A solid of density D is floating in a liquid of density d....

A solid of density $D$ is floating in a liquid of density $d$. If $v$ is the volume of solid submerged in the liquid and $V$ is the total volume of the solid, then $v / V$ is equal to
(a) $\frac{d}{P}$
(b) $\frac{D}{d}$
(c) $\frac{D}{(D+d)}$
(d) $\frac{D+d}{D}$

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Key Concepts

Archimedes’ Principle

Archimedes’ Principle states that any object immersed in a fluid experiences a buoyant force equal to the weight of the fluid it displaces. This fundamental concept is key in understanding how and why objects float or sink and is used to derive relationships between the object's weight and the volume of fluid displaced.

Density

Density, defined as mass per unit volume, is a critical property in fluid mechanics that determines the behavior of materials when immersed in a fluid. In buoyancy problems, comparing the density of a floating object with that of the surrounding fluid allows us to determine the fraction of the object that will be submerged.

Equilibrium of Floating Bodies

When a body floats, it reaches an equilibrium where the downward gravitational force equals the upward buoyant force. This balance implies that the weight of the displaced fluid (which depends on the submerged volume and the fluid's density) must equal the weight of the object. This equilibrium condition leads to the relation between the submerged volume ratio and the densities of the object and the fluid.

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