Variation of pressure with height
Variation of Pressure with Height
The variation of pressure with height in a fluid is a fundamental concept in fluid mechanics and is essential for understanding atmospheric pressure, hydrostatics, and various applications in engineering and physics. This concept is based on the principle that pressure in a fluid decreases with an increase in height.
Hydrostatic Pressure
The pressure at any point within a static fluid is known as hydrostatic pressure. It is caused by the weight of the fluid above the point of measurement. The hydrostatic pressure increases with depth because the weight of the fluid above the point increases.
Formula
The hydrostatic pressure at a depth ( h ) in a fluid of density ( \rho ) under the influence of gravity ( g ) is given by:
[ P = P_0 + \rho g h ]
Where:
- ( P ) is the pressure at depth ( h )
- ( P_0 ) is the atmospheric pressure or the pressure at the surface of the fluid
- ( \rho ) is the density of the fluid
- ( g ) is the acceleration due to gravity
- ( h ) is the height (or depth) within the fluid
Atmospheric Pressure
Atmospheric pressure is the pressure exerted by the weight of the air in the Earth's atmosphere. It decreases with altitude because the density of air decreases with height, resulting in less air mass above a given level.
Formula
The variation of atmospheric pressure with height can be approximated by the barometric formula:
[ P = P_0 \left(1 - \frac{Lh}{T_0}\right)^{\frac{gM}{RL}} ]
Where:
- ( P ) is the pressure at height ( h )
- ( P_0 ) is the standard atmospheric pressure at sea level
- ( L ) is the temperature lapse rate
- ( T_0 ) is the standard temperature at sea level
- ( g ) is the acceleration due to gravity
- ( M ) is the molar mass of Earth's air
- ( R ) is the universal gas constant
- ( h ) is the height above sea level
Differences and Important Points
| Aspect | Hydrostatic Pressure | Atmospheric Pressure |
|---|---|---|
| Definition | Pressure within a static fluid | Pressure exerted by the atmosphere |
| Variation with Height | Increases with depth | Decreases with altitude |
| Governing Factors | Fluid density, gravity, depth | Air density, gravity, temperature |
| Formula | ( P = P_0 + \rho g h ) | ( P = P_0 \left(1 - \frac{Lh}{T_0}\right)^{\frac{gM}{RL}} ) |
| Applications | Hydraulics, submarines, diving | Aviation, meteorology, mountaineering |
Examples
Example 1: Hydrostatic Pressure
Calculate the pressure 10 meters underwater in a lake, assuming the density of water is ( 1000 \, \text{kg/m}^3 ) and atmospheric pressure is ( 101325 \, \text{Pa} ).
Using the hydrostatic pressure formula:
[ P = P_0 + \rho g h ] [ P = 101325 \, \text{Pa} + (1000 \, \text{kg/m}^3)(9.81 \, \text{m/s}^2)(10 \, \text{m}) ] [ P = 101325 \, \text{Pa} + 98100 \, \text{Pa} ] [ P = 199425 \, \text{Pa} ]
The pressure 10 meters underwater is ( 199425 \, \text{Pa} ).
Example 2: Atmospheric Pressure
Estimate the atmospheric pressure at an altitude of 5000 meters, assuming standard atmospheric conditions.
Using the barometric formula (simplified for small height changes):
[ P = P_0 \left(1 - \frac{Lh}{T_0}\right)^{\frac{gM}{RL}} ] [ P \approx P_0 \exp\left(-\frac{Mgh}{RT_0}\right) ] [ P \approx 101325 \, \text{Pa} \exp\left(-\frac{(0.029 \, \text{kg/mol})(9.81 \, \text{m/s}^2)(5000 \, \text{m})}{(8.314 \, \text{J/(mol·K)})(288.15 \, \text{K})}\right) ] [ P \approx 101325 \, \text{Pa} \exp\left(-0.190\right) ] [ P \approx 54019 \, \text{Pa} ]
The atmospheric pressure at an altitude of 5000 meters is approximately ( 54019 \, \text{Pa} ).
Understanding the variation of pressure with height is crucial for many practical applications, including designing pressurized systems, predicting weather patterns, and ensuring the safety of structures and vehicles operating at various altitudes or depths.