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Part I: To determine the density of two solids, one of which is heavier and the other lighter than an equivalent volume of water.
Part II: To measure accurately the density of various liquids by means of the Westphal balance.
Part III: To measure the specific gravity of these liquids by means of a hydrometer.
METHOD: A body is weighed in air and then immersed in a liquid. The apparent loss in weight of the body when immersed in the liquid is, by Archimedes’ principle, equal to the weight of liquid displaced by the body. From these measurements the density and specific gravity of either the solid body or the liquid may be determined.
THEORY: The density of a substance is defined as its mass per unit volume and in the metric system of units is measured in grams per cubic centimeter. The specific gravity of a substance is defined as the ratio of the mass or weight of the substance to the mass or weight of an equal volume of water and is a pure number having zero dimensions.
A convenient method for determining densities or specific gravities is one which uses the principle of Archimedes, namely, that when a body is immersed in a fluid there is exerted on the body a vertical upward force equal to the weight of fluid displaced.
Proof of Archimedes’ Principle: Suppose aright circular cylinder of height h is immersed in a liquid of density d gm per cm3. The bottom of the cylinder is at a depth of h2 cm and the top at a depth h cm below the surface of the liquid (Fig. 1) where h = (h2 − h1) .
The pressure on the liquid at depth h1 is p1 = B + h1gd dynes per cm2, where B is atmospheric pressure. The total downward force on the upper surface of the cylinder is F1 = p1A dynes where A is the area of cross section of the cylinder. Similarly the total upward force on the bottom of the cylinder is F2 = p2A dynes.
Since the pressure in a fluid acts at right angles to any surface in contact with it, the pressure on the sides of the cylinder is everywhere horizontal and has no vertical component. Hence the total upward force in dynes on the cylinder is
F2−F1 = p2A = (h2 − h1)Agd = hAgd (1)
But hA = V, the volume of the cylinder, so that
F2−F1 = Vgd (2)
or the resultant upward force in dynes is equal to the weight of fluid displaced.
While this proof of Archimedes’ principle is for a particularly simple geometrical object, the principle is true for an object of any shape which is immersed in a fluid.
Application of Archimedes’ Principle: 1. Suppose a body has a density of D grams per cm3 and amass of M grams. The volume of the body is V = M/D and its weight in air is Mg dynes. The weight (MLg) of the body when immersed in a liquid of density d is, by Archimedes’ principle, MLg = Mg – Vdg dynes. Thus ML = M – Vd or ML = M – Md/D.
Since ML and M may be measured by means of a balance, it follows that if the density d of the liquid-say water-is known, the density D of the body may be calculated from
D = Md/(M − ML) (3)
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Proof of Archimedes’ Principle اثبات قانون ارشمیدس-