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Let's consider a non-inertial frame with an acceleration of $a$ relative to an inertial frame, if $a$ is really small and we don't need extreme accuracy, can we ignore this acceleration and treat this non-inertial frame as an inertial frame because the acceleration is very small?

In other words, a table is fixed to a accelerating car where $a$ is really small, I put a ball on this table, is it reasonable to assume that this ball will not move significantly and I can apply Newton's Law to this ball?

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    $\begingroup$ Writing $a\ll 1$ doesn't make sense, because $a$ has units of acceleration, so you can't compare it with a unitless number. $\endgroup$
    – Lenard Kasselmann
    Commented Jul 24 at 11:02
  • $\begingroup$ Presumably the "inertial system" you are thinking about is fixed to earth? $\endgroup$
    – Toffomat
    Commented Jul 24 at 11:20
  • $\begingroup$ Yes, the Earth's surface for example experiences a small centripetal acceleration due to the rotation which is often ignored on tabletop experiments. $\endgroup$
    – jalex
    Commented Jul 24 at 13:31

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Yes, this is often done in practice. For example, student lab exercises may study conservation of momentum treating the lab as an inertial frame despite the Coriolis force being present.

The key is not the size of the acceleration, but the size of the errors introduced. A frame can be treated as inertial for the purposes of some experiment if the errors introduced into the measurements are much smaller than the uncertainty of the measurements. So the same lab may be perfectly fine to consider as inertial for one experiment but not for another.

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    $\begingroup$ This is often a useful principle in general. Air friction, for example, is often assumed to be negligible in many experiments when other sources of measurement uncertanty would be much more significant, e.g. when using a hand stopwatch to measure how long it takes a bowling ball to fall 3 meters in open air. $\endgroup$
    – supercat
    Commented Jul 25 at 20:28

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