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Pressure: Fundamental Principles and Applications in Fluids and Gases
Defining Pressure
Pressure (
) quantifies force distribution over a surface area:
![\[ P = \frac{F_{\perp}}{A} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-9e43756dc0e031942179cf0b0fd8d3bb_l3.png "Rendered by QuickLaTeX.com")
Where:
![F_{\perp}]( "Rendered by QuickLaTeX.com")
: Normal force component (N)![A]( "Rendered by QuickLaTeX.com")
: Contact area (m²)- 1 Pa = 1 N/m²
: Normal force component (N)
: Fluid Pressure
Hydrostatic Pressure
![\[ P = P_0 + \rho gh \]](https://on22.com/wp-content/ql-cache/quicklatex.com-230989b36376b874f7e41359d35641f5_l3.png "Rendered by QuickLaTeX.com")
Where:
![P_0]( "Rendered by QuickLaTeX.com")
: Atmospheric pressure (101.3 kPa)![\rho]( "Rendered by QuickLaTeX.com")
: Fluid density (kg/m³)![g]( "Rendered by QuickLaTeX.com")
: Gravitational acceleration (9.81 m/s²)![h]( "Rendered by QuickLaTeX.com")
: Depth (m)
: Atmospheric pressure (101.3 kPa)
: Fluid density (kg/m³)
: 
: Depth (m)Pascal’s Principle
Pressure transmission enables hydraulic force multiplication:
![\[ \frac{F_1}{A_1} = \frac{F_2}{A_2} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-20a65dbce4b4cf2e89ae2824971b561d_l3.png "Rendered by QuickLaTeX.com")
Gas Pressure
Ideal Gas Law
![\[ PV = nRT \]](https://on22.com/wp-content/ql-cache/quicklatex.com-bb511a7e2134ac88fc0036dbb2c95e8a_l3.png "Rendered by QuickLaTeX.com")
Constants:
![R = 8.314 \, \text{J/(mol·K)}]( "Rendered by QuickLaTeX.com")
- STP: 273.15 K, 101.325 kPa
Practical Applications
Engineering Systems
- Hydraulic lifts: Achieve 100:1 force amplification
- Scuba diving: Pressure increases ~101 kPa per 10m depth
Meteorology
- Barometric pressure: 98-104 kPa typical range
- Weather fronts: ~5 kPa pressure differences
Worked Example
Water Pressure at 5m Depth:
![\[ P = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 5 \, \text{m} = 49.05 \, \text{kPa} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-a9cd2b2e5a4400c589969b62941e09b3_l3.png "Rendered by QuickLaTeX.com")
Note: Total pressure = 49.05 kPa + 101.3 kPa = 150.35 kPa
Common Errors
- Using gauge pressure instead of absolute pressure
- Neglecting temperature in gas law calculations
- Confusing density units (kg/m³ vs g/cm³)
Practice Problems
- A hydraulic press has pistons with 10:1 area ratio. What input force lifts a 500 kg mass?
- Calculate the moles of oxygen in a 0.02 m³ tank at 300 kPa and 293 K.
- Explain why submarines have maximum operating depths.
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