Conversion of Motion: Cams.

145

It is a geometrical fact that, with these six links or bars so arranged, if the corner or pole, Q, of the diamond be made to describe any curve, the other pole, P, will describe what is called the inverse of that curve; and in general, if Q move in a circle, P will move in a circle also. But, curiously enough, if Q be made to move in a circle passing through E (which is easily effected by jointing Q with another link or bar, Q F, whose length is half the distance between Q and E, and moveable about F as a pivot), P no longer moves in a circle, but in a straight line, A B, perpendicular to the line of centres, F E. Thus, a rotatory movement (within limits) of the bar, Q F, is converted into a perfectly rectilinear motion of the point P; and in this way an oscillatory movement of the arm, QF, becomes a perfectly straight alternate movement of the joint P. The long-sought-for perfect parallel motion, without guides, has thus been obtained.

Important applications of this principle to various constructions in machinery-such as the steam-engine, planing and polishing machines, millwrights' work, &c.-have already been proposed and even carried into operation.

265. Lastly, continuous circular motion may be converted into a reciprocating, or a variable, motion of any desired nature, by the contrivance known as a cam. This is a plate with a curved edge or groove, which communicates motion to another piece pressing against its curved edge.

Fig. 57 represents a heart-shaped cam, which is of frequent employment in mechanism.

The up and down motions of the steel punch in a punching machine are regulated by a cam of this description; the punch is thus brought down on the plate with the needed velocity at the proper instant, while by the wheel falling into the

Fig. 57.

hollow of the cam, it is kept raised for a sufficient length of time to allow the workman to shift the plate for the punching of another hole.

In printing-machinery, the extreme precision required in the motions of the sheets of paper is insured by the employment of cams; and the delicacy and rapidity of movement so obtained far surpass the highest efforts of skill which the most practised workman could command.

The pallets or teeth on a turning wheel, which so act on the handle of a great forge hammer, that every one in passing shall lift the hammer and produce a blow, are a simple form of the principle of the cam.

We need not bere multiply examples of the thousand artifices employed for the conversion of motion. In this great manufacturing country, with our railways, and steamboats, and power-looms, we are all so accustomed and familiarised with the wonderful results of mechanical inventions, that we have come to regard as very commonplace, what former generations would have looked on as miracles of art.

266. The modification of mere motion, without any ultimate reference to the transmission of Energy, is the object of the important class of mechanical contrivances known as watches, clocks, chronometers, &c.

In these the sole aim (see Art. 184) is the production of a perfectly uniform motion, to serve as a motion-scale for the measurement of other motions, which, as we have pointed out (Art. 155), is the real meaning of time. The inestimable value of such contrivances is too patent to require comment; but it is of interest and importance that we should understand the simple laws upon which their principle depends. As typical of the whole class of such motion-regulators we shall explain the leading features of

"The Pendulum."

We have already seen (Art. 184) that any freely swinging mass may be called a pendulum. Usually it consists of a ball or

b

bob, a (fig. 58), suspended by a length of wood or metal from a fixed point, b. Now, the most remarkable property of such a body, the discovery of which may be Isaid to have created the art of clockmaking, is its isochronism; that is to say, it takes the same time to make a small swing as a large swing (within certain limits) The reason is, that if the pendulum start from c, in place of e (fig. 58), the beginning of its slope of fall is steeper in the former case, and it will therefore fall the

Fig. 58.

Laws of Pendulum Motion.

147

faster, and sweep through the larger arc, ca, just with proportionally greater speed.

Galileo is said to have discovered this property when a student at Pisa. When in the cathedral there he remarked the singularly uniform vibrations of a chandelier hanging from the roof of the building; and, on comparing the swings with the beat of his pulse, the idea occurred to him that such a simple instrument would be valuable for medical observations on the pulse.

A common clock is merely the application of this uniform vibration of a pendulum to regulate the turning of a wheel, and the consequent motion of a train of wheels, by allowing one tooth of the guiding or crown wheel to pass or escape for each of its vibrations.

The pendulum is isochronous, however, only so long as its length remains unaltered, for long pendulums swing more slowly.than short ones; and clocks (or watches) go slower in summer than in winter, if they be not regulated, owing to the expanding effect of the heat on their governing pendulum (or balance spring).

267. Let us see why this should be so, and what is the exact relation between the length and the time of vibration of any pendulum.

If a pendulum, bc (fig. 59), be four times as long as another one, bd, it has just four times as far to travel in its descending arc, ca as the other in its similar arc, de, while in

corresponding parts of the two arcs the slope or inclination is always equal.

The ball of the long pendulum, therefore, may be considered as having rolled four times as far down a given slope as the ball of the short pendulum. Now, a body to fall four times as far, either directly or down any uniform smooth slope, will just take double the time (see Art. 139). Hence the pendulum, bc, which is four times as long as bd, will just take twice the time to each swing or vibration; and generally, in order that one pendulum may swing two, three, four, five, &c., times as slowly as another, the length of the former must be four, nine, sixteen, twenty-five, &c., times as great as that of the latter.

Fig. 59.

A pendulum which vibrates once every second is called a seconds pendulum, which will have an invariable length at any place. For

148

Length of the Seconds Pendulum..

London the length of the seconds pendulum is a little more than thirty-nine inches (39°13 inches); and it has been proposed that such a measure might be adopted as a universal standard of length.

It is remarkable that the metre, or French unit of linear measure, is almost exactly a quarter of an inch longer than the seconds pendulum; but this coincidence is merely accidental, for the metre standard was chosen by the French government in the end of last century, as being the ten-millionth part of the earth's surface measured from the equator to the North Pole through Paris.

For different places on the surface of the globe, however, the seconds pendulum has different lengths, because the power of gravity varies with the distance from the centre of the earth. At the equator, in consequence of the bulging out of our globe there, to beat seconds, the seconds pendulum for London would have to be shortened, and at the poles it would have to be lengthened.

It has, in fact, been found by actual experiment, that a pendulum beating seconds at the equator has to be lengthened by as much as one-fifth of an inch to beat seconds at Spitzbergen. In this way the seconds pendulum may be employed as a means of comparing the intensity of the force of gravity at different places. For a like reason it will be found that a pendulum beating seconds at the level of the sea will beat longer periods when taken to the top of a high mountain; and likewise at the bottom of a mine, where it is attracted by the matter above it, as well as by the matter beneath.

The popular notion that a heavy body falls quicker than a light one (see Art. 135) is confuted by the fact that the time of vibration of a pendulum is unaffected by the weight or material of which it is composed. Equal pendulums of lead, or ivory, or glass, or wood, or iron are all alike in this respect; and a hollow ball vibrates at the same rate, whatever be the nature of its contents-whether air, or water, or mercury.

A

B

Fig. 60.

It has to be noted, however, that the length of a pendulum is not to be measured by the distance of its centre of gravity, or of its end, from its point of suspension, except in the simple case where the pendulum is a ball hung by a fine thread.

Thus, if from a pin, C, we hang any swinging body, A B, whose centre of gravity lies at G; and if we hang from the same pin a ball, P, by a fine thread, we shall find that, in order that the two may swing together or isochronously, the length of the string, C P, will require to be adjusted somewhere

Centre of Oscillation: the Metronome.

149 about the position marked in fig. 60. . The corresponding point in A B is called the centre of oscillation. It was discovered by the celebrated Dutch philosopher Huyghens that if the pendulum, A B, were now to be suspended from this centre of oscillation instead of its former centre of suspension, it would still vibrate at exactly the same rate as before, and would keep time with the simple pendulum, C P. This property is sometimes called the reciprocity or exchangeability of the centres of oscillation and suspension.

a

There is a small pendulum called a metronome, used by musicians for marking time; which, although very short, may still be made to beat whole seconds, or even longer intervals. The reason of its slow motion is, that its rod is prolonged upwards, to b, beyond its axis of support, at a, and has a ball upon the top, at b, as well as on the bottom, at c. This upper ball prevents the under one from moving so fast as it otherwise would, just as a smaller weight attached to one end of a weighing-beam, prevents a greater weight attached to the other end from falling so fast as it would if there were no counterpoise. The rod, ab, is marked with numbers corresponding to the number of beats made per minute when the ball, b, is moved to that number.

"Friction."

Fig. 61.

268. In estimating the effects of machines by the rule of the comparative velocities of the power and resistance, an important deduction has to be made, on account of the friction between the moving parts. Thus in some forms of steam-engine, where the rubbing parts are numerous, the loss from friction may amount to one-third of the whole Energy applied to the machine.

Friction seems to arise from a decree of adhesive attraction between the touching substances, and from the roughness of their surfaces, even where, to the eye, they appear smooth.

The roughnesses, or little projections and cavities, especially in two pieces of the same substance, mutually fit each other, as the teeth of similar saws would, so as to allow the bodies in a degree to