A rectangular block floats in pure water with 0.5 inch above...

A rectangular block floats in pure water with 0.5 inch above the surface and 1.5 inches below the surface. When placed in an aqueous solution, the block of material floats with 1 inch below the surface. Estimate the specific gravities of the block and the solution. (Suggestion: Call the horizontal crosssectional area of the block $A$. $A$ should cancel in your calculations.)

Key Concepts

Archimedes' Principle

Archimedes' Principle states that a body immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid that it displaces. This principle provides the fundamental basis for understanding the floating or sinking behavior of objects. In problems of floating equilibrium, this principle is used to relate the weight of the object (or block) to the weight of the displaced fluid, which in turn depends on the submerged volume of the object.

Floating Equilibrium

Floating equilibrium occurs when the upward buoyant force acting on an object exactly balances the downward gravitational force (the object's weight). For a floating body, the fraction of the total volume that is submerged is directly related to the ratio of the object’s density (or specific gravity) to the density of the fluid. This relationship allows one to determine either the density of the object or the fluid provided the submerged volume fraction is known.

Specific Gravity

Specific gravity is the ratio of the density of a substance to the density of a reference substance—typically water for liquids. It is a dimensionless quantity that indicates whether an object will float or sink in the fluid, and by comparing the specific gravitational values, one can infer the relative buoyancy. In this context, knowing the submerged fractions allows us to set up equations that relate the specific gravity of the block and the fluid.

Relationship of Volume Displacement to Density

In problems involving floating bodies with uniform cross-sectional areas, the actual dimensions (such as area) cancel out, leaving a relationship purely between the heights (or submerged fractions) and the densities. This simplification occurs because the volume of the displaced fluid, which determines the buoyant force, is directly proportional to the depth of submersion when the cross-sectional area is constant. Thus, ratios of these depths can be directly used to estimate the specific gravities.

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