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Re: [Phys-l] accelerated frames

The two effects are combined quite naturally:
Begin by defining R = Ro + r and then take 2 time derivatives.
Since you obviously understand what happens to each term, the rest is also apparent.

I don't follow your qualms about combining the two effects. We are simply referring kinematic quantities to a moving origin and a changing coordinate system. Can you base your objections on some flaw in the definition and development of R = Ro + r ?

As far as usefulness is concerned - first and foremost it is to me very helpful conceptually to distinguish the two different effects.

An example of a useful application:
Inter-continental ballistics calculated from an earth-bound lab frame, with its origin and coordinate system fixed on the earth's surface will have to include both effects (the origin is accelerating - it is travelling in a circle). Of course, one can take the earth's center as the origin and eliminate the effect of an accelerating origin - one would do this precisely because he has recognized the two different effects.

Solar system excursions (and returns) will also have to include the orbital motion of the earth about the sun. Both conceptually and calculationally, it will be helpful to analyze the problem into the effects of an accelerating origin plus the effects of a changing coordinate system. Again one could locate the origin at the sun's center and eliminate the accelerating origin.


- Bob Sciamanda
Physics, Edinboro Univ of PA (Em)
http://www.winbeam.com/~trebor/
trebor@winbeam.com
----- Original Message ----- From: "John Denker" <jsd@av8n.com>
To: "Forum for Physics Educators" <phys-l@carnot.physics.buffalo.edu>
Sent: Friday, October 27, 2006 12:30 PM
Subject: Re: [Phys-l] accelerated frames


On 10/27/2006 10:19 AM, Bob Sciamanda wrote:

It may be useful to recall that kinematics from a wide class of
"accelerated frames" is usefully analyzed into two different effects:

1) effects that arise from referring kinematical quantities to an
ACCELERATING ORIGIN, and
2) effects that arise from referring kinematical quantities to a ROTATING
COORDINATE SYSTEM anchored to the above accelerating origin.

It's not obvious to me why it is "useful" to emphasize this
distinction, especially since a straight-line acceleration
can be considered a rotation about an infinitely-distant
center (infinitesimal rotation rate, infinite lever arm)
and as a corollary, acceleration+rotation can be considered
just another rotation (with some new rate and new center).

I can see how it is sometimes useful to use different language
in the different limiting cases, but I don't see how it is
useful to combine the two languages.

R'' = r'' + Ao + 2w x r' + w' x r + w x (w x r) [1]

The notation should be apparent - consult Fowles' text for details (or ask).

I'll ask.

I understand that if we define
R := r + C + V t + Ao t^2 [2]

then we get
R'' = r'' + Ao [3]
which is
-- independent of C (translational invariance) and
-- independent ov V (Galilean relativity).

I also understand that if we redefine
R := rotation(w) r [4]

then we get
R'' = r'' + 2w x r' + w x (w x r) + w' x r [5]
Coriolis centrifugal

which does not exhibit translational invariance, because
[4] and hence [5] assume that the rotation is a rotation
_about the origin_.

I'm puzzled about how to combine [3] and [5] to obtain [1].
In particular, were we supposed to accelerate and then rotate,
or rotate and then accelerate? They don't generally commute.

Why/when is this useful?

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