In an experiment, a rectangular block with height h is...

In an experiment, a rectangular block with height $h$ is allowed to float in four separate liquids. In the first liquid, which is water, it floats fully submerged. In liquids $A, B$, and $C$, it floats with heights $h / 2,2 h / 3$, and $h / 4$ above the liquid surface, respectively. What are the relative densities (the densities relative to that of water) of (a) $A,(\mathrm{~b}) B$, and $(\mathrm{c}) C ?$

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Key Concepts

Archimedes' Principle

This principle states that any object wholly or partially immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. It is foundational in understanding why an object floats, sinks, or remains neutrally buoyant.

Buoyancy and Equilibrium

In a state of floating equilibrium, the upward buoyant force precisely balances the gravitational force on the object. This balance allows one to relate the submerged volume (or fraction) of the object to the densities of the object and the fluid, serving as the basis for calculations in buoyancy problems.

Relative Density

Relative density is the ratio of the density of a substance to a reference density, commonly that of water. In buoyancy problems, it helps determine whether an object will float and, if so, how much of it will be submerged, as the object’s weight must equal the weight of the displaced fluid.

Submerged Fraction Concept

The fraction of an object's volume that is submerged in a fluid is directly linked to the ratio of the object's density to the fluid's density. This concept allows one to calculate the density of an unknown fluid when given the submerged depth of an object and knowing the object's own density relative to a reference fluid.

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