Wind blows at the speed of 30m/s across a 175m^2 flat roof if a house.
what is the pressure difference between inside the house and the outside of the house just above the roof? (assume the air pressure inside the house is atmospheric pressure)
what is the force on the roof due to pressure difference?
Not bad
To calculate the pressure difference between the inside and the outside of the house just above the roof, we need to consider Bernoulli's principle.
According to Bernoulli's principle, the total pressure in a fluid system is a sum of the static pressure and the dynamic pressure. The dynamic pressure is given by the formula:
Dynamic pressure = 1/2 * density * velocity^2
Given that the wind is blowing at a speed of 30 m/s, we can calculate the dynamic pressure.
Dynamic pressure = 1/2 * density * (30 m/s)^2
However, to calculate the pressure difference, we need to subtract the atmospheric pressure inside the house. Let's assume the atmospheric pressure is 101,325 Pascals.
Pressure difference = Dynamic pressure - Atmospheric pressure
Now, let's calculate the force on the roof due to the pressure difference. The force can be calculated using the formula:
Force = Pressure difference * Area
Given that the area of the roof is 175 m^2, we can calculate the force on the roof.
Force = Pressure difference * 175 m^2
Now, let's substitute the values and calculate the pressure difference and the force.
To calculate the pressure difference between inside and outside the house just above the roof, we need to consider the effect of the wind blowing across the roof. The Bernoulli's equation is commonly used to calculate such pressure differences.
1. Calculate the kinetic energy of the wind:
The kinetic energy (KE) of the wind can be calculated using the formula KE = 0.5 * m * v^2, where m is the mass of air passing through a given area and v is the velocity of the wind.
Given that the wind is blowing at a speed of 30 m/s across a 175 m^2 flat roof, we can assume that the wind is passing through a rectangular cross-section directly above the roof. Therefore, the area perpendicular to the wind is equal to the roof area, which is 175 m^2.
The mass of air passing through the roof during a certain time interval can be calculated using the formula m = ρ * A * v, where ρ is the density of air and A is the area perpendicular to the wind.
Assuming the density of air is 1.225 kg/m^3 (at sea level and 20°C), we can calculate the mass of air passing through the roof:
m = 1.225 kg/m^3 * 175 m^2 * 30 m/s = 6412.5 kg.
Now, let's calculate the kinetic energy:
KE = 0.5 * m * v^2 = 0.5 * 6412.5 kg * (30 m/s)^2 = 5761125 J.
2. Calculate the pressure difference:
The pressure difference between inside and outside the house above the roof can be determined using Bernoulli's equation:
P1 + 0.5 * ρ * v1^2 = P2 + 0.5 * ρ * v2^2,
where P1 is the pressure inside the house (considered to be atmospheric pressure), P2 is the pressure outside the house above the roof, and v1 and v2 are the velocities of air inside and outside the house, respectively.
Since the wind speed outside is 30 m/s, we can assume the velocity of air inside the house is negligible (as there is no wind blowing inside). Thus, the equation simplifies to:
P1 = P2 + 0.5 * ρ * v2^2.
The pressure difference (ΔP) is given by:
ΔP = P2 - P1 = 0.5 * ρ * v2^2.
Substituting the values,
ΔP = 0.5 * 1.225 kg/m^3 * (30 m/s)^2 = 547.875 Pa.
Therefore, the pressure difference between inside and outside the house just above the roof is approximately 547.875 Pa.
Now, let's calculate the force on the roof due to the pressure difference:
3. Calculate the force on the roof:
The force (F) on the roof can be calculated using the formula F = ΔP * A, where ΔP is the pressure difference and A is the area of the roof.
Given that the pressure difference (ΔP) is 547.875 Pa and the roof area (A) is 175 m^2, we can calculate the force on the roof:
F = ΔP * A = 547.875 Pa * 175 m^2 = 95940.625 N.
Therefore, the force on the roof due to the pressure difference is approximately 95940.625 Newtons.
rho = about 1.2 kg/m^3
pressure diff = (1/2) rho v^2 = 0.6 (900) = 640 Newtons/m^3 or Pascals
force = 640 * 175 Newtons