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In lecture no. 1 (Units, Dimensions, and Scaling Arguments) Professor Lewin does a dimensional analysis in which he insists that, if there is only time (t) on one side of the equation, you can’t have length (L), mass (M), or acceleration (g) on the right side. He states this as an axiom. Yet, he eventually derives the equation t = C (h/g)^-2. In that equation, time (t) is on one side, and length (h) and acceleration (g) are on the other. There is no height/distance on the left side of the equation, so how can there be height and acceleration on the right?

MIT 8.01 Physics I Classical Mechanics, Fall 1999
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Its not that he created the components of the equation from thin air. You are allowed to have anything you please on either side of the equation as long as you find it that the units cancel each other out. He had time on one side, and \[\sqrt{h/g}\] which, if you speak in strictly dimensions, is \[\sqrt{meters/meters/seconds ^{2}}\] you clean up that equation \[\sqrt{meters \times seconds ^{2}/meters}\] which is finally equal to seconds. The two sides of the same units.

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