Pressure in a Liquid

• The Fundamental Equation: The pressure exerted by a column of liquid is calculated using the formula:

$p = h \rho g$

• Variable Definitions:

  • $p$: Pressure in Pascals (Pa).
  • $h$: The vertical depth (or height of the column) from the surface of the liquid to the point of interest, measured in meters (m).
  • $\rho$ (rho): The density of the liquid in kilograms per cubic meter ($kg/m^3$).
  • $g$: The gravitational field strength, usually $9.8\text{ N/kg}$ on Earth.

• Linear Relationship: The formula demonstrates that pressure is directly proportional to both depth and density. If the depth doubles in a uniform liquid, the pressure also doubles.

• Depth ($h$): Pressure is independent of the total volume of the liquid; it only depends on the vertical distance from the free surface. A narrow pipe 10 meters tall exerts the same pressure at its base as a massive lake 10 meters deep.

• Density ($\rho$): Denser liquids exert more pressure at the same depth because a column of a denser liquid has more mass (and thus more weight) than an identical volume of a less dense liquid.

• Gravitational Field ($g$): On planets with higher gravity, the same liquid at the same depth would exert significantly more pressure because the weight of the liquid column increases.

• Gauge Pressure: This is the pressure calculated solely from the liquid column ($h \rho g$). It represents the pressure relative to the atmospheric pressure at the surface.

• Absolute (Total) Pressure: To find the actual pressure experienced by a submerged object, one must add the atmospheric pressure ($p_{atm}$) acting on the surface of the liquid to the gauge pressure.

• Formula for Total Pressure: $p_{total} = p_{atm} + h \rho g$

• Practical Application: At sea level, atmospheric pressure is approximately $101,000\text{ Pa}$. When diving, every 10 meters of water depth adds roughly another $101,000\text{ Pa}$ (1 atmosphere) of pressure.

• Check Your Units: Always ensure density is in $kg/m^3$ and depth is in meters. If density is given in $g/cm^3$, multiply by $1000$ to convert to $kg/m^3$.

• Identify 'h' Correctly: In exam diagrams, 'h' is always the vertical distance from the top surface of the liquid down to the point. Do not use the height from the bottom of the container unless specifically asked for a height difference.

• Atmospheric Pressure: Read the question carefully to see if it asks for 'pressure due to the liquid' (gauge pressure) or 'total pressure'. If it asks for total pressure, you must add the atmospheric value (usually provided).

• Sanity Check: Remember that water has a density of approximately $1000\text{ kg/m}^3$. If your calculated pressure for a few meters of water is in the millions of Pascals, you likely have a unit conversion error.