A mercury barometer reads 744 mm on the roof of a building and 769 mm on the ground. Assuming a constant value of 1.29 kg/m3 for the density of air, determine the height of the building.
To determine the height of the building, we can use the concept of pressure difference and the density of air.
The pressure difference between the roof and the ground can be calculated by taking the difference between the readings on the barometer:
Pressure difference = Height difference × Density of air × Acceleration due to gravity
Let's assign variables to the given values:
Pressure difference = Pₒ (reading on the ground) - Pᵣ (reading on the roof)
Density of air = 1.29 kg/m³
Acceleration due to gravity = 9.8 m/s²
Therefore, the equation becomes:
Pressure difference = (Pₒ - Pᵣ) × Density of air × Acceleration due to gravity
Now substitute the given values:
Pressure difference = (769 mm - 744 mm) × (1.29 kg/m³) × (9.8 m/s²)
To convert the millimeters to meters, divide the pressure difference by 1000:
Pressure difference = (25 / 1000) m × (1.29 kg/m³) × (9.8 m/s²)
Simplifying the equation:
Pressure difference = 0.025 m × (1.29 kg/m³) × (9.8 m/s²)
Now, we can calculate the pressure difference:
Pressure difference = 0.025 m × 1.29 kg/m³ × 9.8 m/s² = 0.318 m²/s²
We know that pressure difference is equal to the height difference times the density of air times the acceleration due to gravity. Solving for height difference:
Height difference = Pressure difference / (Density of air × Acceleration due to gravity)
Substituting the known values:
Height difference = 0.318 m²/s² / (1.29 kg/m³ × 9.8 m/s²)
Simplifying the equation:
Height difference = 0.318 m²/s² / 12.642 kg·m⁻²/s²
Finally, calculating the height difference:
Height difference = 0.025 m
Therefore, the height of the building is approximately 0.025 meters or 25 centimeters.