The case of the missing energy
Just a note to amplify a previous post which takes a look at the energy deficit problem for the twin towers.
Correction (April 26, 2007): A mass estimate has been revised to 7 x 108 kilograms per building. This is quite a trivial matter, since the numerical mass is irrelevant. It is the energy ratio that is important.
We may regard the energy associated with the buoyant force as the binding energy of the lower structure of the 417-floor WTC2. Most of this energy went into the construction such that the structure could bear the load above. We could think of this energy as internal energy.
That is, if the entire structure collapses, how much energy should be released?
We feel safe to say that the energy must be at least as much as is required to raise a block to a specified height. That is, it must be at least mgy for a specific block and height.
Though there may be some justification for a discrete summation, which I used in a previous version of this post, I have decided that a routine integral suits our purposes nicely. So here goes.
The mass supported at some height y we estimate as roughly
M - (y/H)M = M(1-y/H)
where H is the height of the building.
The potential energy specific to that height is then
gM(1-y/H)y = gM(y - y2/H)
So, for the sum of all potential energies between ground level and y we have
(using S for the integral sign)
gM S y dy - 1/H S y2 dy
-- for the interval between height 0 and n, the story at which the upper block fell --
= gM(y2/2 - y3/(3H))
Based on a stated 770,000 tons of steel per building, we estimate building mass at 7x108kg. For WTC2, we put n at 81 and H at 417 meters, with about 3.79 meters between floors.
Plugging in those numbers, we get an internal energy of at least
However, the kinetic energy from the top block's crash is given by
1/2mv2 = mgy = 0.25 x 7 x 108 x 9.8 x 3.79 = 6.5 x 109 J, which is five orders of magnitude below the opposing internal energy.
For WTC1, we use H = 420 and n = 92, with 3.82 meters between floors.
Plugging in the numbers, we obtain
1.873 x 1014 in internal energy for the lower structure, versus an impact energy of 6.5 x 109.
Now it is conceivable that a small amount of energy can bring about the release of a much larger amount of energy -- if it is well positioned. However, the binding energy of the lower structure that concerns us is all vectored so as to resist gravitational collapse. Hence, it is plausible for such a small amount of energy to topple a tower -- if it is released near the base of the structure.
The official idea that there was enough energy to hammer the lower structure down to near-ground level is simply not tenable.
Addendum: I took a look at the kinetic energies provided by the impacts of the jets and found that they were miniscule with respect to the "buoyant force energy" of each building. The NIST was somewhat ambivalent about its view of the jet impact damage.