(a) Derive an expression for the buoyant force on a...

(a) Derive an expression for the buoyant force on a spherical balloon, submerged in a freshwater lake, as a function of the depth below the surface, the volume of the balloon at the surface, the pressure at the surface, and the density of the water. (Assume that the water temperature does not change with depth.) (b) Does the buoyant force increase or decrease as the balloon is submerged? (c) At what depth is the buoyant force one-half the surface value?

Key Concepts

Buoyant Force as a Function of Variable Volume

In cases where the displaced volume changes with depth due to compressibility effects, the buoyant force must be expressed as a function of depth. By combining Archimedes' principle with the pressure-volume relationship from Boyle's Law and the hydrostatic pressure variation, one obtains a depth-dependent expression for the buoyant force on a compressible body.

Boyle's Law (Isothermal Compression)

When the temperature is held constant, the pressure and volume of a gas are inversely related according to Boyle’s law. This relationship is crucial when deriving the volume of a submerged, compressible balloon as a function of depth, as the increased pressure underwater leads to a decrease in the balloon's volume.

Archimedes' Principle

This principle states that any object wholly or partially submerged in a fluid experiences an upward buoyant force equal to the weight of the fluid displaced by the object. It is fundamental to understanding why objects float or sink and is the starting point for deriving expressions for buoyant force in varying conditions.

Hydrostatic Pressure Variation

Hydrostatic pressure describes how pressure in a fluid increases with depth due to the weight of the overlying fluid. In problems like this, the pressure at a given depth is used both to determine the volume changes of compressible objects and to calculate the buoyant force, since increased external pressure can compress a gas-filled balloon.

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