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Re: non-inetial: part IV



On Tue, 30 Apr 1996, Rauber, Joel Phys wrote:

...
I claim that one can apply the 2nd law in a non-inertial frame
as long as one includes terms that represent what I call
"kinematical" forces. Others call them "fictitious" , others
"pseudo" others "apparent". In order to be neutral between
Marlow and me, I'll call them "apparent" for the time being.
...

...
Note: the rhs of (2) is the total work done by all the forces and
includes work terms from the apparent forces. In fact, if there
are no "real"forces presents then this is work done solely
by apparent forces.
...

And you must CLEARLY label it APPARENT work (and don't let the
energy company bill you for it). If we're going to be making these
distinctions, then don't let yourself slip into fudging work and
apparent work together. I thought that much was at least agreed
upon.

...
in others words

Delta (kinetic energy)' = work done by real and apparent forces,
...

And so you have a mixture of real and apparent kinetic energy, a useless
concept if I have ever heard one!

What you really want to do is compute REAL kinetic energy changes produced
by REAL forces, even though you may choose to compute these REAL dynamical
quantities in terms of the positions, velocities and accelerations measured
relative to a noninertial reference frame. This latter is a perfectly
acceptable procedure, and always starts by applying Newton's laws in an
inertial reference frame, precisely to avoid the introduction of spurious
dynamical quantities, and only then transforms the kinematical quantities
of position, velocity and acceleration to the corresponding quantities in
any other reference frame you choose. In this procedure there is no room
for any pseudowork, pseudoenergy or pseudoforce to appear, so you don't
have to worry about correcting for them or keeping track of them.

But, I agree, if you start from the assumption that Newton's laws are
equally valid relative to noninertial frames, with the introduction of
pseudoforces, than you definitely have to introduce pseudowork,
pseudoenergy, Etc., and keep careful track of it.

...
Note: in order for this to be correct and consistent we must
include in the work terms of the apparent forces. i.e. those
terms produce measurable effects (effects measured in the
prime frame only!!). Namely, they contribute to the quantity
that is the lhs of (4).

Conclusion: applying the 2nd law as the equations of motion
in the non-inertial frame correctly produces the trajectory
of the object as measured in the non-inertial frame as long
as apparent forces are included in the sum of forces term.
...

I've already conceded that your method can compute purely kinetic
quantities ( == position, velocity, acceleration); it's dynamic
quantities (== work, energy, force, momentum) where you must introduce
pseudo quantities.

... I'll give a specific example, worked out in detail, of the work
produces the correct change in "kinetic" energy in a future post.
...

I presume you will be calling any term of the form 1/2 m*v**2 "kinetic
energy" whether or not it corresponds to any actual energy or not.

A correct calculation would start with K.E. == 1/2 m*v**2, where v is
required to be relative to some inertial frame, so that we start off
knowing we are dealing with only actual kinetic energy, and then we
are free to use any transformation we want, say v = v'-a*t, where v'
and acceleration a might be measured relative to some noninertial frame, to
get K. E. = 1/2 m*(v'-a*t)**2. Note that no pseudo concepts are
introduced anywhere, and yet you end up computing REAL kinetic
energy in terms of REAL kinematical quantities relative to a noninertial
frame of your choice. You also find (on carrying out the square above)
that the real K.E. contains a term m*v'*a*t, so the simple Newtonian
formula for K.E. no longer holds relative to the noninertial frame --
i.e., Newtonian formulas are simply not guaranteed to be valid except
relative to inertial frames!

Joel Rauber
rauberj@mg.sdstate.edu




A. R. Marlow E-MAIL: marlow@beta.loyno.edu
Department of Physics PHONE: (504) 865 3647 (Office)
Loyola University 865 2245 (Home)
New Orleans, LA 70118 FAX: (504) 865 2453