ured in terms of a magnitude of the same kind, so that among our known forces we must pick out one in terms of which to measure all the rest.

The force which we select for this purpose is one due to gravitation, and the system of measurement is accordingly called the Gravitation System-the unit force being called the Gravitation Unit.

Gravitation Unit.-This unit force is the force exerted by gravity on, or the weight of, a certain lump of platinum kept in the Exchequer Office in London, and defined as one pound.

Our unit force then is the weight of one pound, and we measure forces in terms of this unit, or in ordinary language, in pounds. Thus, for example, if we hang up to a peg a piece ten times as heavy, we should say the pull on the peg is 10 pounds.

Now, there is a theoretical drawback to the use of the gravitation unit, which is, that the force exerted by gravity on the lump of metal is not a constant one at different parts of the earth's surface, being at the poles 177 times as much as at the equator. We thus have a variable unit, and we should, for definiteness, insert the particular position on the earth's surface at which the force is to be measured. If this be done we have a quite definite unit.

For scientific purposes this may be done, but usually for such purposes another unit is used, depending on the known laws governing matter and motion. For the purposes of the engineer, however, it is quite unnecessary to consider such refinements, since that absolute accuracy which must in scientific matters be attained, is not only not necessary, but cannot possibly be obtained in results which depend for their accuracy on that of the instruments by which they are obtained.

Suppose, for instance, a column 4 inches in diameter to be sufficient to support certain material at the equator, what difference should there be at the pole ?

The area should be increased in the ratio 177, diameter in the square root of this ratio,

poles

:. Correct diameter at equator=4x V

=4.012.

or the

Now it is not at all unlikely that such a column though intended to be 4 inches in diameter, would be quite 4.012 inches, this being quite within ordinary limits of accuracy, and, even if not, there would have been an ample margin of strength allowed in the original design to admit of such a small increase of load without prejudicial effect.

For absolute accuracy of expression also we should not speak of a force of 10 or 20 pounds, but of 10 or 20 pounds' weight, since the unit is not the lump of platinum but its weight. No harm, however, will ensue, and time is saved by the abbreviation, so long as the student clearly comprehends that the unit is the force or weight, and not the lump of metal.

We will now see then how to measure the various forces in order.

Elasticity of a Fluid is measured by allowing it to push out a piston of known area, compressing a spring before it.

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Fluid

Fig. 38.

We then measure the compression of the spring, and knowing by experiment the weight which will compress it to the same

extent, we know the amount of the effort or pressure on the piston.

For example—let the piston be 2 sq. ins. in area, and let the spring be found compressed to the same extent as would be done by a force of 60 lbs. Then

The force on any other part of the surface is proportional to the area, and we most easily find it by first finding the pressure on each square inch. The elasticity

is then measured by the pounds pressure on a square inch.

In the present example

Pressure on I sq. in. = 60 = 30 lbs.

and we say the fluid is at 30 lbs. per sq. in. pressure. Then the pressure on any number, say n sq. in. of the surface, is 30 n lbs.

66

EXAMPLE.-Steam is admitted to a cylinder 20 ins. diameter at a pressure of 30 lbs. by gauge; what is the effort on the piston? The pressure is " 30 lbs. by gauge"; here per sq. in." is omitted, which is common in actual practice; also we must ask-Does " 30 lbs. by gauge" mean that the actual pressure of the steam on a surface in contact with it is 30 lbs. on the sq. in.? The answer is that it does not. For a boiler pressure gauge is so marked as to show, not the pressure, but the difference between the steam pressure inside and the air pressure outside. This is common to most pressure gauges. For another case we have so-called vacuum gauges, which are attached to spaces in which the pressure is less than that of the atmosphere; these show the amount by which the inside pressure falls short of the outside.

The actual pressure on the inside surface is called the Absolute Pressure. We have then

Absolute pressure per sq. in. =30+ atmospheric pressure.

The latter varies, and must for any given case be measured by the barometer at the particular time considered. It does not, however, vary much from 14.7 lbs. per sq. in., which value can generally be taken as quite accurate enough,

When we have large forces as here to deal with, we often use a larger unit, viz. the Ton of 2240 lbs.

Thus in above

Effort=14043=6.027 tons.

Gravitation Efforts, etc.-These are, of course, the easiest of measurement, although we took Elasticity of Fluids first, as we had before given it the first place.

To determine the resistance gravity offers to the lifting of a body, we can either actually weigh it in a weighing machine, or if this be not convenient or possible, e.g. when we have to estimate the weight of a body from the drawing of it before it is made, we calculate its volume; and then knowing the weight of a known volume of the material of which it is composed, we can easily deduce its weight. Thus the resistance offered to the lifting of a boiler plate 12 ft. by 6 ft. by 1 inch is found thus— Volume = 12×6×12 c. ft.,

[The work is facilitated by leaving all the arithmetic to the last, unless the intermediate results be required. For example, the volume above is 10368 c. ins., but since we do not require it we shorten the work by not calculating it out.]

Laws of Resistance-Spiral Spring.—The resistance offered by a spiral spring to extension or compression can, for any given spring, be determined for any given alteration of length by determining, by actual experiment, the weight which will cause the particular alteration. Similarly during the compression of a given volume of fluid behind a piston, the resistance at any instant can be determined by a pressure gauge. It is found, however, that the resistances so found are, in the first example, connected by a certain law with the alterations in length of the spring, and in the second connected by a similar law with the change of volume.

Thus knowing one value of the resistance we can determine any others we require. Take first the spiral spring.

The figure shows a spring in three positions. In the centre it is in its natural state, length

7, while in the other two it is represented as compressed and extended respectively through a distance

The law then is

If x be small compared to 7, then the force P required, either to com

Fig. 39.

press or extend the spring, varies directly as x. may say

We

0

where Po is the force which would double the length or compress it to zero, if the law held good, but this it does not do when r becomes large.

We can conveniently represent the law graphically thus

Take OA = / and produce it.

[The line is broken since we want to use a fairly large scale.] Take points 1, 2, 1', 2', etc., and at each point set up an ordinate representing,

B 2

H

Fig. 40.

C

on a selected scale, the force -0 required to compress or extend the spring to the said point.

We thus get a curve DAE through the tops of the ordinates, and the law says that so long as we do not go too far from A the curve is a straight line. What its shape is farther away we do not discuss. To prove that this agrees with the law, we have by similar triangles

i.e. the force varies as the extension or compression in each case.

Elastic Fluid.—The magnitude of the effort exerted,