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I am hoping to dig a little deeper into what Taylor says on page 9 of Classical Mechanics. I've provided an excerpt just below:

A more important difference arises when two frames are in relative motion; that is, when one origin is moving relative to the other. In Section 1.4 we shall find that not all such frames are physically equivalent. In certain special frames, called inertial frames, the basic laws hold true in their standard, simple form. (It is because one of these basic laws is Newton's first law, the law of inertia, that these frames are called inertial.) If a second frame is accelerating or rotating relative to an inertial frame, then this second frame is noninertial, and the basic laws — in particular, Newton's laws — do not hold in their standard form in this second frame.>

The emphasis on origin is mine.

My question is:

The sentence including origin seems to imply that, when classifying a reference frame, our procedure begins with determining whether the origin of a given reference frame is moving relative to an inertial frame. If it is not, then the frame is inertial. Otherwise (if the origin of this other frame is moving with respect to the original), we must examine whether the two frames are physically equivalent (presumably evaluating the derivatives of the basis vectors of the other frame in terms of the inertial frame). Is this implication true? It seems to fly in the face of what he says two sentences later - "If a second frame is accelerating or rotating relative to an inertial frame, then this second frame is noninertial." Essentially, is a frame that has a coincidental origin with an inertial frame but which is rotating with respect to that frame inertial (does the introduction of the Coriolis acceleration count as a "fudge factor")?

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  • $\begingroup$ He was trying to say that for non-inertial reference frames you need to include effect of pseudo-forces (such as Coriolis or centrifugal or "inertial force") into Newton laws for them to work. What part you did not get about that ? $\endgroup$ Apr 23, 2020 at 16:19

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A more important difference arises when two frames are in relative motion; that is, when one origin is moving relative to the other.

Taylor is just defining relative motion here. We haven't gotten to inertial frames yet at this sentence. This does not contradict anything, although I do agree the set up could have been better. So for this question

Essentially, is a frame that has a coincidental origin with an inertial frame but which is rotating with respect to that frame inertial?

the answer is no. It is not inertial. But if you want to take the first sentence as what it is, then this means in this case the frames just aren't in relative motion.

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  • $\begingroup$ Thanks very much, this clears things up nicely for me. One quick last question before I give it the checkmark - does the first part of your answer (and Taylor's first sentence as quoted) mean that relative motion between two frames is when their origins move with respect to each other (and that a rotating frame with a coincident origin to a non-rotating frame is not in relative motion to that non-rotating frame)? $\endgroup$
    – EE18
    Apr 23, 2020 at 17:01
  • $\begingroup$ I'm also not sure I fully grasp what you mean by "But if you want to take the first sentence as what it is, then this means in this case the frames just aren't in relative motion." They are in relative motion aren't they? $\endgroup$
    – EE18
    Apr 23, 2020 at 17:05
  • $\begingroup$ @1729_SR According to Taylor, it looks like he does not consider the frames in your scenario to be in relative motion. It seems like he only considers relative motion to be when the origins are moving relative to each other. That is what my answer is saying. $\endgroup$ Apr 23, 2020 at 17:23
  • $\begingroup$ Hmm, that seems like a curious definition. I will have to read further. Thanks again! $\endgroup$
    – EE18
    Apr 23, 2020 at 23:35
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A non-inertial reference frame is one that is accelerating with respect to an inertial (non-accelerating) reference frame; a rotating frame is accelerating and hence is a non-inertial frame. "Ficticious" or pseudo forces appear in a non-inertial frame.

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