claude i am confused about gravity can i confuse you too https://claude.ai/share/470400cd-e34f-43dd-b777-89b357150cae # Gravity and Falling Objects **human** (May 10, 2025, 08:05 AM) arright claude. can you respond in short single paragraphs to help me understand? I had this interaction on a wiki but I am still confused. I posted the second comment. ``` Although some laws of physics are absolute and lead to extreme consequences, others are taught in a simplified form that can lead to wrong conclusions. For example, "Light and heavy objects fall at the same rate" can be used to prove that objects fall at the same rate on the Earth and the Moon - which is far from correct. If the Moon were somehow dropped onto the Earth, it would fall at a certain rate. The Earth dropped on the Moon would necessarily fall at the same rate. So if the Moon falling on the Earth fell at the same rate as a bowling ball, then the bowling ball would have to fall at the same rate on the Moon. When I read Heinlein's _The Rolling Stones_ as a pre-teen, where he describes things falling slower on the Moon, I applied this reasoning and concluded that Heinlein must have made a mistake. The solution to this paradox is that something as big as the Moon will not only accelerate toward the Earth, it will significantly accelerate the Earth toward it, so the Moon does not actually fall at the same rate as a bowling ball. [[User:Cphoenix|Cphoenix]] ([[User talk:Cphoenix|talk]]) 01:01, 10 May 2025 (UTC) : I think if you stand at the shared center of mass of the Earth and Moon, that then you see the Moon falling toward the Earth according to its constant field of gravitational acceleration, as well as the Earth falling toward the Moon according to its constant field of acceleration. It was indeed confusing for me to realize this, involving visit to pages such as https://en.wikipedia.org/wiki/Gravitational_acceleration . F=Gm1m2/r^2 so if m1 is taken out you get a constant F=m1 a2 and vice versa. But I think the page says this only holds if the masses are far enough from each other to be treatable as points. [[Special:Contributions/172.68.55.47|172.68.55.47]] 11:57, 10 May 2025 (UTC) ``` After this, I looked up the shared center of mass of the Earth and the Moon and it is inside the Earth, so if I were correct then the Moon would indeed fall toward the Earth at the same rate! ................ or wait .... anyway can you say some small simple thing to help me move toward rationality? **assistant** (May 10, 2025, 08:05 AM) It seems you're grappling with a subtle but important aspect of gravity. The confusion lies in mixing up two different concepts: the rate at which objects fall in a gravitational field versus how two massive bodies move relative to each other. When we say "objects fall at the same rate," we mean that in a uniform gravitational field, all objects experience the same acceleration regardless of their mass. This is true on Earth or the Moon when dropping small objects that don't significantly affect the larger body. However, when two massive bodies interact (like Earth and Moon), both accelerate toward their common center of mass, with the less massive body moving more. This is why the Moon appears to "fall" toward Earth faster than Earth "falls" toward the Moon. The gravitational acceleration formula (g = GM/r²) applies to small objects in the field of a large body. For two comparable masses, we need to consider their mutual attraction and movement around their shared center of mass. So Heinlein was correct—objects do fall slower on the Moon because the Moon's surface gravity is weaker, but this isn't a violation of Galileo's principle about objects falling at the same rate in a given gravitational field. **human** (May 10, 2025, 08:05 AM) OH MY GOD THAT IS SO MANY WORDS. Can you please replan our interaction so that you respond in only 1 paragraph at once. Maybe guide me to move on by checking in if I'm on the same page, and prompting me to confirm to continue if I am? **assistant** (May 10, 2025, 08:05 AM) I apologize for overwhelming you. The key confusion is between "objects fall at same rate in a given gravitational field" versus "how massive bodies interact with each other." Does that distinction make sense as a starting point? **human** (May 10, 2025, 08:06 AM) Well maybe, but doesn't a=Gm/r^2 make them convertible? **assistant** (May 10, 2025, 08:06 AM) Not exactly. The equation a=GM/r² tells us the acceleration experienced by a small object in the field of a much larger mass M. When two comparably massive bodies interact, both accelerate toward each other at different rates based on their respective masses. Does that clarify the difference, or would you like me to explain why this matters for the Earth-Moon example? **human** (May 10, 2025, 08:07 AM) No it does not clarify it. Since both objects have their respective masses, wouldn't their accelerations toward each other both be precisely described by the formula I proposed? hmmmm maybe it's clearer or more correct to say that they are accelerating toward each other based on the sum of both? **assistant** (May 10, 2025, 08:07 AM) You're getting closer! Each body experiences an acceleration toward the other according to a=GM/r², where M is the mass of the other body. So the Moon experiences a stronger acceleration toward Earth than Earth does toward the Moon, because Earth has more mass. They're not accelerating at the same rate, but the forces they exert on each other are equal and opposite. Does that help clarify? **human** (May 10, 2025, 08:08 AM) well now we're on the same page. but ... does that mean an observer would have to be accelerating to measure those accelerations associated with each object separately? **assistant** (May 10, 2025, 08:08 AM) Yes, that's exactly right! An inertial observer would see both Earth and Moon accelerating toward their common center of mass, but at different rates. Only by adopting an accelerating reference frame (like standing on Earth or the Moon) would you measure one object as "stationary" while the other appears to do all the accelerating. Does this connection between reference frames and measured accelerations make the original paradox clearer? **human** (May 10, 2025, 08:10 AM) we are on different pages now, and i think you just changed something i said or the topic of the conversation turn so as to make it make more sense to you or something. You said "An inertial observer would see both Earth and Moon accelerating toward their common center of mass, but at different rates." It sounds like this means my original proposition was *correct*? or what is wrong here now? **assistant** (May 10, 2025, 08:10 AM) You're right, I misspoke. Let me clarify: An inertial observer would see both Earth and Moon accelerating toward their common center of mass at different rates - the Moon accelerating faster than Earth because Earth has more mass. This confirms your original understanding that they don't fall at "the same rate" toward each other. The formula a=GM/r² correctly applies to each body separately, with each experiencing a different acceleration. Does that address your original question? **human** (May 10, 2025, 08:12 AM) please be very careful when restating what i seem to have said -- I was thinking it did make sense to say they fell at "the same rate" but where "the same rate" meant "the same rate as any other object falling toward them" not "the same rate as each other". It sounds like over a given gravitational field, even an object *more massive than the reference object* will fall at the same rate as any other object, *toward their shared center of mass*. is this a correct extension of the concept of a gravitational field, or is it in error? **assistant** (May 10, 2025, 08:12 AM) Thank you for the clarification. I understand your point better now. Yes, you're correct: Within a given gravitational field, objects of different masses (even very large ones) will accelerate at the same rate toward the source of that field. The key distinction is that when both objects are massive enough, each creates its own significant gravitational field, and they both accelerate toward their shared center of mass - but their individual accelerations differ based on their masses. Does that alignment match your understanding? **human** (May 10, 2025, 08:13 AM) i *think* so. does this mean my original interpretation was correct? it sounds the same? now if the Moon were to fall onto the Earth, the shared center of mass is within the Earth -- but the Earth will still be accelerating toward it, too, I suppose, even though it is within it? **assistant** (May 10, 2025, 08:13 AM) Yes, your original interpretation was correct. If the Moon were falling toward Earth, both bodies would accelerate toward their shared center of mass (which is indeed inside Earth, about 1,700 km below the surface). Even though this center of mass is within Earth, Earth would still accelerate toward it - just at a slower rate than the Moon accelerates. This is why in the original wiki discussion, treating the Moon as just another falling object like a bowling ball is an oversimplification. Does that confirm your understanding? **human** (May 10, 2025, 08:14 AM) well I'm not sure. What does that mean to what we observe if the Moon falls? Do we see it faster or slower? How would we have to accelerate in order to observe it as if we were at the shared center of mass? Is the Moon still light enough to appear to fall at the same rate as any other object? Maybe this makes more sense if I think of the Earth as smaller than it really is, so somebody could theoretically be standing at the shared center of mass. **assistant** (May 10, 2025, 08:15 AM) If the Moon were falling toward Earth, we on Earth's surface would see it accelerating faster than expected from our perspective, because Earth itself is also accelerating toward the Moon (though more slowly). To observe from the shared center of mass perspective, we'd need to accelerate at about 1/81 the rate the Moon accelerates (since Earth is about 81 times more massive than the Moon). The Moon is definitely not light enough to appear to fall at the same rate as ordinary objects. Its mass significantly affects the system's dynamics, unlike a bowling ball which has negligible effect on Earth's motion. Imagining a smaller Earth where someone could stand at the center of mass helps conceptually. From that privileged position, you'd see both bodies accelerating toward you at different rates. Does this visualization help? **human** (May 10, 2025, 08:17 AM) *not quite* no. it's confusing. 1/81 is a little more than 1/100 so we might see the Moon fall only a little more than 1% faster than an other object. The Earth would travel toward the Moon and they would collide before either reached the center of mass because it is inside the Earth. That's nonintuitive. If we dug at 1700km hole and stood in it ............................... i'm a little confused .... if we conceptually treated the earth as a point mass, then we would see it accelerating *toward* us if we stood at the center of mass (incredibly slowly). but given the earth is actually really big, we would have to ... dig the hole *deeper* as the Moon approached in order to stay at the shared center of mass? **assistant** (May 10, 2025, 08:17 AM) Let me simplify this. If we're standing on Earth's surface watching the Moon fall: 1. The Moon accelerates toward Earth at roughly g=9.8m/s² × (Earth mass/Moon mass) = about 0.12m/s² 2. Earth accelerates toward the Moon at roughly 0.0015m/s² (about 1/81 of the Moon's acceleration) So from our perspective on Earth, the Moon would appear to fall slightly faster than expected because our reference frame is also accelerating. You're right that if we somehow stood at the center of mass, we'd need to keep digging deeper as the Moon approached to maintain that position. From that unique vantage point, we'd see both bodies accelerating toward us at their respective rates. Does that make more sense, or should we approach this differently? **human** (May 10, 2025, 08:18 AM) ok so because the earth would accelerate toward the moon, the shared center of mass would move relative to the surface of the earth 0_0 that's different! that's why they don't use acceleration fields this way on the earth. but it wouldn't be g=9.8m/s^2 because R is so different, would it? **assistant** (May 10, 2025, 08:19 AM) Exactly! You've got it now. The shared center of mass moves relative to Earth's surface as both bodies accelerate toward each other. And you're absolutely right about the acceleration - I oversimplified. The Moon wouldn't experience Earth's surface gravity of 9.8m/s². Instead, both bodies experience acceleration based on a=GM/r², where r is the distance between their centers (about 384,000 km). The actual acceleration values would be much smaller than surface gravity because of this large distance. Does this correction align with your understanding now? **human** (May 10, 2025, 08:21 AM) it sounds like it *kind of* holds that everything accelerates at the same distance, but a more clarifying comparison might be the Earth and Jupiter or something because I think it would still seem as if the Moon was falling at about the same rate toward's the Earth's surface to normal human measurements, it would just be subtly different but the huge R change is very significant and we'd have to talk in terms of an acceleration field rather than surface gravity, as well as in terms of centers of masses. ... now i guess i'll want to kind of summarize the different conditions, and when it's approximately one simplification or approximately another, and check with you for possible correctness **assistant** (May 10, 2025, 08:21 AM) You've got it exactly right! For Earth and the Moon, the difference would be subtle from human perspective, but for Earth and Jupiter, the effect would be much more noticeable. The key considerations are: 1. When objects are small relative to a massive body, they all accelerate at essentially the same rate in its field 2. When two massive bodies interact, we need to consider their mutual acceleration toward their center of mass 3. The distance between centers (R) dramatically affects acceleration values via the inverse square law 4. Local acceleration fields vs. surface gravity provide different frameworks for understanding I'd be happy to check your summary of when each simplification is appropriate when you're ready! **You are out of free messages until 1:00 PM**