For a static fluid, the pressure at a depth h is given by P = P0 + rho * g * h, where rho is the fluid density and P0 is the pressure at the free surface. This expression is associated with which basic law or principle in fluid mechanics?

Introduction / Context:
The expression P = P0 + rho * g * h describes how pressure increases with depth in a static fluid under gravity. This relationship is fundamental in hydrostatics and is consistent with Pascal's law about transmission of pressure in an enclosed fluid. Understanding this formula is essential for analysing dams, submarines, deep sea diving, and many engineering applications involving liquids at rest.

Given Data / Assumptions:
• A fluid is at rest under the influence of gravity. • P0 is the pressure at the free surface of the fluid (often atmospheric pressure). • rho is the density of the fluid. • g is the acceleration due to gravity. • h is the depth below the free surface.

Concept / Approach:
In a static fluid, pressure at a point depends only on the depth and the density of the fluid, not on the shape of the container. The basic hydrostatic equation dP/dh = rho * g integrates to P = P0 + rho * g * h. This result reflects Pascal's law, which states that pressure applied to an enclosed fluid is transmitted undiminished to all parts of the fluid and to the walls of its container. Pascal's law underlies the uniform distribution of pressure in a static fluid and is the conceptual foundation for this pressure depth relationship.

Step-by-Step Solution:
Step 1: Consider a small fluid element at depth h in a fluid at rest. Step 2: The vertical forces on the element include pressure forces on the top and bottom surfaces, and the weight of the fluid element. Step 3: For equilibrium (no acceleration), the sum of forces must be zero, leading to the differential relation dP/dh = rho * g. Step 4: Integrate this equation from the free surface (h = 0, P = P0) to depth h, giving P = P0 + rho * g * h. Step 5: This result shows that pressure increases linearly with depth in a uniform, incompressible, static fluid. Step 6: This hydrostatic pressure distribution is one of the primary consequences of Pascal's law.

Verification / Alternative check:
We can test the formula with a simple example. In water (rho ≈ 1000 kg/m^3), at a depth of 10 m, the gauge pressure is approximately rho * g * h ≈ 1000 * 9.8 * 10 ≈ 98 000 Pa, which is close to one extra atmosphere of pressure. This matches what is taught in basic physics and engineering courses and confirms that the relationship is correct. The fact that the same depth in different shaped containers gives the same pressure further illustrates Pascal's law and the independence from container shape.

Why Other Options Are Wrong:
Option b (Newton's law): Newton's laws of motion describe the relationship between forces and acceleration, not directly the static pressure distribution in a fluid. Option c (Bernoulli's principle): Bernoulli's principle relates pressure, kinetic energy per unit volume, and potential energy per unit volume in a moving fluid, not a static one. Option d (Archimedes' principle): Archimedes' principle deals with buoyant force equal to the weight of displaced fluid, not the pressure depth equation. Option e (Hooke's law): Hooke's law applies to elastic solids, relating stress to strain, and does not govern hydrostatic pressure in fluids.

Common Pitfalls:
Students sometimes mistakenly associate any fluid formula with Bernoulli's principle, ignoring whether the fluid is moving or at rest. Another confusion is between Pascal's law and Archimedes' principle, since both involve fluids. A quick way to remember is that Pascal's law is about pressure transmission and hydrostatics, while Archimedes deals with buoyancy. Always check whether the fluid is static or dynamic when deciding which principle applies.

Final Answer:
The pressure depth relation P = P0 + rho * g * h is associated with Pascal's law in fluid mechanics.