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non-inertial: ad infinatum




...
The measurements I'm referring to are the ones that you can predict from
Newton's laws; namely the postion, velocity and acceleration of objects;
this is all Newton's laws can tell when you solve them;
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This is the nub of the disagreement -- you seem to restrict mechanics to
computing and predicting correctly KINEMATICAL quantities only (position,
velocity and acceleration). But from measurements of kinematical
quantities mechanics must predict also accurate values of the important
dynamical quantities: real work done, real energy changes, both losses
and gains (the sort of things the energy companies make us pay real money
for), real mass values, real momentum changes.

I think it is partly the nub of disagreement, and in so far as the first
statement is concerned it is philosophical. Perhaps it would be helpful if
Marlow would distinguish kinematics and dynamics in a precise way and in a
way that doesn't amount to self defining his position as the only valid
position possible. I take a very utilitarian philosophy to Newton's laws.
They are a system of equations that one uses to make predictions.
Predictions about what? acceleration, velocity, position, momentum, kinetic
energy, mass, force etc etc. And yes, I do make a distinction between the
first three (kinematical quantities) and the others dynamical quantities;
but I calculate them by using the 2nd and 3rd law. That is how people use
them in practice!!

If you self-define "real work done", "real energy changes" as being only
those quantities calculated from an inertial frame of reference, then you
have simply defined my position to be wrong and have not shown it to be in
error or inconsistant.

For example, I've claimed that the work-energy theorem applies in the
non-inertial frame of reference.

paragraph 1):
What I mean is the following:
compute the quantity change in 1/2 mass v'^2 and one finds that it equals
the work done by sum of interaction forces and kinematical forces. And if
there are no interaction forces present then it is a work done solely by the
kinematic forces.

Proof: Take as a starting point any of the following:

Goldstein equation 4-129 or
Marion (equation at middle of page 342) (3rd ed.)
Baierlein equation 6.1-17 after multiplying by mass
Landau & Lifschitz equation 39.7

and then go through the same manipulations to arrive at the work-enrgy
theorem that one does to take Newton's 2nd law in an inertial frame to
derive the work-energy theorem.

The math is the same and therefore the conclusion that I expressed above in
words in paragraph 1 follows: Where is the error in this?

Interestingly enough Landau even goes so far as to compute a potential
energy for the centrifugal force (I shudder, a dynamical concept for a
fictitious force). Incidently, one can also compute a valid lagrangian for
it, which doesn't prove anything since lagrangian mechanics is derivable
from Newtonian mechanics and vice versa; but I think of lagrangian ideas as
being very dynamical.

All Marlow has said, "Doesn't apply". But the calculations will fully agree
with what Marlow does if he first calculates the work done by interaction
forces present in an inertial frame and then transforms the change of
kinetic energy to the non-inertial frame. Note: I'm not saying the change
in kinetic energy is the same in the two frames, its not because it is not
an invariant concept.

dynamical quantities: real work done, real energy changes, both losses
and gains (the sort of things the energy companies make us pay real money
for), real mass values, real momentum changes.

As far as I'm aware the power companies do all of their measurements in a
non-inertial frame of reference, and charge me real money for it.
Furthermore oceanographers correctly predict the motion of ocean currents
doing what I say, civil engineers correctly (usually) build structures that
don't fall down by applying Newton's laws in a non-inertial frame; if they
don't include the effects of the kinematical forces, the structures don't
work. Why? they are doing all of their calculations in a non-inertial frame
of reference.

It is quite permissable to use dynamic ideas in a non-inertial frame of
reference, and except for stupid math errors, I always get the same answers
anybodyelse does.

By introducing fictitious
dynamical quantities, such as a centrifugal force inside a turning car,
you completely obfuscate the issue of computing real dynamical quantities,

But you don't obfuscate the issue of computing those quantities as measured
in the non-inertial frame

such as real energy expended, and then if and when you decide you want to
compute such quantities, you are going to have to go back and remove the
fictions (i. e., get back to an inertial frame) to see what real dynamical
quantities you actually have.


Of course! if you wish to compare calculations as done in one frame of
reference to those done in another you will have perform the
transformations, when I say we get the same answers I obviously mean we have
to transform our answers to the same; or else we can't compare them to see
if we get the same answer. I could equally well say you must first remove
erroneous idea that kinematic forces aren't present in non-inertial if you
wish to convert over from the inertial frame to compute quantities such as
energy , momentum, kinetic energy has measured in the non-inertial frame.

Instead, simply get rid of the rather strange idea that every acceleration
relative to any arbitrarily accelerated reference frame must have a force
associated with it.

Its not a strange notion. In fact one uses it everytime they use a spring
balance to measure a force. You balance the spring force against the force
you are trying to measure. The idea being you set the acceleration equal to
zero and use the notion that that implies zero net force: and voila, the
spring balance reading then equals the magnitude of the force you trying to
measure and the direction of the force you were trying to measure is in the
opposite direction of the spring force on the object.

That is you used this strange notion of associating acceleration with a
force!

This is not what Newton's second law says -- Newton's
second law only says accelerations correspond to forces in inertial
reference frames -- and then when you come to need to compute accurate
values of dynamical quantities, you will not have to sort out fiction from
fact.


Again, Newton's laws expressed in a non-inertial frame, with kinematic and
interaction force terms present are formally the same as Newton's laws in
inertial frames; therefore any idea derived from them in inertial frames
have their equivalent in the non-inertial frame; even the dynamical ones.

Al Clark wrote:

I think the point is that some people find it easier to handle problems
of motion in a rotating reference frame with the use of the apparent
centrifugal and coriolis "forces", and there is no reason to deny them
the freedom to do that as long as they know what they are doing, and that
the apparent kinetic energy may be just that, apparent only.

I basically agree with this statement. However, I would ask , are we being
told that when I compute 1/2 m v^2 for a baseball thrown to the batter that
we start calling that "apparent kinetic energy" rather than "kinetic
energy". I vote no. because it has all the properties of kinetic energy,
namely the ability to do work. It will drive a nail as far into a plank of
wood as the same quantity(experiment) would if calculated (performed) in an
inertial frame.


I've asked Marlow to send me a copy of the relevent posts here, so I can
address the
question.

No computations of the correct dynamical quantity were ever produced -- so
there's nothing to send. The claim, however, was that you could compute
the mass of the Sun using a frame in which both Earth and Sun are at rest
using fictitious forces to balance the gravitational attraction. I'd
like to see how that is done, but no one has done it yet.

I'll give it a shot, but I would like some clarifications. All I have to do
is compute the mass of the sun by working in a non-inertial frame of
reference? I'll interpret this to mean solve for "mass of sun". Now explain
what you meant in the other post about the correction "m+M". I wasn't privy
to original statements.

Also, is it sufficient for purposes of this problem to assume the center of
mass of the system is for all practical purposes at the center of sun and
that the earth orbits about that center of mass in a circle? More
precisely, I want to know the ground rules of what it is I'm supposed to
calculate.

Joel Rauber
rauberj@mg.sdstate.edu