Archimedes’ principle can be used not only to determine the specific gravity of a solid using a known liquid (Example 10–8); the reverse can be done as well. (a) As an example, a 3.40-kg aluminum ball has an apparent mass of 2.10 kg when submerged in a particular liquid: calculate the density of the liquid. (b) Derive a formula for determining the density of a liquid using this procedure.

EXAMPLE 10–8

FIGURE 10–14 (a) A scale reads the mass of an object in air—in this case the crown of Example 10–8. All objects are at rest, so the tension FT in the connecting cord equals the weight w of the object: FT = mg. We show the free-body diagram of the crown, and FT is what causes the scale reading (it’s equal to the net downward force on the scale, by Newton’s third law). (b) Submerged, the object has an additional force on it, the buoyant force FB. The net force is zero, so F′T + FB = mg (= w). The scale now reads m′ = 13.4 kg, where m′ is related to the effective weight by w′ = m′g. Thus F′T = w′ = w – FB.

TABLE 10–1 Densities of Substances†