# Force on an airplane window, given altitude and area of the window.

1. Jan 27, 2013

### monnapomona

1. The problem statement, all variables and given/known data
The modern Boeing 747 passenger jet airplane cruises at a speed up to 920 km/hr, at an altitude of 11,500 m. The cabin pressure is maintained equivalent to the atmospheric pressure at an altitude of 2,000 m. This is called the "cabin altitude".

I couldn't find information on the area of the passenger windows in the 747, so let's assume they're about 0.085 m2.
If the airplane were stationary at its cruising altitude, what would be the force on the window?

Air density = 1.29 kg/m^3

2. Relevant equations
ΔF = ΔP*A

3. The attempt at a solution
Needed to find the pressure difference so I tried using this formula to find pressures at the given altitudes:
P2 = P1 + ρgh
But it gave me an unrealistic value at height 11000 m (since pressure decreases with increasing height). I'm not really sure which formula to use to find the pressure at the given altitudes. :s

2. Jan 27, 2013

### rollingstein

Density of air won't be constant. Air thins as you go up.

3. Jan 27, 2013

### fgb

For clarity, I would like to point out that, in the formula P2 = P1 + ρgh, h in this equation it is not really the height of the point whose pressure you want to know - e.g., at sea level it is not zero, h here is the height of the column of air above that point, which is smaller for higher altitudes (since there is less air above the point). I am under the impression that you got that mixed up, sorry if I am wrong.

Besides that, the formula you mention, P2 = P1 + ρgh, is only valid when the density of the fluid is constant. As stated above, density of air is not a constant (i.e. it is not the same for the entire atmosphere, it changes with height. Air thins as you go up, as stated above), it is also a function of height

Neglecting temperature change in the atmosphere and assuming air as an ideal gas, the formula for the pressure is

$P = P_o\exp{(-\dfrac{Mgh}{RT})}$,

where Po is atmospheric pressure at sea level (101325 Pa), M is the molar mass of dry air (0.0289 kg/mol), R the universal gas constant (8.13 J/molK), T sea level standard temperature (around 290K) and h the height. (in this formula, h is really the height above the ground)

If you are interested in the derivation of that formula, just google up "isothermal atmosphere", it is pretty straightforward :)

Last edited: Jan 27, 2013