Final answer:
The hydrostatic paradox describes the uniform pressure at any horizontal level within a fluid at rest, which only depends on the depth of the fluid. The expression of pressure variation with depth in a fluid is given by P = ρgh, where ρ is the density of the fluid, g is the gravitational acceleration, and h is the depth. This principle is encapsulated by Pascal's Law, which states that pressure changes in a confined fluid are transmitted undiminished.
Step-by-step explanation:
Hydrostatic Paradox and Pressure Variation with Depth The term hydrostatic paradox refers to the phenomenon that within a fluid at rest in a gravitational field, the pressure is the same at any given horizontal level, regardless of the shape or volume of the fluid above that level. While it may seem paradoxical, it is actually a result of the fact that the pressure at any point in a static fluid depends only on the depth at that point, and not on the overall volume of the fluid.
To derive an expression for the variation of pressure with depth in a fluid, consider a fluid of density ρ within a gravitational field. The pressure at a depth h is given by P = ρgh, where g is the acceleration due to gravity. This indicates that as you go deeper into a fluid, the pressure increases linearly with depth due to the weight of the fluid above. The factors on which this variation depends include the density of the fluid (ρ), the acceleration due to gravity (g), and the depth (h) within the fluid.
According to Pascal's Principle, any change in pressure applied to a confined fluid at rest is transmitted undiminished throughout the fluid. This principle is vital in understanding how pressure operates within static fluids and has significant applications in hydraulic systems.