Problem Statement
Determine the gauge and absolute pressure at a point which is 2.0 m below the free surface of water. Take atmospheric pressure as 10.1043 N/cm².
Given Data & Constants
- Depth below free surface, \(h = 2.0 \, \text{m}\)
- Atmospheric pressure, \(P_{\text{atm}} = 10.1043 \, \text{N/cm}^2\)
- Density of water, \(\rho = 1000 \, \text{kg/m}^3\) (standard value)
- Acceleration due to gravity, \(g = 9.81 \, \text{m/s}^2\) (standard value)
Solution
1. Calculate Gauge Pressure (\(P_{\text{gauge}}\))
Gauge pressure is the pressure due to the column of fluid above the point. It is calculated using the formula:
2. Convert Atmospheric Pressure to Standard Units (N/m²)
For calculations, all units must be consistent. We convert atmospheric pressure from N/cm² to N/m².
3. Calculate Absolute Pressure (\(P_{\text{abs}}\))
Absolute pressure is the sum of the gauge pressure and the atmospheric pressure.
Gauge Pressure: \( P_{\text{gauge}} = 19620 \, \text{N/m}^2 \) or \(19.62 \, \text{kPa}\)
Absolute Pressure: \( P_{\text{abs}} = 120663 \, \text{N/m}^2 \) or \(120.66 \, \text{kPa}\)
Explanation of Pressure Types
Gauge Pressure is the pressure measured relative to the surrounding atmospheric pressure. It is the pressure exerted by the weight of the fluid itself. Most pressure gauges, like a tire pressure gauge, measure gauge pressure (reading zero in open air).
Absolute Pressure is the total pressure at a point. It is the sum of gauge pressure and the atmospheric pressure. It represents the true pressure, including the weight of the atmosphere pressing down on the fluid's surface.
Physical Meaning
Imagine a diver swimming 2 meters below the surface of a lake. The diver's eardrums feel pressure from two sources:
- The weight of the 2-meter column of water directly above them. This is the gauge pressure of 19,620 N/m².
- The weight of the entire Earth's atmosphere pushing down on the lake's surface, which is then transmitted through the water. This is the atmospheric pressure of 101,043 N/m².
The total, or absolute pressure, experienced by the diver is the sum of these two, which is 120,663 N/m². This is why deep-sea submersibles must be built with incredibly strong hulls to withstand the immense absolute pressure at great depths.
