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Body equal weight of the displaced liquid

Experimental Demonstration of Archimedes’ Principle. — If a rec­tangular block ABCD (Figure 29) is immersed in a vessel of liquid, the pres­sures on the vertical sides are equal and in opposite directions. These forces will not therefore tend to move the block in the liquid. Upon the upper face of the block, there is a downward force equal to the weight of the column of liquid having this face as a base and having a height h. On the lower face, there is an upward force equal to the weight of a column of liquid which has an area equal to the area of the lower base and a height H equal to the depth of this face below the surface of the liquid. The upward force exceeds the downward force by the weight of a column of liquid having a base equal to the area of the cross section of the block and a height equal to the height of the block. The volume of this column is just equal to the volume of the liquid displaced by the immersed block, and the weight of this column is equal to the weight of the displaced liquid. The same sort of reasoning will hold for a body of any shape in any liquid. Hence, a body immersed in a liquid is light­er by the weight of the volume of liquid that it has displaced.

Density and Specific Gravity. — In order to determine the density of a body, it is necessary to determine its mass and its volume. The density is then found by dividing the mass by the volume. The mass of the body is easily determined by weighing, but it is sometimes difficult to find the volume, especially when the body has an irregular shape. In such cases, the volume may be determined by an application of Archimedes’ principle. Since the body displaces a volume of water equal to its own volume and since each cubic centimeter of water weighs 1 g, the loss of weight in water is numerically equal to the volume of the immersed body.

The numerical value of the density of a body depends on the units which the mass and the volume are measured. In the cgs system, the density is the number of grams per cubic centimeter. In the British system, it is the number of pounds per cubic foot.

The specific gravity of a body is the ratio of its density to the density of water at 4 °C. Since in the cgs system a gram is defined to be the weight of a cubic centimeter of water at 4 °C, the numerical values of the density and the specific gravity in this system are the same. In the British system, however, they are very different.

Density of Solids Heavier than Water. — When a body is heavier than an equal volume of water and is insoluble in water, its volume can be determined by finding its loss in weight when weighed in water. This loss of weight is equal to the weight of the water displaced, and if this loss of weight is expressed in grams, it is numerically equal to the volume of the body in cubic centimeters. By dividing the mass of the body by this volume, the density is obtained.

Density of Solids Lighter than Water. — If the body is lighter than water but insoluble, its volume may still be determined by this method by fastening to the body a sinker large enough to force it below the surface of the water. In this case (Figure 30), the combined weight of the body and the sinker is first determined when the sinker is immersed in water and the body is above the sur­face of the water.

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