Lectures on Physics has been derived from Benjamin Crowell's Light and Matter series of free introductory textbooks on physics. See the editorial for more information.... 
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Scaling and OrderofMagnitude
Why can't an insect be the size of a dog? Some skinny stretchedout cells in your spinal cord are a meter tall  why does nature display no single cells that are not just a meter tall, but a meter wide, and a meter thick as well? Believe it or not, these are questions that can be answered fairly easily without knowing much more about physics than you already do. The only mathematical technique you really need is the humble conversion, applied to area and volume.
Area and volumeArea can be defined by saying that we can copy the shape of interest onto graph paper with 1 cm × 1 cm squares and count the number of squares inside. Fractions of squares can be estimated by eye. We then say the area equals the number of squares, in units of square cm. Although this might seem less "pure" than computing areas using formulae like A = πr^{2} for a circle or A = wh/2 for a triangle, those formulae are not useful as definitions of area because they cannot be applied to irregularly shaped areas. Units of square cm are more commonly written as cm^{2} in science. Of course, the unit of measurement symbolized by "cm" is not an algebra symbol standing for a number that can be literally multiplied by itself. But it is advantageous to write the units of area that way and treat the units as if they were algebra symbols. For instance, if you have a rectangle with an area of 6m^{2} and a width of 2 m, then calculating its length as (6 m^{2})/(2 m) = 3 m gives a result that makes sense both numerically and in terms of units. This algebrastyle treatment of the units also ensures that our methods of converting units work out correctly. For instance, if we accept the fraction
as a valid way of writing the number one, then one times one equals one, so we should also say that one can be represented by
which is the same as
That means the conversion factor from square meters to square centimeters is a factor of 10^{4} , i.e., a square meter has 10^{4} square centimeters in it. All of the above can be easily applied to volume as well, using onecubiccentimeter blocks instead of squares on graph paper. To many people, it seems hard to believe that a square meter equals 10000 square centimeters, or that a cubic meter equals a million cubic centimeters  they think it would make more sense if there were 100 cm^{2} in 1 m^{2}, and 100 cm^{3} in 1 m^{3}, but that would be incorrect. The examples shown in figure b aim to make the correct answer more believable, using the traditional U.S. units of feet and yards. (One foot is 12 inches, and one yard is three feet.)
→ Solved problem: converting mm2 to cm2 page 66, problem 10 → Solved problem: scaling a liter page 67, problem 19 Discussion Question


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