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dimensional analysis question...

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In the first lecture of OCW's physics 1 Lewin derives the time for a falling object as \[t\propto\sqrt{\frac hg}\]from the assumption that time is proportional to the height of falling h, mass of the object m, and the acceleration of gravity g\[t\propto h^\alpha m^\beta g^\gamma\]He then goes on to show how, though seemingly a reasonable starting point, that assuming proportionality to the mass of the earth instead of gravitational acceleration as such:\[t\propto h^\alpha m^\beta M^\gamma\]leaves you unable to draw a meaningful conclusion. The main reason I can think that is is that my assuming time is proportional to the mass of the earth M and not acceleration is that none of the variables contain units of time, and therefore it is impossible to get a meaningful expression from it. Is there more to this that I am missing, or is that essentially the main problem with the second analysis?
Hey @TuringTest , Your point is certainly half the way to the complete reasoning. The technical half is that - as you pointed, there is no time in the 2-nd product, and since time is a FUNDAMENTAL, IRREDUCIBLE UNIT it cannot be "combined" of other units. But the missing half - which is much deeper, and in fact even not-fully understood by current physics to this very day, is WHY g is PROPORTIONAL in 1-st degree to the planet-s mass. Or , in Einstein's language WHY IS THE GRAVITATIONAL MASS DIRECTLY PROPORTIONAL TO THE INERTIAL MASS . Einstein states this AS A POSTULATE. However neither Einstein nor anybody since that time has not given a FUNDAMENTAL REASON to this equality of the two properties of mass. The property of resisting acceleration, and the property of attracting other masses.
May I send you my question which is non-scientific but about this site ?

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Sure, and thank you very much for the clear and enlightening explanation. I had an intuition that there was a deeper issue that made the assumption problematic that I was missing, and you explained it artfully :)
I have sent you the question as a privt. message - will be very much obliged for any guidance. Thank you in advance. Mikael

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