Problem Statement
The barometric pressure at sea level is 760 mm of mercury while that on a mountain top is 735 mm. If the density of air is assumed constant at 1.2 kg/m³, what is the elevation of the mountain top?
Given Data
- Pressure head at sea level, \(h_0 = 760 \, \text{mm of Hg}\)
- Pressure head at mountain top, \(h = 735 \, \text{mm of Hg}\)
- Density of air, \(\rho_{air} = 1.2 \, \text{kg/m}^3\)
- Specific gravity of mercury, \(S_{Hg} = 13.6\)
- Density of water, \(\rho_w = 1000 \, \text{kg/m}^3\)
Solution
1. Define Density of Mercury
2. Calculate Pressure at Sea Level (\(p_0\))
Convert the pressure head from mm of Hg to N/m².
3. Calculate Pressure at Mountain Top (\(p\))
4. Calculate the Elevation of the Mountain
The pressure difference is due to the weight of the column of air. Assuming constant air density:
The elevation of the mountain top is approximately \( Z \approx 283.3 \, \text{m} \).
Explanation of the Principle
Atmospheric pressure is caused by the weight of the column of air above a certain point. As one goes up in altitude (e.g., climbing a mountain), the height of the air column above decreases, and therefore, the atmospheric pressure also decreases.
For relatively small changes in elevation, the density of air can be assumed to be constant. In this case, the relationship between the pressure drop (\(p_0 - p\)) and the change in elevation (\(Z\)) is linear, following the basic hydrostatic equation: $$ \Delta p = \rho g Z $$ This problem uses this simplified model to estimate the mountain's height based on the measured difference in barometric pressure.
Physical Meaning
The result shows that a drop in barometric pressure of 25 mm of mercury (from 760 mm to 735 mm) corresponds to an elevation gain of approximately 283.3 metres under the assumed conditions.
This principle is the basis for how an altimeter works in an aircraft or for a hiker. An altimeter is essentially a sensitive barometer (an instrument that measures air pressure). By measuring the change in atmospheric pressure, it can calculate and display the corresponding change in altitude. This calculation is a simplified example of how such a device functions.
