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93 9393 932010-05-13 00:43:32 / rev b667c9e4c1f1+5Dimensions5.1 Power of multinational corporations 855.2 Dimensionless groups 875.3 Hydrogen atom 925.4 Bending of light by gravity 975.5 Buckingham Pi theorem 1045.6 Drag 1055.1 Power of multinational corporationsCritics of globalization often make the following comparison [17] to provethe excessive power of multinational corporations:In Nigeria, a relatively economically strong country, the GDP [gross domesticproduct] is $99 billion. The net worth of Exxon is $119 billion. “When multi-nationals have a net worth higher than the GDP of the country in which theyoperate, what kind of power relationship are we talking about?” asks LauraMorosini.Before continuing, explore the following question:What is the most egregious fault in the comparison between Exxon and Nigeria?The field is competitive, but one fault stands out. It becomes evident afterunpacking the meaning of GDP. A GDP of $99 billion is shorthand fora monetary flow of $99 billion per year. A year, which is the time forthe earth to travel around the sun, is an astronomical phenomenon thathas been arbitrarily chosen for measuring a social phenomenon—namely,monetary flow.94 9494 94862010-05-13 00:43:32 / rev b667c9e4c1f1+Suppose instead that economists had chosen the decade as the unit oftime for measuring GDP. Then Nigeria’s GDP (assuming the flow remainssteady from year to year) would be roughly $1 trillion per decade andbe reported as $1 trillion. Now Nigeria towers over Exxon, whose punyassets are a mere one-tenth of Nigeria’s GDP. To deduce the oppositeconclusion, suppose the week were the unit of time for measuring GDP.Nigeria’s GDP becomes $2 billion per week, reported as $2 billion. Nowpuny Nigeria stands helpless before the mighty Exxon, 50-fold larger thanNigeria.A valid economic argument cannot reach a conclusion that depends onthe astronomical phenomenon chosen to measure time. The mistake liesin comparing incomparable quantities. Net worth is an amount: It hasdimensions of money and is typically measured in units of dollars. GDP,however, is a flow or rate: It has dimensions of money per time andtypical units of dollars per year. (A dimension is general and independentof the system of measurement, whereas the unit is how that dimension ismeasured in a particular system.) Comparing net worth to GDP comparesa monetary amount to a monetary flow. Because their dimensions differ,the comparison is a category mistake [] and is therefore guaranteed togenerate nonsense.Problem 5.1 Units or dimensions?Are meters, kilograms, and seconds units or dimensions? What about energy,charge, power, and force?A similarly flawed comparison is length per time (speed) versus length:“I walk 1.5 m s−1—much smaller than the Empire State building in NewYork, which is 300 m high.” It is nonsense. To produce the opposite butstill nonsense conclusion, measure time in hours: “I walk 5400 m/hr—much larger than the Empire State building, which is 300 m high.”I often see comparisons of corporate and national power similar to ourNigeria–Exxon example. I once wrote to one author explaining that Isympathized with his conclusion but that his argument contained a fataldimensional mistake. He replied that I had made an interesting pointbut that the numerical comparison showing the country’s weakness wasstronger as he had written it, so he was leaving it unchanged!A dimensionally valid comparison would compare like with like: eitherNigeria’s GDP with Exxon’s revenues, or Exxon’s net worth with Nige-ria’s net worth. Because net worths of countries are not often tabulated,95 9595 95872010-05-13 00:43:32 / rev b667c9e4c1f1+whereas corporate revenues are widely available, try comparing Exxon’sannual revenues with Nigeria’s GDP. By 2006, Exxon had become ExxonMobil with annual revenues of roughly $350 billion—almost twice Nige-ria’s 2006 GDP of $200 billion. This valid comparison is stronger than theflawed one, so retaining the flawed comparison was not even expedient!That compared quantities must have identical dimensions is a necessarycondition for making valid comparisons, but it is not sufficient. A costlyillustration is the 1999 Mars Climate Orbiter (MCO), which crashed intothe surface of Mars rather than slipping into orbit around it. The cause,according to the Mishap Investigation Board (MIB), was a mismatch be-tween English and metric units [18, p. 6]:The MCO MIB has determined that the root cause for the loss of the MCOspacecraft was the failure to use metric units in the coding of a groundsoftware file, Small Forces, used in trajectory models. Specifically, thrusterperformance data in English units instead of metric units was used in thesoftware application code titled SM_FORCES (small forces). A file called An-gular Momentum Desaturation (AMD) contained the output data from theSM_FORCES software. The data in the AMD file was required to be in metricunits per existing software interface documentation, and the trajectory mod-elers assumed the data was provided in metric units per the requirements.Make sure to mind your dimensions and units.Problem 5.2 Finding bad comparisonsLook for everyday comparisons—for example, on the news, in the newspaper,or on the Internet—that are dimensionally faulty.5.2 Dimensionless groupsDimensionless ratios are useful. For example, in the oil example, the ratioof the two quantities has dimensions; in that case, the dimensions of theratio are time (or one over time). If the authors of the article had used adimensionless ratio, they might have made a valid comparison.This section explains why dimensionless ratios are the only quantities thatyou need to think about; in other words, that there is no need to thinkabout quantities with dimensions.To see why, take a concrete example: computing the energy E to producelift as a function of distance traveled s, plane speed v, air density ρ,96 9696 96882010-05-13 00:43:32 / rev b667c9e4c1f1+wingspan L, plane mass m, and strength of gravity g. Any meanginfulstatement about these variables looks likemess+mess=mess,where the various messes mean ’a horrible combination of E, s, v, ρ, L,and m.As horrible as that statement is, it permits the following rewriting: Divideeach term by the first one (the triangle). Thenmessmess+messmess=messmess,The first ratio is 1, which has no dimensions. Without knowing the indi-vidual messes, we don’t know the second ratio; but it has no


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MIT 6 055J - Dimention

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