Liquid Pressure at Depth and Manometric Measuring Instruments

Pressure at a point inside a static liquid is physically defined as the normal compressive weight of the liquid column resting above that point per unit cross-sectional area: P = F / A = (m × g) / A = (ρ × V × g) / A = (ρ × A × h × g) / A = ρgh. When the liquid vessel is exposed to ambient atmospheric air, total absolute pressure is: P = P_atm + ρgh. The gauge pressure is simply: P_gauge = P - P_atm = ρgh.

• Characteristics of Hydrostatic Pressure: (1) Liquid pressure at a given depth acts uniformly in all spatial directions (omnidirectional).
• (2) In connected vessels of varying geometric shapes, the liquid levels in all branches stabilize at the exact same horizontal height regardless of vessel geometry or cross-sectional area, because pressure depends solely on vertical depth h, not base surface area or liquid volume.

The U-Tube Manometer: A U-shaped glass tube partially filled with liquid. When two immiscible liquids (e.g., water and oil) are poured into opposite arms, at equilibrium, the pressure at the separating interface level must balance: P1 = P2 -> P_atm + ρ1·g·h1 = P_atm + ρ2·g·h2 -> ρ1·h1 = ρ2·h2. The column heights above the common interface are inversely proportional to their respective densities.

Torricellian Mercury Barometer: Evangelista Torricelli balanced atmospheric air weight against a dense mercury column. At sea level and 0°C, atmospheric pressure balances exactly 76 cm of mercury (0.76 mHg). The vacuum above the mercury column is known as Torricellian Vacuum, containing only trace mercury vapor. Standard atmospheric pressure: P_atm = ρ_Hg × g × h = 13,600 kg/m³ × 9.8 m/s² × 0.76 m = 1.013 × 10⁵ N/m² (Pascals) = 1.013 bar = 760 Torr (mmHg).

Sphygmomanometer Blood Pressure Dynamics: Arterial pressure cycles between Systolic Pressure (~120 mmHg, maximum pressure during left ventricular systole when aortic valves open) and Diastolic Pressure (~80 mmHg, minimum resting pressure during ventricular diastole).