Derive the hydrostatic pressure at depth h for a liquid of density ρ and acceleration g (reference gauge pressure).

Derive the hydrostatic pressure at depth h for a liquid of density ρ and acceleration g (reference gauge pressure).

• A. P = ρ g h_c
• B. P = ρ g h^2
• C. P = ρ g h_c^2
• D. P = ρ g h

Answer: (D)

The main idea is that hydrostatic pressure grows linearly with depth because every additional layer of liquid adds its weight to the pressure below it. If you look at a thin slice of fluid at depth h with thickness dh, the pressure increase across that slice is dP = ρ g dh. Integrating these increments from the surface (where gauge pressure is zero) down to depth h gives P(h) = ∫0^h ρ g dh = ρ g h. Since this is gauge pressure, atmospheric pressure is excluded, so the pressure at depth h is simply ρ g h. The other forms would imply a non-linear relationship or use a height term that isn’t defined here, which isn’t correct for a uniform liquid in hydrostatic equilibrium.

The main idea is that hydrostatic pressure grows linearly with depth because every additional layer of liquid adds its weight to the pressure below it. If you look at a thin slice of fluid at depth h with thickness dh, the pressure increase across that slice is dP = ρ g dh. Integrating these increments from the surface (where gauge pressure is zero) down to depth h gives P(h) = ∫0^h ρ g dh = ρ g h. Since this is gauge pressure, atmospheric pressure is excluded, so the pressure at depth h is simply ρ g h. The other forms would imply a non-linear relationship or use a height term that isn’t defined here, which isn’t correct for a uniform liquid in hydrostatic equilibrium.