Pressure Knowledge-Based Study

Study notes

1. Pressure is defined as the magnitude of the normal force exerted per unit area on a surface:

   $P = \frac{F_{\perp}}{A}$

   where $P$ is pressure, $F_{\perp}$ is the component of force perpendicular to the surface, and $A$ is the surface area over which the force is distributed.

   In the International System of Units (SI), force is measured in newtons (N) and area in square meters ($\text{m}^2$). Therefore, the SI unit of pressure is the pascal (Pa), defined as:

   $1\text{ Pa} = 1\text{ N/m}^2$

   Pressure is a scalar quantity. At any given point within a static fluid, it acts with equal magnitude in all directions against any surface it contacts.

   In a stationary, incompressible fluid of uniform density, pressure increases linearly with depth due to the weight of the fluid column above.

   The hydrostatic pressure at depth $h$ below the free surface of a fluid with density $\rho$ is given by:

   $P_{\text{fluid}} = \rho g h$

   where:

   • $\rho$ is the fluid density (in $\text{kg/m}^3$),
   • $g$ is the acceleration due to gravity (approximately $9.81\text{ m/s}^2$),
   • $h$ is the vertical depth beneath the fluid surface (in $\text{m}$).

   When the fluid surface is open to the atmosphere, the total or absolute pressure $P$ at depth $h$ is:

   $P = P_0 + \rho g h$

   where $P_0$ represents atmospheric pressure at the surface.

   Hydrostatic Paradox: Hydrostatic pressure depends solely on fluid density, gravity, and vertical depth. It does not depend on the shape, total volume, or base area of the container.

   Pascal's Principle states that an externally applied pressure change to an enclosed, incompressible fluid at rest is transmitted undiminished to every portion of the fluid and to the walls of the containing vessel.

   This principle is the foundation of hydraulic machinery, including hydraulic lifts and hydraulic brakes. In an ideal hydraulic system with two connected pistons of cross-sectional areas $A_1$ and $A_2$:

   $\Delta P_1 = \Delta P_2 \implies \frac{F_1}{A_1} = \frac{F_2}{A_2}$

   Rearranging for the output force gives:

   $F_2 = F_1 \left(\frac{A_2}{A_1}\right)$

   Because $A_2 > A_1$, a relatively small input force $F_1$ produces a much larger output force $F_2$. By conservation of energy, the mechanical work remains equal (assuming no friction or fluid compression):

   $F_1 d_1 = F_2 d_2$

   Consequently, the smaller piston must move through a proportionally larger displacement $d_1$ than the output displacement $d_2$.

   Atmospheric pressure arises from the weight of air molecules in Earth's atmosphere. At sea level, standard atmospheric pressure is defined as:

   $1\text{ atm} = 101{,}325\text{ Pa} \approx 1.013 \times 10^5\text{ Pa}$

   Fluid pressure is commonly categorized and measured in two ways:

   • Absolute Pressure ($P$): The total pressure relative to an absolute vacuum (zero pressure).
   • Gauge Pressure ($P_g$): The pressure measured relative to local atmospheric pressure:

   $P_g = P - P_0$

   Common pressure-measuring instruments include:

   • Mercury Barometer: Measures atmospheric pressure by balancing it against a vertical column of mercury of height $h$, where $P_0 = \rho_{\text{Hg}} g h$ (equal to $760\text{ mm Hg}$ at standard sea level).
   • Open-Tube Manometer: Measures the gauge pressure of a confined gas by observing the difference in fluid column heights between the two arms of a U-tube.