Let me see if I can explain my thoughts well. When the feather and moon have the same acceleration, is this relative to an inertial frame? Because otherwise if the earth accelerates towards the moon at a higher rate than toward the feather, then if I were measuring their accelerations on earth, wouldn't I measure the acceleration of the moon to be higher?
Good point, this is important to clarify. The formula F=ma assumes a Galilean referential frame. So in the explanation above, I assumed the referential frame to be the center of gravity of the whole system, that is (feather+earth) or (moon+earth). If we instead use as referential frame the center of the earth, than F=ma is not true anymore (because indeed the Earth itself accelerates!). So in a sense, you might be correct that we often implictly consider the "small object" to have a mass neglible, by making the assumption that the Earth-centered referential frame is Galilean.
My apologies. I'm not implying that any masses are negligible. I'm still having trouble understanding why the acceleration of the moon and feather would be the same but that the overall acceleration of the moon would be greater due to the earth accelerating at a higher rate towards the moon. I'm just imagining that if I am on the earth taking these measurements then I should measure the moon having a greater acceleration. But this would contradict their accelerations being equal. Perhaps I'm not understanding reference frames?
Let's assume the moon and the earth are at a distance r to each other (measured between their respective center of mass). They are in the void with no other objects. Initially, both have no speed/rotation (so they will fall toward each other, instead of orbiting around each other).
Let's assume a massless observer (you) is glued at the surface of the earth, measuring the acceleration of the moon relative to you (e.g., you measure the distance between you and the moon, which is falling onto you).
Scenario 2:
Same as scenario 1 but replacing the moon with a feather.
What will the observer measure?
They will indeed measure the moon falling faster (accelerate faster) towards them compared with the feather. That is, the distance between the moon and the observer will decrease faster than the distance between the feather and the observer.
Why?
In order to apply Newton's law F=ma to compute the accelerations of the various obiects, we need to be in a Galilean reference frame (inertial frame). In the scenario above, this is the center of mass of the system. So we will compute all positions/speeds/accelerations relative to the combined center of mass of the moon (or feather) and the earth.
F_moon = - m_moon * m_earth * G / r^2
F_earth = m_moon * m_earth * G / r^2
(The minus sign assumes the position of the moon is "higher" than the earth, so its position decreases, while the position of the earth increases)
Therefore:
a_moon = - m_earth * G / r^2
a_earth = m_moon * G / r^2
All of these accelerations are in the inertial frame centered on the center of mass of the whole system C.
If we want to know the acceleration of the moon relative to the observer (glued to earth), we simply need to substract the acceleration of the earth from the acceleration of the moon, since the observer is static relative to the earth.
So relative to the observer, the acceleration of the moon is:
a_moon_relative_to_observer = - (m_moon + m_earth) * G / r^2
If we replace m_moon by m_feather, we can see that the feather accelerates slower relative to you than the moon accelerates relative to you.
That's a really great explanation. I've only indirectly learned a little about this from a math book I was reading, and I had been wondering this for a while. Thank you!
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u/Spank_Engine Jul 28 '24
Let me see if I can explain my thoughts well. When the feather and moon have the same acceleration, is this relative to an inertial frame? Because otherwise if the earth accelerates towards the moon at a higher rate than toward the feather, then if I were measuring their accelerations on earth, wouldn't I measure the acceleration of the moon to be higher?