If a point is to be at the free surface of a liquid open to the atmosphere, where the pressure is the atmospheric pressure P_atm, then the gauge pressure at a depth ‘h’ from the free surface becomes:

Concept:

Gauge pressure is the pressure relative to atmospheric pressure. When dealing with fluids, gauge pressure can be calculated using the depth of the fluid, the density of the fluid, and the acceleration due to gravity.

The pressure at a depth h in a fluid can be calculated using the formula:

\(P = P_{\text{atm}} + \rho gh\)

Where:

• P is the absolute pressure at depth h.
• P_atm is the atmospheric pressure.
• ρ is the density of the fluid.
• g is the acceleration due to gravity.
• h is the depth below the free surface.

However, gauge pressure ( P_{\text{gauge}} ) is defined as the absolute pressure minus the atmospheric pressure. Therefore, the gauge pressure at depth h is:

\(P_{\text{gauge}} = P - P_{\text{atm}}\)

Substituting the expression for P :

\(P_{\text{gauge}} = (P_{\text{atm}} + \rho gh) - P_{\text{atm}}\)

\(P_{\text{gauge}} = \rho gh\)