gram is the usual unit used, and hence when we know the volume in cubic centimetres of any quantity of water at 4° C., we know also its weight in grams.

FIG. 9.-The Kilogram. From Aldous's Course of Physics (Macmillan).

It is advisable to remember that there are 453 59 grams in a pound; that 1 gram = 15 432 grains; and a kilogram = 2 lbs.

Weight. The weight of a body is the attractive force which the earth exerts on the body at or near its surface. It varies with the position of the body on the earth's surface, being greatest at the poles and least at the equator.

Hence the weight of a pound is a quantity of force, viz., the force equal to the tension of a string supporting a mass of 1 lb., or, as it is commonly called, a pound weight. The force of gravity varies with the distance from the earth's centre; thus, at a place where gravity is nil, such as at the centre of the earth, the weight would also be nil.

The weight of the body, equal to the tension of the string supporting it, could be ascertained by attaching the string to a spring balance. If the mass attached to the balance could be carried from the pole to the equator, although the mass remains constant, its weight, as indicated by the balance would vary continuously.

By means of an ordinary balance or pair of scales the mass of a body in terms of the unit mass, or the weight of a body in terms of the weight of the unit mass, can be obtained at the particular place where the estimation is made.

The operation of finding the weight of a body is called weighing. Thus the body whose weight is to be found is placed in one pan of a balance, and known weights are placed in the other pan until the two are balanced; the sum of the weights used is the weight of the body.

Density. The density of a substance is the mass of the unit volume of it. Assuming the density to be uniform, the density of a substance, when the unit of mass is one pound and the unit of volume one cubic foot is the number of pounds in a cubic foot of the substance.

In the cases where metric units are adopted, the density is the number of grams in a cubic centimetre of the substance.

Density of Water.-It has been found by means of careful experiments that water does not contract regularly down to the freezing point, but begins to expand before it reaches this temperature. The temperature at which a given mass of water occupies the smallest volume is 39-2° on the Fahrenheit Scale, or 4° on the Centigrade; both below and above. this temperature its volume increases, hence at this temperature water has its

maximum density, and the weight of a cubic foot is nearly 1000 oz. or 62.5 lbs. The weight of a cubic centimetre of water at its maximum density is 1 gram. It is a matter of common experience that the weights of equal volumes of different substances are not the same. For example, three balls or cubes of the same volume, but made of wood, iron, and lead respectively, would, when placed in a balance, have different weights. The wooden ball is comparatively light in comparison with the heavier iron, and the lead the heaviest of all three.

A vessel containing a cubic foot of water, the mass of which is 62.3 lbs., would, when the water is replaced by the same volume of mercury, contain 13:596 × 62.3 lbs. ; the weight of 1 cubic centimetre of water is 1 gram, the weight of the same volume of mercury is 13.596 grams.

Relative Density.-The relative density of a substance is the ratio of its weight to the weight of an equal volume of a standard substance.* It is obvious that the standard substance should be at any place easily obtainable and in a pure state; pure distilled water fulfils these conditions. The only precaution necessary is with regard to temperature. On account of the expansion and contraction due to heating and cooling, the weight of a cubic unit will depend on its temperature, hence the standard temperature is 4° C.

The relative density is usually called the specific gravity; the specific gravities of various substances are tabulated in Table III. Thus, if the specific gravity of cast iron is 7·22, then the weight of a cubic foot is 7-22 times the weight of a cubic foot of water=7·22 × 62.3=450 lbs.

The weight of a cubic centimetre of cast iron will evidently be 7.2 grams.

The weight of V cubic feet of water = Vw, where w is the weight of unit volume.

Hence, if V denote the volume of a body in cubic feet, and S its specific gravity, the weight of the body is VSxw (1). If w be the weight of unit volume, then the weight of the body is VSw= VS × 62.3 lbs. [Eq. (1)]. In this manner it is customary

* If necessary portions of this section, more especially those examples dealing with area and volume, may be postponed until the chapters on area and volume have been read.

to define specific gravity as the ratio of the weight of a given volume of a substance to the weight of the same volume of water.

If the volume of the body is obtained in cubic inches then in Eq. (1) w will denote the weight of one cubie inch (the weight of one cubic inch of water=62·3÷1728=036 lbs.).

The method of obtaining the specific gravity of a solid (not soluble in water) depends on what is known as the "Principle of Archimedes”: When a body is immersed in a liquid it loses weight equal to the weight of the liquid which it displaces; that is, if the weight of a body is obtained, first in air, and next when immersed in water, the difference in the weights is the weight of an equal volume of water:

weight in air

.. specific gravity of body=weight in air-weight in water

Ex. 1. A piece of ivory weighing 6 lbs. is found to weigh 2.8 lbs. when immersed in water. Find its specific gravity.

Ex. 2. in water.

6

specific gravity=6-28 = 1.875.

A piece of metal weighs 62-63 grains in air and 56 grains
Find its specific gravity.

S.G.=

62.63 62.63-56

=9'4.

Ex. 3. A piece of metal of specific gravity 9.8 weighs in water 56 grains. What is its true weight?

Let w denote its true weight.

Ex. 4. If 28 cubic inches of water weigh a pound, what will be the specific gravity of a substance, 20 cubic inches of which weigh 3 lbs. ?

3

As 20 cubic inches weigh 3 lbs., 1 cub. in. = 20;

Ex. 5. A piece of round steel wire 12 inches long weighs 0·65 lbs., its specific gravity is 7-8, find the area, also the diameter of the wire. If A denote the area of the wire in square inches,

Volume in cubic inches = 12× A.

Weight=124 × 7.8 × 036, but this is equal to 65 lbs. ;
.. 124 x 7.8 × '036 = '65;

inch.

From this (p. 244) the diameter is found to be

When the specific gravity of a body lighter than water is required, what is called a sinker is used, i.e. a piece of lead or metal of such a size as to secure complete immersion of the body.

Ex. 6. A piece of wood which weighs 10 lbs. is attached to a piece of metal weighing 20 lbs. ; the whole weighs 14 lbs. in water; the metal alone weighs 18 lbs. in water. Find the specific gravity of the metal and the wood.

Exc. 7. Three pints of a liquid of specific gravity 6 are mixed with four pints of a liquid of specific gravity 81; if there is no contraction, find the specific gravity of the mixture.

(A pint of water weighs 14 lbs.)

Weight of 3 pints of liquid of specific gravity 6

=3× 11 × 6=2.25 lbs.

Weight of 4 pints of liquid of specific gravity '81

= 4 × 1 × 81=4.05 lbs.

Hence the weight of the 7 pints=2.25 +405=6.3 lbs.
But the weight of 7 pints of water=7×14=8.75 lbs.

.. specific gravity of mixture= ='72.

6.3 8.75

The principle of Archimedes applies to all bodies whether they are heavier or lighter than the liquid in which they are immersed. Thus, a floating body displaces a volume of water equal in weight to the weight of the body itself.

Hence

S.G.=

volume of body immersed

whole volume