The issue isn't so much the sensitivity (although that is a significant issue), it's all the other crap going on. You'll probably be able to filter out the Mains Hum, but every anything moving in the same axis as the test will confount the data.

I'm thinking we might set up the instuments near counterweight energy storage or pumped hydro, and some on the exact opposite side of the planet, and try to measure the movement of the earth that way.

We can already see a change in the length of a day after big earthquakes and dam construction/destruction, but I don't think the acceleration has ever been measured directly.

because of two bodies can not occupy the same space, the feather and the ball will be in different position when you drop them. And therefor gravitation will pull the earth slightly more toward the ball and slightly less toward the feather.

The acceleration from gravity would be the same no matter the object mass (~9.8m/s²).

The above is just referring to the fact that the standard "feather vs. bowling ball" question assumes the earth/moon/ground is immovable. In that case, Newton says they fall the same.

The fact that the ground is not immovable is what's being referenced

in this picture, things don't "fall," they are each accelerated towards each other.

That’s only be true if the feather was in the same position as the ball. Otherwise, the earth is moving ever so slightly more towards the ball.

No, because the earth is accelerating towards the bowling ball and the feather is next to the bowling ball, the force vector is (ever so slightly) greater towards the bowling ball than the feather, thus the bowling ball drops faster

You got it.

Yes. And that force is proportional to acceleration so it accelerates earth

And is the Sun was replaced with a black hole of the same mass, the Earth would just keep on rotating around it without issues, if slightly frozen

Just to add some formality to this, the original commenter might want to look up the shell theorem for classical mechanics and Birkhoff's theorem for general relativity.

Heavy is a subjective term based on the force of gravity. You are heavier if we weigh you on the earth compared to if you are weighed on the moon.

Your mass in those two examples is unchanged. The amount of mass you have is finite and not subjective like weight.

But how would that make the bowling ball fall faster? F = G × m₁ × m₂ / r² and F = m₁ × a ⇒ a = F / m = G × m₂ / r², where m₁ is the mass of the ball and m₂ the mass of the planet. So the gravitational acceleration of a bowling ball is independent of its mass (assuming the planet has way more mass than a bowling ball).

Ah but the earth doesn't just attract the ball or feather. The bowling ball attracts the earth as well, and since it has more mass, it will pull the earth towards it faster than the feather.

But if you drop them at the same time, that's moot.

So why does the bowling ball fall faster in a vacuum? Does it appear faster locally because the heavier object makes local time slower than the lighter object compared to a distant observer? I'm trying to understand what the meme is getting at.