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 (a) the glycerin and (b) the sphere.

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Key Concepts

Density Relationships

Understanding the relationship between the densities of the object and the fluid is key. The fraction of the object submerged is directly related to the ratio of the object’s density to the fluid’s density. By comparing behaviors in different fluids, one can deduce unknown densities using the equilibrium conditions established by buoyancy.

Archimedes' Principle

This principle states that the buoyant force exerted on an object immersed in a fluid is equal to the weight of the fluid that the object displaces. It is essential for determining how much of the object’s volume needs to be underwater for it to float, based on the densities involved.

Buoyancy Equilibrium

When an object floats, the weight of the object is exactly balanced by the buoyant force acting upward. This condition leads to a direct relationship where the fraction of the object submerged equals the ratio of its density to that of the fluid. This concept is fundamental in solving problems involving floating objects in different fluids.

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