The modern Boeing 747 passenger jet airplane cruises at a speed up to 900 km/hr, at an altitude of 11,000 m. The cabin pressure is maintained equivalent to the atmospheric pressure at an altitude of 2,000 m. This is called the "cabin altitude".
The pressure as a function of altitude is obtained via a simple formula that depends on the temperature and a variety of other physical parameters.
Let's assume the area of the passenger windows are about 0.082 m^2.
If the airplane were stationary at its cruising altitude, what would be the force on the window?
To determine the force on the window of the stationary Boeing 747 at cruising altitude, we need to calculate the air pressure exerted on the window.
The pressure at an altitude of 2,000 meters can be determined using the formula for atmospheric pressure as a function of altitude:
Pressure = Pressure at sea level * e^(-altitude/Scale Height)
Where:
- Pressure at sea level is approximately 101.3 kPa (kilopascals)
- e represents the mathematical constant e (approximately 2.71828)
- Altitude is the difference between the cruising altitude and the cabin altitude, which is (11,000 m - 2,000 m) = 9,000 m
- Scale height is approximately 8,500 m
By substituting the values into the formula, we can calculate the pressure at the cruising altitude:
Pressure = 101.3 kPa * e^(-9,000 m / 8,500 m)
Calculating this value gives us the pressure at the cruising altitude.
Next, we can calculate the force on the window using the formula:
Force = Pressure * Area
Where:
- Area is the area of the window, which is given as 0.082 m^2.
Substituting the calculated pressure and the given area into the formula will give us the force on the window.
To calculate the force on the window of the stationary Boeing 747 airplane at its cruising altitude, we need to consider the atmospheric pressure difference between the inside and outside of the cabin.
The formula to calculate pressure (P) is given by:
P = ρ * g * h,
where ρ represents the density of air, g is the acceleration due to gravity, and h is the height or altitude.
First, we need to convert the altitude to meters by dividing it by 1000:
Altitude = 11,000 m
Next, we calculate the pressure difference by subtracting the cabin altitude from the cruising altitude:
Pressure difference = 11,000 m - 2,000 m = 9,000 m
Now, we need to find the density of air at the cruising altitude. The density of air decreases with increasing altitude due to lower pressure and temperature. However, for simplicity, let's assume a constant air density for the calculation.
The density of air at sea level is approximately 1.225 kg/m^3.
Therefore, we can use this value for the density (ρ) in our calculation.
Finally, we can calculate the force (F) on the window using the formula:
F = P * A,
where A is the area of the window.
Given that the area of the window is 0.082 m^2, we have:
F = (P * A) = (ρ * g * h * A) = (1.225 kg/m^3 * 9.8 m/s^2 * 9,000 m * 0.082 m^2).
Now we can calculate the force on the window:
F = (1.225 kg/m^3 * 9.8 m/s^2 * 9,000 m * 0.082 m^2) ≈ 8,484.15 Newtons.
So, if the Boeing 747 airplane were stationary at its cruising altitude, the force on the window would be approximately 8,484.15 Newtons.