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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms . 6.055 / Art of approximation 52 7.3 Dimensionless groups Dimensionless ratios are useful. For example, in the oil example, the ratio of the two quan tities has dimensions; in that case, the dimensions of the ratio are time (or one over time). If the authors of the article had used a dimensionless ratio, they might have made a valid comparison. This section explains why dimensionless ratios are the only quantities that you need to think about; in other words, that there is no need to think about quantities with dimensions. To see why, take a concrete example: computing the energy E to produce lift as a function of distance traveled s , plane speed v , air density , wingspan L , plane mass m , and strength of gravity g . Any true statement about these variables looks like mess + mess = mess , where the various messes mean a horrible combination of E , s , v , , L , and m . by the first one (the triangle). Then mess mess + mess mess = mess mess , The first ratio is 1, which has no dimensions. Without knowing the individual messes, we dont know the second ratio; but it has no dimensions because it is being added to the first As horrible as that true statement is, it permits the following rewriting: Divide each term ratio. Similarly, the third ratio, which is on the right side, also has no dimensions. So the rewritten expression is dimensionless. Nothing in the rewriting depended on the particular form of the true statement, except that each term has the same dimensions....
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This note was uploaded on 02/24/2012 for the course MECHANICAL 6.055J taught by Professor Sanjoymahajan during the Spring '08 term at MIT.
 Spring '08
 SanjoyMahajan

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