A 45 m/s (100 mph) wind blows across the top of a flat-roofed house. Calculate the net force on the roof due to the pressure differential from inside the house to the outside. Assume that the pressure in the house is atmospheric pressure and that the roof has an area of 200 square meter. The density of air equals 1.3 kilogram per cubic meter.
Force = (roof area)*(1/2)*(rho)*V^2
Rho is the air density
To calculate the net force on the roof due to the pressure differential, we need to determine the pressure difference across the roof and then use that to calculate the force.
Step 1: Calculate the pressure difference
The pressure difference is caused by the wind blowing across the top of the house. This creates a region of lower pressure on the windward side and a region of slightly higher pressure on the leeward side.
Given that the wind speed is 45 m/s, we can assume that the air flows over the roof without turbulence. In this case, we can use Bernoulli's equation to calculate the pressure difference.
Bernoulli's equation states that the total pressure in a fluid system is the sum of the static pressure and the dynamic pressure. In this case, the dynamic pressure is due to the motion of the wind.
P_total = P_static + P_dynamic
The dynamic pressure (P_dynamic) is given by the equation:
P_dynamic = (1/2) * density * V^2
Where:
- P_dynamic is the dynamic pressure
- density is the density of air (1.3 kg/m^3)
- V is the velocity of the wind (45 m/s)
Step 2: Calculate the net force
The pressure difference across the roof is the difference between the pressure inside the house (atmospheric pressure) and the pressure on the outside of the roof.
Using the pressure difference and the area of the roof, we can calculate the net force (F_net) exerted on the roof using the equation:
F_net = pressure difference * area
Where:
- F_net is the net force on the roof
- pressure difference is the difference in pressure between the inside and outside of the house
- area is the area of the roof (200 m^2)
Let's calculate:
First, calculate the dynamic pressure:
P_dynamic = (1/2) * density * V^2
P_dynamic = (1/2) * 1.3 kg/m^3 * (45 m/s)^2
Next, calculate the pressure difference:
P_difference = P_dynamic
Since the pressure inside the house is assumed to be atmospheric pressure, we can consider it as the reference pressure (P_static = atmospheric pressure = 0).
P_difference = P_dynamic - P_static
P_difference = P_dynamic - 0
P_difference = P_dynamic
Finally, calculate the net force:
F_net = P_difference * area
F_net = P_dynamic * area
Now, substitute the values and calculate:
F_net = (1/2) * 1.3 kg/m^3 * (45 m/s)^2 * 200 m^2
F_net = 2.925 * 10^4 Newtons
Therefore, the net force on the roof due to the pressure differential is approximately 2.925 * 10^4 Newtons.