Hydrostatics Fundamentals
📌 Hydrostatics studies fluids in a static (non-moving) state, primarily focusing on liquids and gases that can flow.
💧 A homogeneous fluid is defined as one that flows and possesses a constant density ($\rho$) regardless of its position within the volume.
🌊 Pressure ($P$) is defined as the magnitude of the force applied perpendicular to a surface area ($A$): $P = F/A$. Common units include $\text{N/m}^2$ (Pascal, Pa), atm, $\text{cmHg}$, and psi.

Hydrostatic Pressure and Gauge Pressure
⚖️ Hydrostatic Pressure ($P_h$) is the pressure exerted due to the weight of the fluid above a point, calculated using the formula $P_h = \rho g h$, where $h$ is the depth from the surface.
💨 Absolute Pressure ($P_{abs}$) is the total pressure at a point, which is the sum of hydrostatic pressure and the external atmospheric pressure ($P_0$): $P_{abs} = \rho g h + P_0$.
📊 Gauge Pressure ($P_{gauge}$) is the pressure relative to the external atmospheric pressure, calculated as $P_{gauge} = P_{abs} - P_0$, which simplifies to $P_{gauge} = \rho g h$.

Principle of Hydrostatics
🔗 The Principle of Hydrostatics states that at the same vertical depth ($h$) within the same continuous fluid, the pressure is equal along any horizontal line.
📐 This principle is applicable only if two conditions are met: the points must be at the same depth and within the same continuous fluid.
🌡️ A practical application is the barometer, which measures external air pressure by balancing it against the hydrostatic pressure of a fluid column (like Mercury, $\text{Hg}$). For a simple barometer with vacuum above the fluid column, the external pressure $P_0$ equals the hydrostatic pressure: $P_0 = \rho_{Hg} g h$.

U-Tube Manometer Application
🛠️ In a U-tube manometer containing multiple fluids (e.g., gasoline, water, oil), the pressure equality principle is used by drawing a horizontal line across points at the same depth within the same connecting fluid.
💡 For the example problem involving three fluids, the balancing equation equating pressures on the left and right sides of the interface yields a relationship solved for the height difference ($\Delta h$):
$$\rho_2 g \Delta h_2 + \rho_1 g h_1 = \rho_3 g h_3$$
The calculation determined the height difference of the water columns to be $1.6 \text{ cm}$.

Key Points & Insights
➡️ Understanding the distinction between absolute pressure ($\rho g h + P_0$) and gauge pressure ($\rho g h$) is crucial for fluid mechanics problems.
➡️ When applying the Principle of Hydrostatics, ensure the reference line is horizontal and connects points within the same continuous fluid.
➡️ In multi-fluid systems like the U-tube, systematically calculate the pressure contribution ($\rho g h$) from each fluid column above the reference points.

📸 Video summarized with SummaryTube.com on Nov 25, 2025, 13:41 UTC