You Can't Fight in Here, This is the War Room!

Yesterday I watched one of my favorite movies, Dr. Strangelove.  During the scene where General Jack D. Ripper pulls a machine gun out of his golf bag and starts firing out his office window, I started wondering how much the barrel would heat up, and whether he'd really be able to hold it with his bare hands.  I remember a scene in some classic war novel (possibly All Quiet on the Western Front) where a group of soldiers cool their machine gun by pouring urine on it, so it seems like a pretty serious concern.

I'm not sure whether the heat comes more from the exploding gunpowder, or from the friction of the bullet against the barrel, but I don't know anything about the chemistry involved in gunpowder, so I'll stick to the friction.  The force of friction is given by

where μ is the coefficient of friction, and N is the normal force on the object being considered.  The coefficient of friction for steel against moving steel is 0.42.  The normal force will be determined by how hard the bullet is pushed against the barrel.  For simplicity, we'll assume it's just gravity causing the contact, although there are probably more complicated forces contributing too, like the rifling of the barrel.  In this case, the normal force is

where m is the mass of the bullet.

To get the total energy deposited by the friction, we integrate over the length of the barrel:

We can turn this into a change in temperature using the specific heat capacity, c:

where M is the mass of the body being heated, the barrel in this case.  Plugging in the numbers for an M2 Browning gives an energy of 0.21 Joules, and a temperature change of 4.6 x 10^-5 Kelvin.  This is a pretty tiny amount, but remember that this is only for a single bullet.  The gun fires approximately 10 bullets every second, so its temperature goes up by 4.6 x 10^-4 Kelvin per second of firing.  I guess that's still pretty tiny; it would take 20 minutes of firing to raise the temperature by 1°F.  Sounds like either most of the heat comes from the gunpowder, or there's much more friction than I estimated.