From: crb7q@kelvin.seas.Virginia.EDU (Cameron Randale Bass)
Subject: Re: gravity question help!!
Organization: University of Virginia
Date: Thu, 17 Dec 1992 20:35:49 GMT

In article <1992Dec17.153237.833@nhgs.vak12ed.edu> rpurman@nhgs.vak12ed.edu (Richard I. Purman) writes:
>One of my students would like a response to this question.
>PLEASE RESPOND PERSONALLY TO;
>rwright@nhgs.vak12ed.edu
>
>
>Subject: gravitational Potential energy for an object far above
>the earth's surface:  U = - GmM/r
>
>IF U is defined this way then standard textbook formulas for
>conservation of mechanical energy look like: K2 + U2 = K1 + U1
>provided only gravity does work on the object (like a satellite)
>
>__Issue #1:___
>
>
>Students point out that U = -GmM/r does NOT simplify to mgy
>when object is close to earth's surface.  Yet many
>texts/teachers argue that mgy is a special case of -GmM/r

      No, because mgy is the definition of \Delta U from the earth's
      surface, as in (TeX notation)

 1)          \Delta U = U_y - U_surf = -GMm/(r_e+y) - (-GMm/r_e)
   
      where r_e is the earth's radius.  We can taylor expand 1/(r_e+y)
      to get 

 2)           1             1        y
            -----     =    ---(1 -  ---   + higher order terms in y/r_e)
           r_e + y         r_e      r_e
  
      Putting 2) into 1) we get
                                 2
            \Delta U = GMmy/(r_e)  + higher order terms in y/r_e
                     
                    2
      Since GM/(r_e)  ~= g = 9.8 m/s^2, we have
   
            \Delta U = mgy

      With all this said, for mgy to be U, you basically throw away 
      a -GMm/r_e term in the first order expression for U near the earth's
      surface.

      This is acceptable near the earth since a) classical potential is 
      a relative quantity, and differences are the important physical
      quantity, b) y/r_e is small.

>My students feel there is a contradiction in how U is defined.
>If an object is close to the earth then U > 0 if one uses mgy
>but U < 0 if one uses -GmM/r.

      No contradiction, the sign on U is of no real import in 
      classical physics.  In both cases U increases as one crawls
      out of the well,  the choice on the earth is simply a convenience
      so as to not have to carry around the large negative expression for
      the potential at the earth's surface in every calculation.

      The thing to stress is that the *difference* in potentials
      is what is important, not the sign of some 'absolute' potential.

>
>Issue # 2
>
>
> My college-level physics students feel that
>defining U = -GmM/r is very arbitrary (definitions are such, of
>course) in the sense that they feel physicists are "forcing"
>mechanical energy E = K + U to be conserved, where E is
>"defined" with a + sign between the two energy terms.  My students
>basically want to know if it is "ok" FROM A PHYSICS VIEWPOINT
>to have K2 - U2 = K1 - U1 with U defined as U = +GmM/r AND if
>this is NOT "ok" then WHY NOT?

      They feel like potential should *decrease* when going away from
      the center of the earth?  Why would you want to define potential
      that way?  That destroys the intuitive concept of swapping
      energy from KE to PE depending on position in a gravitational well.
      That way, as you went deeper into the well, you gained both 
      KE and 'potential'.

      Here again, it is differences that are important, not signed values.

      Basically, potential is defined so that it is zero at r=infinity
      and smaller (i.e. large negative numbers) as one approaches 
      r=0.  It is done this way because a) it makes sense in the intuitive
      sense that potential should increase as you get 'higher', and
      b) it is convenient to have it zero at infinity.

      It is a convention, but it is one that everyone uses.  I don't think
      it very useful to have a potential that decreases as you gain
      intuitive 'potential'.

      Have a nice day.

                              dale bass
-- 
C. R. Bass                                          crb7q@virginia.edu        
Department of Mechanical, 
     Aerospace and Nuclear Engineering
University of Virginia                              (804) 924-7926