The Slammer

With the warmer weather here, Marika and I have been opening up patio doors and windows to get some air moving. I've noticed though that when I go to close the front door when I'm going in or out, the amount of force I'm used to using makes the door slam. My hypothesis was that this was because in a (nearly) sealed apartment, closing the door reduces the volume of air, creating negative pressure. In an open apartment, new air flows in from the windows, but otherwise the pressure from outside stops the door from closing.

For simplicity, we'll assume the door shuts quickly enough that no air leaks around the edge. That lets us relate the angle the door has swung through to the volume of air that's been removed:

where r and h are the width and height of the door. For an ideal gas, pressure is inversely proportional to volume:

where V is the total volume of the apartment, and P_atm is the atmospheric pressure outside (and initially inside). As the door is swinging shut, the volume is increasing and the pressure is decreasing. We can multiply these quantities to find the energy needed to close the door:

Substituting in from above, this comes to

Let's get some numbers to find out how plausible this is: According to my lease, the apartment is 47.88 square meters, and this site suggests a ceiling height of 2.3 meters, which gives a volume of 110.1 m^3. This page gives the door dimensions as 0.9 x 2.15 meters. Plugging all that in, along with an angle of 90° gives 136.9 kilojoules. It's a little hard to picture an amount of energy, so we can convert it by considering the kinetic energy of the door:

We can find the door's rotational inertia I from the mass, and the dimensions from earlier. Putting everything together, the speed necessary to close the door with the resisting pressure would close it without pressure in only 7.366 milliseconds! This seems pretty extreme, but I imagine it's due to the assumption I made at the very start about no air leaking around the edges. It's impressive how much difference that bit of air can make out of a whole apartment.