinds of vibrations into which a stretched string may be thrown: transverse, longitudinal, and twisted. These may be illustrated by the instrument represented at Fig. 186. It consists of a piece of spirallytwisted wire, stretched from a frame by a weight. If the lower end of the wire be secured by a clamp, on pulling the wire in the middle, and then letting it go, it executes transverse vibrations, a. If the weight be gently lifted, and then let fall, the wire performs longitudinal vibrations; and if the weight be twisted round, and then released, we have rotatory vibrations, b.

If we take a string, a b, Fig. 187, and having stretched it between two fixed

с

f

Fig. 187.

points, a and b, draw it aside, and then let it go, it executes transverse vibrations, as has already been described. The cause of its motion, from the position we have stretched it to, is its own elasticity. This makes it return from the position, a b c, to the straight line, a fb, with a continually accelerated velocity; but when it has arrived in a fb, it cannot stop there, its momentum carrying it forward to a d b with a velocity continually decreasing. Arrived in this position, it is, for a moment, at rest; but its elasticity again impels it as before, but in the reverse direction, to a f b; and so it executes vibrations on each side of that straight line, until it is finally brought to rest by the resistance of the air. One complete movement, from a cb to a d b‍and back, is called a vibration, and the time occupied in performing it the time of an oscillation.

The vibratory movements of such a solid are isochronous, or performed in equal lines. They increase in rapidity with the tension-that is, with the elasticity-being as the square root of that force. The number of vibrations in a given time is inversely as the length of the string, and also inversely as its diameter.

The vibrations of solid bodies may be studied best under the division of cords, rods, planes, and masses. The laws of the vibrations of the first are such as we have just explained.

In rods the transverse vibrations are isochronous, and in a given time are in number inversely as the squares of the lengths of the vibrating parts. Thus, if a rod makes two vibrations in one second, if its length be reduced to half, it will make four times as many-that is, eight; if to one-fourth, sixteen times as many-that is, thirty-two, &c. The motion performed by vibrating rods is often very complex. Thus, if a bead be fastened on the free extremity of a vibrating steel rod, Fig. 188, it will exhibit in its motions a curved path, as is seen at c. Rods may be made to exhibit nodal points. The space between the free extremity and the first nodal point is equal to half the length contained between any two nodal points, but it vibrates with the same velocity. Thus a, Fig. 189, being the

fixed, and b the free end of such
a rod, the part between b and c
is half the distance, c c'.

Fig. 189.

in

When elastic planes vibrate they exhibit nodal lines, answering to the nodal points in linear vibrations; and if the plane were supposed to be made up of a series of rods, these lines would answer to their nodal points. Fig. 188. By them the plane is divided into spaces--the adjacent ones being always in opposite phases of vibrations, as shown by the signs + and Fig. 190, where A B is the vibrating plane. The dimensions of these spaces are regulated in the same way as the internodes of vibrating rods-that is, the outside ones, a b a b, are always half the size of the interior. The A relation of these spaces, and positions of the nodal lines, may be determined by making a glass plate covered with dry sand vibrate.

[When we wish to make plates vibrate we use a vice, such as is shown in Fig. 191, and having placed the plate between the cylinder, a, and the screw, b, both of which are tipped with leather or

Fig. 191.

+

+

+

+

Fig. 190.

+

cork, the latter is turned until the plate is fixed firmly; and then, when a bow is drawn along the edge of the plate once or twice, it vibrates sufficiently to cause the sand upon its surface to rise and fall during the tone produced, and to accumulate upon the nodal lines, so as to form the sound-figures, as they were called by Chladui, their discoverer. These figures vary in form according as the bow is moved more or less rapidly or violently, and the point of support and action is changed..]

When the surface of a liquid, as water, is touched, a wave arises at the disturbed point, and propagates itself into the unmoved spaces around, continually enlarging as it goes, and forming a progressive undulation. [This is easily proved by filling a tumbler with water, and letting a small