At the top a mountain the temperature is \(-5^{\circ} \mathrm{C}\) and \(\mathrm{a}\) mercury barometer reads \(566 \mathrm{~mm}\), whereas the reading at the foot of the mountain is \(749 \mathrm{~mm}\). Assuming a temperature lapse rate of \(0.0065 \mathrm{~K} \cdot \mathrm{m}^{-1}\) and \(R=287 \mathrm{~J} \cdot \mathrm{kg}^{-1} \cdot \mathrm{K}^{-1}\), calculate the height of the mountain. (Neglect thermal expansion of mercury.)

The temperature lapse rate is a key factor when calculating altitude based on pressure and temperature differences. It represents the rate at which temperature decreases with an increase in altitude. In many atmospheric conditions, this rate is standardized at 0.0065 K/m, meaning for every meter you ascend, the temperature drops by 0.0065 Kelvin.
For example, if you start at sea level with a temperature of 15°C and climb 1000 meters, you would experience a decrease in temperature by approximately 6.5 K, ending up around 8.5°C. Understanding this rate helps us accurately determine the changes in atmospheric pressure and, consequently, in altitude.

Pressure difference is a crucial component in altitude calculations. In the given problem, the pressures at the top and bottom of the mountain were 566 mmHg and 749 mmHg, respectively. Atmospheric pressure decreases with altitude due to the thinning of air.
By measuring the pressure difference and utilizing the barometric formula, we can infer the height difference between two points. It's important to convert pressure units consistently. For instance, 1 mmHg (millimeter of mercury) is equivalent to approximately 133.322 Pa (Pascal). The relative decrease in pressure as altitude increases is essential for determining height accurately.

Altitude calculation involves combining temperature and pressure data to estimate height differences. Using the barometric formula \[P_2 = P_1 \times \text{exp}\bigg(-\frac{g \times \bigtriangleup h}{R \times T}\bigg)\], we link atmospheric pressure changes to altitude.
Here, \[P_2\] and \[P_1\] are the pressures at two altitudes, \[g\] is the gravitational constant (9.81 m/s²), \[R\] is the specific gas constant for dry air (287 J/kg/K), and \[T\] is the average temperature. Converting pressures to the same units and solving for \[\bigtriangleup h\] using the logarithmic form of the equation helps us determine the height of the mountain, which in this case is approximately 2199.6 meters.

Fluid mechanics principles are embedded in the barometric formula calculation. The atmosphere can be considered a fluid, and we analyze its behavior under varying conditions of pressure, temperature, and density. Fluid dynamics and statics govern how atmospheric pressure changes with altitude.
Understanding fluid mechanics in this context allows us to apply principles such as hydrostatic equilibrium, where the weight of the atmosphere at a given height balances the pressure forces. This balance is what leads to the exponential relation in the barometric formula, enabling us to link pressure differences with altitude changes accurately.