## TuringTest 3 years ago dimensional analysis question...

1. TuringTest

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?

2. Mikael

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.

3. Mikael

4. TuringTest

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 :)

5. Mikael

I have sent you the question as a privt. message - will be very much obliged for any guidance. Thank you in advance. Mikael