Friday, September 10, 2010

GPE of building WTC7

The mass of of building WTC7 had potential energy in the gravitational field (GPE) given by the formula

E = m * g * h

The values of m, g, and h are not all known, but can be estimated as follows, using measurements taken from Wikipedia and NIST's revised technical briefing of November 19, 2008:
  • g: That value is known. It is about 9.805 m/s2 in New York
  • h: Is the height of the center of mass of the building above ground level, minus the hight of the center of mass of the rubble after the collapse. The latter may have been up to 5m. The former would be somewhat lower than half the hight of the roof of the building, considering these factors: a) floors would have about equal masses from ground to top b) load-bearing structure (columns, mainly) would be much heavier in the lower part than the upper part. In my blog post "Potential energy of the mass of each twin tower" I showed that Gregory H. Urich, in his paper "Analysis of the Mass and Potential Energy of World Trade Center Tower 1", assumes that the center of mass would be about at 41% percent of the height of the building. Since WTC7 had a height of 186m, the center of mass might have been near a height of 76m. Deduct from this up to 5m to account for the height of the rubble pile, the total mass descended by a height of h= 71m.
  • m: Is the total mass of WTC7. As I write this (september 9, 2010), I know no good source that gives that mass. In my post on the GPE of the twin towers, however, I found that a good estimate would be one sixth of the displacement of water. The volume of WTC7 can be computed by its height (186m), width of the south front (247ft = 75m), north front (329ft = 100m), and depth (140ft = 42.7m): V = 186m * (75+100)m/2 * 42.7m = 694942.5m3. Total mass would therefore be 694942.5/6 tons or about m = 1.16 * 10E8 kg
Therefore, I estimate the total GPE of WTC7 that was converted into other forms of energy to be about

E = 1.16*10E8kg * 9.805m/s2 * 71m = 80,753,980,000J
I will round that down to

GPE = 8 * 10E10J

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