Friday, September 10, 2010

Potential energy of the mass of each twin tower in the gravitational field (GPE)

When mass is lifted against the force of gravity, it acquires potential energy in the gravitational field (GPE) according to the formula

E = m * g * h

Where

  • m is the mass, measured in g or kg.
  • g is the (nearly) constant acceleration of earth's gravity - in New York, that value is about 9.805 m/s2
  • h is the difference in hight, in meters, measured between the "ground" level from which the mass was lifted to where it was lifted. In the case of an object like a building, one would have to consider the height of its center of mass

What was the total mass m of a twin tower?

If you google this, you may find various quoted figures. One often given is 500,000 metric tons. This appears too round a number to be accurate.

One back-of-the-envelope estimate occurred to me the other day: Compare a Twin Tower to ship, and estimate its displacement. Looking at steel ships. I'd think that they displace about one sixth of their total volume. Surely, the WTC would float on water like a ship, if you assumed the windows would not break. The volume of the tower is 415m x 63m x 63m = 1647135m3. Displacement might be 1647135m3/6 = 274522.5m3, or 274,522.5 tons.

A very thorough estimate was given by Gregory H. Urich, B.S. Electrical and Computer Engineering, in his whitepaper "Analysis of the Mass and Potential Energy of World Trade Center Tower 1":

His result: WTC1 had a mass of 288,100 metric tons (in the Abstract, page 1). That's only about 5% off of my back-of-the-envelope, confirming my method wasn't so bad.

Urich also gives us an estimate of GPE in 5.1 Summary of Results (p. 23): 480,600 MJ, or 4.806 * 1011J

With E = m * g * h <=> h = E / (m*g), this corresponds to an estimated hight of the Center of mass of
h = 4.806 * 1011J / (288,100,000kg * 9.805m/s2) = 170m
(rounded to full meters), or 41% of the height of WTC1. This seems to be reasonable, considering that...

  • Most of the 110 floors are evenly spaced and nearly identical to each other, making the mass of them evenly spread
  • The perimeter columns likewise are of the same dimensions from about 3rd floor to roof, making their mass evenly distributed along the total height
  • The massive core columns, on the other hand, are much more massive in the lower floors and become lighter and lighter towards the top. Their center of mass, in isolation, would likely to be much lower than half the hight of the tower; maybe 25% or 30% of its height, or 104-125m

I note in passing that Urich, on page 4 (3.1 Popular numbers) quotes FEMA (Hamburger ed.al.) with an estimated GPE of "> 4 E+11 J" and NIST (Sunder ed.al.) with an estimated total mass of "> 250,000 tons", confirming that both operated with conservative, realistic numbers.

When that mass collapsed, a good portion of it formed a debris heap above ground level. Thus, not all the potential energy was realeased in the fall. It is difficult to estimate the height of the center of mass of the rubble after the fall. It certainly wasn't very much: Much of the rubble actually fell below ground level (into the basement. and very much more assembled  below 1m than beteween, say, 10 and 11 meters above ground. So I have reason to assume that the new center of mass was at most 5m above ground level, or under 3% of the original height of 170m. I will therefore go with Urich's original number, bearing in mind that a 3% deviation is easily possible, given the uncertainties we have about mass and mass distribution.

Conclusion:
I agree with G. Urich that each of the Twin Towers had a

  • mass of about 2.88* 108 kg
  • GPE of about 4.8 * 1011J

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