Derivation of Δp = ρgΔh

• Static Equilibrium: For a fluid at rest, the net force in any direction must be zero ($\sum F = 0$), meaning all vertical forces acting on a volume element must cancel out.

• Definition of Pressure: Pressure is defined as force per unit area ($P = F/A$), which implies that the force exerted by a fluid on a surface is $F = P imes A$.

• Mass-Density Relationship: The mass of a fluid volume is the product of its density and volume ($m = ho V$), where volume for a cylinder is $V = A imes \Delta h$.

• Weight Force: Gravity acts on the mass of the fluid to create a downward force known as weight ($W = mg$), which contributes to the pressure at lower depths.

1. Establish Force Balance

• Consider a vertical column of fluid with cross-sectional area $A$ and height $\Delta h$. The forces acting vertically are the downward force from pressure at the top ($P_1 A$), the upward force from pressure at the bottom ($P_2 A$), and the downward weight of the fluid ($W$).
• Equation: $P_2 A - P_1 A - W = 0$

2. Express Weight in Terms of Density

• Substitute the weight formula $W = mg$ into the equilibrium equation. Since $m = ho V$ and $V = A \Delta h$, the weight becomes $W = ho (A \Delta h) g$.

3. Simplify the Equation

• Substitute the weight expression back into the force balance: $(P_2 - P_1) A = ho A \Delta h g$.

4. Final Result

• Divide both sides by the area $A$. The area cancels out, showing that pressure difference is independent of the surface area: $\Delta p = ho g \Delta h$.

• Depth vs. Height: In fluid statics, $\Delta h$ is usually treated as depth (distance downward). While height usually increases upwards, pressure increases as depth increases (downwards).

• Incompressible vs. Compressible: This derivation assumes $ho$ is constant. In gases over large vertical distances (like the atmosphere), $ho$ changes with height, making this linear formula inaccurate.

• Check Units: Always ensure density is in $kg/m^3$, height in $m$, and $g$ in $m/s^2$ or $N/kg$ to obtain pressure in Pascals ($Pa$).

• Area Independence: Remember that the cross-sectional area $A$ cancels out during the derivation; if a question asks how pressure changes if the container widens, the answer is that it does not change for the same depth.

• Sign Conventions: Be careful with the direction of $\Delta h$. If you are moving 'up' in a fluid, the pressure change is negative; if moving 'down', it is positive.

• Sanity Check: Pressure must always be higher at the bottom of a column than at the top. If your calculation shows otherwise, check your force balance signs.

• The 'Shape' Trap: A common misconception is that a wider container exerts more pressure at the bottom because it holds more water. In reality, pressure only depends on vertical depth, not the total volume or shape of the container.

• Ignoring Atmospheric Pressure: When asked for 'total pressure' or 'absolute pressure,' students often forget to add the atmospheric pressure ($P_{atm} \approx 101,000 Pa$) to the hydrostatic value.

• Density Units: Using $g/cm^3$ instead of $kg/m^3$ is a frequent error. Since $1 g/cm^3 = 1000 kg/m^3$, failing to convert will result in an answer that is off by a factor of 1000.