A plastic sphere floats in water with 50.0% of its volume...

A plastic sphere floats in water with 50.0$\%$ of its volume submerged. This same sphere floats in glycerin with 40.0$\%$ of its volume submerged. Determine the densities of the glycerin and the sphere.

Key Concepts

Volume Displacement

The concept of volume displacement is crucial because it connects the submerged fraction of an object to the weight of the fluid displaced. In floating objects, the proportion of the volume immersed directly determines the amount of displaced fluid, thereby linking the object's density to that of the fluid for equilibrium conditions.

Buoyant Force and Equilibrium

When an object floats, it is in mechanical equilibrium, meaning the upward buoyant force exactly balances the object's weight. By analyzing the fraction of the object’s volume that is submerged, one can relate the densities of the object and the fluid using the principle that the buoyant force equals the weight of the displaced fluid.

Archimedes' Principle

This principle states that any body completely or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid that is displaced by the body. It is key in understanding why and how objects float and is the basis for setting up equations relating the object's weight to the volume of fluid displaced.

Density

Density is the mass per unit volume of a substance and it determines how heavy a given volume of material is. It is fundamental to determining buoyancy, as an object's ability to float depends on the relationship between its density and the density of the fluid it is in.

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