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I will use the explanation in this site as an example of what looks to me as a wrong explanation of why acceleration in uniform circular motion is centripetal.

This site, as other sources, reasons as follows:

  • it vectorially obtains the difference v final minus v original;
  • notes that the resultant arrow more or less points towards the centre of the circle, like here:

enter image description here

  • and concludes: you see, we have proved that the acceleration is centripetal.

I have no doubt that the instantaneous acceleration is centripetal, but here they are showing the average acceleration and I am not sure that the latter is also centripetal.

A clue: make the interval 180 degrees and the same exercise shows that the length of the change of velocity vector is doubling the v initial vector and it is pointing, logically, in the opposite direction; make it 360 degrees and, logically, the change of velocity vector is 0 and pointing nowhere…

I suspect that, even in the original example of the site, the operation, if well done, should not show that the change of velocity vector is pointing towards the center, but in a direction justifying precisely how much the particle has changed its direction from point A to B and that, if their drawing shows a center-seeking arrow, it is only because it has been (unconsciously) re-arranged to that end.

Would you agree to this or did I miss anything?

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The issue that comes up with averages is

At what location along the arc do you compare the average acceleration to the position to decide if it is centripetal or not?

Clearly the acceleration vector in the picture is not centripetal at the beginning of the arc in the picture (where it would need to point straight down) nor at the end of the arc (where it would need to be about 210 degrees).

But it is centripetal half-way along the arc. That turns out to always be the case (at least for arcs less than ninety degrees), and has the desired property in the limit of a short arc as well.

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  • $\begingroup$ Ok, so the arrow should be placed halfway along the arc and then it points to the center... Well, also, if the instantaneous acceleration is always centripetal, I suppose that the average acceleration must also be so. $\endgroup$ – Sierra May 25 '18 at 20:52
  • $\begingroup$ Yet I am still unhappy with this explanation, which as you also say fails when the arc is 90 degrees or more. I prefer to simply say: if you have uniform circular motion, it must be because you pull constantly, you start perpendicularly and you keep doing it perpendicularly, which in turn requires a certain magnitude. Thus centripetal direction is (almost) the assumption, what needs to be derived is just the module. $\endgroup$ – Sierra May 25 '18 at 20:52
  • $\begingroup$ I don't actively say that it fails, I just haven't checked it for oblique angles (it certainly fails for a 180 degree arc). But it doesn't bother me in any case, as any explanation over a finite arc is just preparatory to talking about short time period (either with or without the mathematical infrastructure of limits), so even if it really is limited to acute arcs we don't have a problem. $\endgroup$ – dmckee May 25 '18 at 20:58

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