I have to take issue with the answer given about feathers and hammers hitting the ground simultaneously. I’m no physicist, and I could be wrong, but I think there’s an error in your logic here.
To put it in laymens’ terms my brain can wrap around, the premises are:
The gravitational attraction force between two bodies is a function of their combined mass.
Acceleration of an object by a fixed amount of force is a function of the object’s mass.
Based on these two (valid) assumptions, you have stated that since the hammer has more mass, it will have more gravitational force applied to it, but also because it has more mass, it will take more gravitational force to cause the same acceleration the feather is getting, therefore they accelerate at the same rate towards the earth. So far so good. You drop the ball by then jumping to the conclusion that they will strike the earth at the same time.
Two bodies, attracted by gravitational force, are both moved towards each other’s center of gravity. Equal force is applied by gravity to both objects. Because of the acceleration problem mentioned above, the more massive of the two will accelerate less and thus not move as far as the less massive of the pair.
Therefore… the earth is also accelerated towards both the feather and the hammer. Assuming the feather and hammer are not allowed to occupy the same spot (they are next to each other, seperate by some small distance), then the earth will in fact accelerate towards the hammer at a higher rate than it accelerates towards the feather.
Granted, the difference in the timing of the feather and hammer striking the earth would be so minute as to be unmeasurable… but replace “feather” and “hammer” with bigger objects and it becomes more obvious. Lets say our moon and another moon twice as massive, starting stationary in a vacuum 10,000 miles out… which strikes earth first due to gravitational force? - the more massive of the two does, of course.
Well, you’re certainly correct in a hypothetical universe where the Earth, the feather and the hammer are the only objects in existence.
In the real universe where the Earth orbits a massive star which in turn is part of a moderately large galaxy I’m far from convinced.
IANA physicist, but I’d need to see some evidence before I’d believe that the hammer could overcome angular momentum, gravity of the sun, inertia of the planet etc. I doubt the hammer would have any effect, even an immeasurable one. The forces involved with the celestial bodies are so great and the mass of the hammer so small I don’t believe the hammer effect would even counteract random quantum-level fluctuataions. This means that the difference between dropping a hammer and dropping a feather would be absolutley nothing.
In this universe there really is a leval at which an effect is so slight that it is no longer an effect at all.
Very nearly (so as the difference to be negligible) this same quibble has been broached before. I only want to say, you have not yet reached the height of pure quibbleness exhibited in those previous threads.
Granted it’s a quibble in the case of the hammer and the feather, but I think the point of the thought experiment is to understand what really happens even in the case of larger objects, so I feel the full answer is best. The guy above is probably right that whatever the hammer’s advantage works out to, it’s probably so tiny that it’s beneath some quantum limit where it has no effect. However, column gives a reason based in math for their simultaneous landing, but it ignores that there should be another part to the math, which is the acceleration of earth towards the objects. To correct it all you have to do is plug this in, do the calculation, and then say “However the hammer’s advantage in the mathematical model works out to it arriving at the earth when the feather is still 5.6x10^-245986098039456 (or whatever) milimeters from earth, which is below some quantum limit and therefore has no effect on reality”. Now you’re still saying they hit at the same time, but you’re telling the truth about how to arrive at that conclusion.
Again, if this were a larger example, on the scale of planets and solar systems, and you had 3 bodies, the sun, jupiter, and earth… all starting at a standstill relative to each other in a vacuum, jupiter would impact the sun before earth did, because the whole time jupiter and earth are accelerating towards the sun, the sun would be accelerating in both of their directions, but moreso in the direction of jupiter.
The biggest problem with your analysis is that the objects diameter plays a more important part than any of these factors. But let’s ignore that.
One thing you seem to be missing is the attraction of the feather to the hammer. That was important in some of the other threads on this subject. For instance, rather than dropping them at the same time, what if you dropped each at separate times and just timed the descent? As you point out, the force of attraction between a given mass of the Earth and the hammer is greater than between the same mass and the feather–except that when the feather was dropped the hammer would have to be somewhere on the Earth, and that additional hammer mass cancels out the effect almost completely.
Your example is different, but there is still an effect which may make a difference. I’ll try to calculate it–or someone will.
My point, ph317 is that there may not be any attraction of the Earth to the hammer. Given the forces at play the Earth may not be physically drawn to the hammer at all. It’s not just that the resultant difference would be tiny and swamped by quantum fluctuations, I remain to be convinced that there would be any difference whatsoever, even on paper.
You can’t get away with saying ‘To correct it all you have to do is plug this in, do the calculation…’. The calculation involves a situation where a stationary hammer is drawn towards the planet. However the planet isn’t staionary while it is drawn towards the hammer. The equation doesn’t take into account the fact that the Earth orbits the sun, rotates on its axis or orbits galactic central point. To calculate the actual movement of the Earth towards the hammer you would need to take into account these far larger forces as well. You couldn’t just reverse the equation used for a stationary hammer.
And I would need to see some evidence before I would beleive that th hammer could overcome all those effects to any degree that would move the planet in any way whatsoever.
Besides the fact that those aren’t forces, the forces normally associated with those motions can be set normal to the experiment, and their effect minimized. In the spirit of the OP.
However, if you dropped them simultaneously, right next to each other, then the Earth is going to be accelerating towards the combination feather/hammer pair, and the hammer would no longer enjoy such an advantage. There would still be the exceptionally nitpicky difference that the Earth would be drawn in a direction towards the hammer more than towards the feather, so would hit the hammer slightly ahead of when it hit the feather. However, this effect would be several orders of magnitude less than the timed-separately scenario - maybe below the planck limits.
That exceptionally nitpicky difference seems to me to be the essence of ph317’s OP, and I maintain that there is still the hammer/feather interaction to contend with. In other words, sure the Earth is drawn to the hammer more than the Earth is drawn to the feather, per se, but the mass of the Earth/hammer attracting the feather is greater than the Earth/feather attracting the hammer. I don’t think that’s been addressed.
In previous examples, most of the nitpick effect that people thought was present was obviated by consideration of the hammer/feather interaction, and I suspect the same will be true here. Just run the numbers, and let’s see.
Good point. There’s another nitpicky effect that I haven’t seen mentioned, which would probably swamp the effect we’ve been talking about.
To perform the experiment, you’d need to hold both objects so that their lowest points are the exact same height off the ground. But the two objects will then have their centers of mass at different heights, therefore will have slightly different accelerations. My gut feel is that this effect would be much greater than the other one we’ve been discussing.
Or one could simply alter the experiment to use two spheres of equal size both having a uniform density (ensuring center of sphere is center of mass). The only difference is one is of a low foam-like density and the other of a heavy lead-like density. If you start from that thought experiment, then the only remaining questions are truly gravitational attraction between the three bodies and the resultant acceleration of them in each others’ direction. I still think the math favors the more dense object striking first, but that the two “falling” objects would need to be much larger relative to the main “earth” object for the effect to be noticeable or measurable in the real world, or for that matter to be high enough to cross those basic quantum cutoffs.
Unfortantely, that’s just a guy feeling, I’m neither a mathematician or a physicist, so if I tried to run the numbers myself here in this post, I’d likely make some big blunders which would become the center of attention instead of the idea. Anyone who can do it off the top of their head care to actually run the numbers and see about planck limits and whatnot for the small case, and then for a bigger (moon-sized or better) case?
The usual formula for the gravitational acceleration of the Earth is GM/r[sup]2[/sup] (M–Earth’s mass, G–gravitational constant, r–distance), whereas a more exact formula, as everyone has pointed out, is GM/r[sup]2[/sup] + Gm/r[sup]2[/sup], where m is the mass of the small object. For small objects like a hammer or feather, the difference between the two formulas is extremely small. As m gets larger, and closer to M, the difference in the two formulas will make a difference.
So go ahead and make one sphere equal to the mass of the Earth–that’ll double the mutual acceleration. Make the other have the mass of a feather. But if you had two such spheres side by side, and dropped them, the larger one would attract both the Earth and the smaller one, and would not fall towards the Earth twice as fast as the smaller one–but I realize you knew that. The Earth is moving generally towards both of them because they are side by side.
Your question is, what would be the effect of the Earth moving a little bit more towards the larger one. As you can see, the difference is going to be small, the race towards the Earth will be nearly a tie–even with a hammer which has the mass of the Earth!
