Air pressure at the see level is approximately 1 atm or 760 mm Hg. The meaning of the mercury units is that if you take a column of atmosphere, say 1 cm2 in cross section and as high as the atmosphere goes, its weight will be equal to the weight of a column of mercury with the same 1 cm2 in cross section but only 760 mm high. If atmospheric pressure was measured in water units, it would be equivalent to 10 m of water. Prove that such a column would weigh approximately 1 kg.
The Correct Answer and Explanation is :
To prove that a column of water 10 meters high weighs approximately 1 kg per square centimeter of cross section, we need to use the relationship between pressure, density, and height.
Step 1: Pressure of a Column of Fluid
The pressure exerted by a column of fluid is given by the formula:
[
P = \rho g h
]
Where:
- (P) is the pressure at the base of the column.
- (\rho) is the density of the fluid.
- (g) is the acceleration due to gravity.
- (h) is the height of the fluid column.
For water, the typical density (\rho) is (1000 \, \text{kg/m}^3), and the acceleration due to gravity (g) is (9.81 \, \text{m/s}^2).
Step 2: Calculate the Pressure Exerted by a 10-meter Water Column
For a 10-meter column of water:
[
P = 1000 \, \text{kg/m}^3 \times 9.81 \, \text{m/s}^2 \times 10 \, \text{m}
]
[
P = 98100 \, \text{Pa} \quad (\text{Pascal is the unit of pressure, 1 Pa = 1 N/m}^2)
]
Step 3: Find the Weight of the Column of Water
The weight of the water column depends on its mass. The volume of the water in a column with a cross-sectional area (A) and height (h) is:
[
V = A \times h
]
For a 10-meter-high column with a cross-sectional area of 1 cm² (or (1 \times 10^{-4} \, \text{m}^2)):
[
V = 1 \times 10^{-4} \, \text{m}^2 \times 10 \, \text{m} = 1 \times 10^{-3} \, \text{m}^3
]
Now, using the density of water ((\rho = 1000 \, \text{kg/m}^3)), the mass of the water is:
[
\text{Mass} = \rho \times V = 1000 \, \text{kg/m}^3 \times 1 \times 10^{-3} \, \text{m}^3 = 1 \, \text{kg}
]
Thus, the weight of the column of water is approximately 1 kg.
Conclusion:
The weight of a 10-meter-high column of water with a cross-sectional area of 1 cm² is 1 kg. Therefore, the atmospheric pressure at sea level, when measured in water units, corresponds to a 10-meter-high water column weighing approximately 1 kg per cm².