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In Renato Brito's book Fundamentals of Mechanics, a property of the non-inertial frame is defined as follows:

Non-inertial referential is any one that presents acceleration in relation to an inertial referential. For this reason, non-inertial frames are also known as accelerated frames.

When considering a statement in the book, it is really necessary to compare two frames of reference to know the inertia or non-inertia of each of them, or it is possible to determine whether a frame of reference is inertial or not just by the apparent acceleration or by the perception that there is a source of force acting about this frame?

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    $\begingroup$ That definition seems circular to me. How do you determine if the second frame is inertial or non-inertial in the first place, compare it to a third frame? etc... $\endgroup$
    – Rodney
    Commented Dec 28, 2021 at 12:34

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Indeed you are correct, it is not necessary to refer to a second frame in order to determine if the first is inertial. You can simply use accelerometers. If the acceleration relative to the reference frame is not equal to the acceleration measured by the accelerometer (for all accelerometers) then the frame is non-inertial.

For example, say we are using a spinning space station as our reference frame. An accelerometer at rest on the space station is not accelerating relative to the reference frame, but the accelerometer measures centripetal acceleration. Therefore it is a non inertial frame. No comparisons to other frames are needed

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It is possible to guess that you are in a non-inertial frame of reference from inside of one. However it is very hard to prove that you are.

So the reason you might guess is that you might see a universal force in your reference frame which scales perfectly proportional to the mass of the objects it acts upon. This is a hallmark of fictitious forces. But, I don't think you can go further in nonrelativistic classical mechanics: there should not be a mathematical difference between an inertial frame with a real force law that scales like mass, and a non-inertial frame with the same force law but regarded as a fictitious force. (The only exception I can imagine is maybe some classes of force law are too spatially weird to be represented by a coordinate transform which could make them disappear?)

Let's give a brief mention of relativity though. This mass-scaling property is also, interestingly enough, present for the gravitational force. This led to a dramatic break with classical tradition of considering a typical laboratory on the Earth as being inertial up to, say, the Coriolis effect. Ignoring Foucault's pendulum, we considered ourselves in an inertial reference frame! But today we know that's not true, the reference frame is inertial only if it is in some form of freefall. This makes a difference in relativistic thought where it didn't make any difference in classical thought.

Why does it make a difference? Special relativity only adds one basic physical mechanism to our repertoire, a slight modification of the Doppler shift. So you know the Doppler shift, when a car is coming towards you it sounds like it has a higher pitch, when it's going away from you that pitch becomes lower. You could equivalently say that sonically, things in the car seem to play out in fast motion as the car approaches, then in slow motion as it recedes. Special relativity adds an anomalous Doppler shift to any accelerating observer: they see clocks ahead of them ticking anomalously faster and clocks behind them anomalously slower, proportional to both their acceleration and the distance to the clock.

With gravity, this is known as gravitational time dilation, we are not freefalling so we are actually accelerating upwards, so we see clocks in the upper atmosphere tick faster than our clocks down here. Equivalently they see our clocks tick slower, they see us in slow motion. So relativity gives a way to detect in some absolute sense whether you are accelerating or not... If we had not observed gravitational time dilation between atomic clocks at the bottom and top of a tower, we would not believe that we were accelerating upwards—but we have, so we do.

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  • $\begingroup$ Two things about your answer: How is a falling object an inertial frame if it is subject to a non-zero net force? And I don't understand the claim that we are gravitationally accelerating upward if the acceleration of gravity points towards the center of the planet. $\endgroup$ Commented Dec 28, 2021 at 0:03
  • $\begingroup$ The falling object is not subject to a force. Someone in the ISS is in a state of freefall: they do not experience the gravitational force. The point is that the gravitational force in general relativity becomes a fictitious force... Where you need complicated math is that, to tell a lie, the planet is sort of sucking up the fabric of space into itself, hence if you are staying at rest you “fall in” with space itself. (Actually the equation of freefall, the geodesic equation, does not look anything like these sorts of drag forces... so that is an oversimplified explanation.) $\endgroup$
    – CR Drost
    Commented Dec 28, 2021 at 9:49
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Well, answer B) is correct in every situation.

All the details, Q1, Q2, Q3, Q4 are totally irrelevant.

Seeing a power source does not mean it is actually doing anything. Imagine that there is a visible nozzle, obviously ejecting hot gas. How is anyone to know that there is not, somewhere hidden, another nozzle that exactly compensates the first one, so the frame really is inertial ? No feeling of acceleration is characteristic of an inertial frame. Well, if the acceleration is very small, one might not actually "feel" it just by one's body, we do not have a very subtle sense of acceleration. Very often, when my train leaves the platform, but I am not sitting facing the platform, but rather facing another train, I sometimes believe the other train is leaving when in fact it is mine. This is because the acceleration is very smooth and I am just not sensitive enough. But finer experiments, like the way an object which is not fixed behaves, is enough to determine if the frame is inertial or not. A pen might start rolling slowly on the table in front of me, proving my frame is not inertial.

The presence of a second, inertial, frame is absolutely not needed.

EDIT

Looks like you changed your question while I was typing my answer.

So my answer does not make a lot of sense. Well, at least you remember what Q1,Q2, Q3 and Q4 meant before yo uedited them out, as well as what answer B) was....

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