or the resistance offered by a certain mass of elastic fluid during expansion or compression, varies according to the circumstances, and the law can only be determined by aid of the principles of Thermodynamics, and then only for certain simple cases. There is, however, one law which by reason of its simplicity is most used, and which applies very nearly to many actual cases.

The law just mentioned is known as Boyle's Law, and is as follows: The pressure per sq. in. exerted by a given quantity of a fluid on the sides of the containing vessel varies inversely as the volume of the vessel.

By quantity we mean weight, not volume, because any quantity of an elastic fluid, however small, will, if allowed, expand, and fill any volume however large.

Thus if we have in a cylinder fluid at 90 lbs. pressure, then by moving the piston till the volume is halved, the pressure will be doubled and vice versa.

This law also is well suited for graphic representation.

The original pressure and volume being given, we proceed as follows :—

Choose a scale for volumes, i.e. I inch to represent

Fig. 41.

B

pressure P2 lbs.

say n cubic feet; and a scale for pressures, i.e. I inch to say m lbs. per square inch.

Set off OA on the volume scale to represent V1, the original volume in cubic feet, then OA=V1/" ins. And also Ar to represent P1 the original pressure, then AI = P1/m ins.

Let now the piston move until the volume is V, cubic feet, and the per square inch. Set off

2

OB=V2, B2=P2,

and so on for a number of other volumes.

We shall thus obtain a number of points similar to

1 and 2, and we then draw a curve through these points as shown.

The curve so drawn represents the relation between . the pressure and volume of the given quantity of fluid. Now the law is

i.e.

P1V1=P2V ̧=any Px the corresponding V,

A1 × OA=B2 × OB=any ordinate x the corresponding abscissa.

Looking back to the Preliminary Chapter, we see that this is a known curve, viz. the Hyperbola.

Hence, then, we need only know one point, say 1, on the curve, and we can then construct it by the method given (page 8), and thus determine graphically the pressure at any given point of the expansion or compression.

The law then gives us, at any point, the pressure per square inch, and we calculate the effort as already explained.

In an actual steam engine cylinder, we cannot find the law of effort, yet we can by means of an instrument called an Indicator make the pressure register itself, and actually draw a curve of the nature of the one we have just been considering. We shall return to this important question in the next chapter.

Friction. We will now see in what way to calculate the values of resistances due to friction.

We shall treat for the present only the sliding pair, leaving the turning pair until we have inquired into the peculiar character of the efforts and resistances in that kind of pair. Also, we leave the case in which one of the bodies is a fluid—e.g. a ship sliding relative to the water to the section on Hydraulics.

Whenever we move one element of a pair relative to the other, the surfaces being pressed together, a certain resistance is offered to the motion; which resistance we know varies with the state of the surfaces, and also with

E

the force with which they are pressed together. resistance is called Friction.

This

The foregoing facts are matters of common knowledge. For example, let us take a sledge loaded with a certain load; then if the road be fairly smooth, the sledge can be drawn along; and the greater the load the greater the force required to draw it. If the road, however, exceed a certain amount of roughness the sledge would not move at all; while, on the other hand, if we smooth the road, less and less effort is required, till, when we come to a smooth surface such as ice, the effort becomes very small indeed.

Now we do not in this country use sledges, but the example is taken because the surfaces rub on each other, and the sliding friction is evident. The friction in the case of a wheeled carriage is of a more complicated nature if we examine it thoroughly; but just as the motion is, on the whole, sliding; so, on the whole, will the friction follow much the same law as sliding friction.

The surfaces we shall principally have to deal with are not like that of a road, but are of metal, and made as smooth as circumstances will admit of.

What we wish to know now then is, What is the law, if any, which connects together the resistance offered and the pressure between the surfaces? And how does it vary for different surfaces?

Experiment only can furnish the answer required, and

D

B

E

accordingly experiment has been used with the following results :—

Let CD represent a small sledge, which can be loaded as required, this sledge slides on a path on a horizontal bed. Attached to CD is a cord passing over a pulley at B, and having on its end a box Suppose now the sledge be

Fig. 42.
E, which can be loaded.

loaded with weights which, together with its own weight, come to W lbs. Then to keep the sledge moving uniformly along its path a certain load, P lbs. say, is required in the box E. The effort causing sliding is the tension of the cord, which, omitting certain small corrections with which we need not concern ourselves, is P lbs. Our results will refer to the corrected value, i.e. the load in the box would not be exactly P lbs. but a little more, so that P lbs. is the pull of the cord on CD.

The ratio of P to W is called the Coefficient of Friction, and we shall see on what the value of it depends. Calling the coefficient ƒ we have

P=ƒ W.

First, keeping the surfaces in contact the same, how does P vary, when we alter W by altering the load in the sledge?

Result. P varies directly as W, so that so far ƒ is

constant.

Next. Is the force P altered by altering the area of the bearing surfaces, still keeping the same materials and load?

Result. P does not alter.

The coefficient then is independent of the area of contact.

Next. Does the speed of sliding affect ƒ?

Result. The sledge being set in motion at various speeds, P was the same for all. Therefore, the coefficient

is independent of the speed.

Next. Does P alter when we alter the condition of the bearing surfaces as regards smoothness or lubrication ?

To this of course we should expect an affirmative answer, and this answer the experiments gave.

Thus

-

The value of the coefficient depends on the nature of the surfaces.

There is only one thing left to vary, viz. the material

of the surfaces; and here we must notice that it is difficult to say that two surfaces of different materials are in exactly the same condition, since condition cannot be measured, but only judged. It is evidently difficult to compare the smoothness, for example, of metals generally with that of wood; or even that of wrought-iron, with that of cast-iron, the grain being different.

Allowing for this, we find from experiment that the material also affects the value of P, which differs for different materials, even although each is finished in the best manner.

We combine all the foregoing results together in

The Ordinary Laws of Friction, viz. — The amount of friction between the elements of a sliding pair is equal to the total pressure between the surfaces multiplied by the coefficient. This coefficient being

Dependent on the material and condition of the rubbing surfaces, but independent of the extent of the surfaces in contact and of the sliding velocity.

The experiments proving the above laws were carried out by Morin in 1831-33 in the manner we have roughly sketched, and they proved the truth of the laws within the limits of pressure and velocity which he used, viz. from 0 to 10 f.s., and from 3 to 128 lbs. per square inch.

In much modern machinery these limits are far exceeded, and the methods of lubrication are so perfect, that the friction is in many cases that of a fluid between the surfaces of two solid bodies, but for the effect of these circumstances we must refer to more advanced treatises.

The experiments involved the measurement of P and W, and hence gave the values of f. Some of the more important of these values are to be found in the appended table.