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The image below shows that a bystander watching the merry-go-round is in an inertial frame of reference. However, to nitpick, wouldn't the observer still be accelerating because it's on Earth? enter image description here

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    $\begingroup$ The earth itself is an approxiamtely inertial frame. This is because the acceleration caused by earth's rotation, is negligible when dealing with newtonian mechanics and forces of regular order. $\endgroup$
    – Mehul
    May 18 at 2:28

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In newtonian mechanics, inertial frames are an equivalence class. They can be defined as frames where Newton's laws are valid.

If you can find one inertial frame, then you automatically get an infinite number of other such frames by trying all galilean transformations from the first one.

There is, however, no true inertial frame in practice. The best we can do is to ask, for a given frame of reference, whether or not treating it as an inertial frame is a good approximation.

For instance, if you're studying the oscillations of a pendulum over the span of a minute or two, the terrestrial frame is a good inertial frame, as is taught (more and more silently) in high school.

But if you're studying the oscillations of a large pendulum for hours or days, then treating that same frame as an inertial frame will give bad results than don't match reality (see Foucault pendulum), because you can't neglect Earth's rotation over such an extended period of time.

In practice, the best approximation for an inertial frame that we have is the Copernicus one. But for most usecases, it's cumbersome to use, so we have a hierarchy of frames that are less and less inertial, but that can be easier to use:

  • Copernicus frame
  • Heliocentric frame
  • Geocentric frame
  • Terrestrial frame
  • local frame
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    $\begingroup$ An important quibble: Earth's surface approximates a frame with a fixed linear acceleration g, which approximates an inertial frame for acceleration regimes in which it's fair to approximate $10m/s^2 \approx 0$. Pendula in inertial frames just float around doing nothing, after all. $\endgroup$
    – g s
    May 18 at 4:36
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    $\begingroup$ In newtonian mechanics (the framework of my answer and, I believe, of the question), a pendulum in an inertial frame has no problem oscillating under the effect of its weight (as an external force). It's only in general relativity that gravity is related to the choice of reference frame (I'm no GR expert, so feel free to correct me if necessary). $\endgroup$
    – Miyase
    May 18 at 8:57
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    $\begingroup$ What's the Copernicus frame? I did a Google search for that and the only definition I found was "the frame where the Sun is at the origin," which sounds like the same thing as the heliocentric frame. $\endgroup$ May 18 at 13:09
  • $\begingroup$ Yes, there's a little subtlety. Both the Copernicus frame and the heliocentric frame share the same set of axis, but their origins slightly differ. It's the center of mass of the Solar System for the Copernicus frame, and the center of mass of the Sun for the heliocentric frame. $\endgroup$
    – Miyase
    May 18 at 13:40

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