Dimensional Analysis: Newtonian Physics Lecture Series Lecture 1: Chapter 1 Section 3

This is a natural continuation of Chapter 1 Section 2.  When we make models, we need to make sure that, when we run the numbers, the dimensions of the solution naturally make sense.  The word "dimension" in physics means something more general than the definition we are used to here in every day life.  We typically think of dimensions as length, width, and height; we do live in a three dimensional world, after all.  The word dimension in science means the units that we are working with.  This could be Distance (like length of a box), or is could be time (duration of a trip), or it could be mass (I weigh 15 stone).  The dimensions of density are kilograms per cubic meter (kg/m3).

When these dimensions are used in mathematical equations, they are treated as if they are variables we've seen in Algebra; they can be canceled and squared like any other value in mathematics.  For example, if we have a room that is 20ft.×30ft.×15ft, then we multiply the numbers together, but we also multiply the three units together to get a volume of 9,000ft3 (nine thousand cubic feet). Division works the same way.  When we are looking for the square footage of an apartment, and only know it's volume of 4,500ft3 (for some reason) and it's height of 12 ft. (assuming that it's a bottom story, so flat cieling), we can divide the two numbers and divide ft3 by ft, and vuala! we have 375ft2.

When we're adding and subtracting, the dimensions do not increase or decrease in their exponent for the same reason variables don't increase or decrease in their exponents when we're combining like terms; we are not going from one dimension to two dimensions, we are merely staying within the same dimension (adding one lines length to another line) or going from one dimension to a completely different dimension (even if they have the same unit, like when we're applying the Pythagorean Theorem).

Next time I will cover conversion of units.  Until then, stay curious.

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