Galileo was technically WRONG

[Trippy]

Here's why I think that RJBeery's assertion that Gallileo was wrong is, in itself, wrong.

Historical aspects:
It is my understanding that Gallileo never actually dropped the balls off the leaning tower of Pisa. Gallileo's experiments involved timing the motion of masses down inclined planes. It is my understanding that the phraseology associated with what Gallileo said, the modern phraseology that is attributed to him, was introduced some time later by a biographer. Gallileo himself, as I understand it, only ever talked about 'falling objects' and how falling objects fell at the same rate. This assertion is supported by considering the state of science at the time. The debate was whether falling masses of different composition fell at the same rate, not whether they hit the ground at the same time - this relates to the point that Bruce P has made repeatedly about the equation of motion being used. The debate as, effectively, couched in the terms of considering falling objects dropped from an infinite height in a uniform gravitationl field, because they were concerned with how fast the objects fell, not how long it took them. In the context of the original discussion being had, Gallileo was, by my understanding of the historical context, correct. This brings me to my next point.

The Physics:
As I said in my previous post, and as Bruce P has pointed out more than once, the equation of motion used in the OP is the wrong one for the problem being considered. The equation of motion in the OP is for the motion of bodies in a uniform gravitational field, which is the correct equation of motion for Gallileo's thought experiments. It is a perfectly adequate approximation for the realms for which it was originally drafted - you're not going to hit that castle more accurately with a cannon-ball if you use Einsteins field equations over Gallileo's equations of motion, nor are you going to hit it more frequently if you take the motion of the earth towards the cannon ball into account. The thing is, as soon as you consider the motion of the earth towards the cannon-ball, you're no longer considering a uniform gravitational field, you've introduced a kink into it, and once you do that, you can't use the equations of motion in the OP any more.

The Reality:
At the end of the day the distinction that RJBeery is trying to draw here is both meaningless and unphysical as well as being wrong. The amount of motion being described here is much less than the radius of a proton. It's a fraction of an atom. It is impossible to measure the center of mass of the earth to the precision required to measure the motion. More than that, no sane physicist (at least not the one's i've met) would agree that if you took a mass of 1,000,000 kg and added 1g to it that you now have 1,000,000.001 kg. Why? Because, effectively we've turned a number that is accurate to one significant figure into a number that is accurate to ten significant figures. What if the weight of the original mass was only recorded to the nearest 1,000 kg's, or even the nearest kg. There's an element of common sense that comes about in situations like this where you have to realize that whilst an answer might be technically correct it's also physically meaningless and un-testable. I'd almost be tempted to calculate the momentum of the earth, based on its average velocity falling towards the bowling ball, and, taking the distance covered into consideration work out whether or not the measurements involved are even possible or if the numbers are too small.

All of which leads me to my final point. Gallileo was not wrong. Gallileo's statement had a context - uniform gravitational field and/or m1 <<< m2 - RJBeery's assertions rest on ignoring one or both elements of that context to construct a strawman hypothesis. If I make the statement "All mamals in New Zealand are Cats", and RJBeery goes to America and finds that some mamals in America are dogs, RJBeery's finding doesn't disprove my original statement because my original statement as only meant to be valid in New Zealand in the first place. Like wise, if Gallileo's statement was that objects in a uniform gravitational field fall at the same rate, and RJBeery goes and proves that if we introduce additional objects our added objects and use those as the gravitational field source and consider their motion towards the dropped objects then they approach each other at different rates, well, that doesn't disprove what Gallileo said. Why? Because you're taking Gallileo's statement outside of the realm it was claimed to be valid in, and we all already know that Gallileo's statement will be wrong if you consider it outside the realm it's applicable in.

So, congratulations on figuring out a piece of highschool physics?

 

[Trippy]

RJBeery gave an example of a black hole falling to earth, and that makes it pretty clear that he is including non-terrestrial objects in his thought experiment. A black hole certainly would cause the earth to accelerate toward it as it fell toward earth, and the accelerations of both objects should be taken into consideration. (Unless the earth is chained to a magic immovable wall.)

A black hole does not come under the caveat of m1 <<< m2, unless it's a microscopic black hole, even then, a black hole most certainly does not have a uniform gravitational field.

 

[brucep]

Galileo did not have a valid reference frame because it was not inertial during the experiment. You're starting to sound like a bit of a blowhard.

 

[brucep]

Sorry about that. I didn't realize that I pushed the post button with no text.

 

[brucep]

Yes, I know. That would be part of the context of Gallileo's original statement that I was referring to.

I wasn't referring to your comments. It's my bad since I don't answer separate quotes. Which is a problem for me sometimes. Otherwise your posts have been exemplary in my estimation.

 

[exchemist]

Remember, this hypothetical is framed in the context of gravity, which when you start comparring time as measured by multiple frames of reference, starts to get very sticky... You know time dilation and how it affects clocks and time measurements...

When you are talking about a small mass, compared to the total mass of the earth, and an initial separation distance of 10 meters{?}, even Farsight's optical clocks would likely disagree within their indepent margins of error... Even if their individual locations in the gravity well did not make comparrisons completely nonsensical. Even measurements of time are FoR dependent.

The point I am trying to make here, is that for say the bowling ball test, for a clock located at the center of mass of the two body problem and a clock located at the center of mass of the earth, or for practical purpose the surface of the earth and an elevation equal to the distance between those two centers of mass, even Farsight's optical clocks would agree on the elapsed time. Which supports the validity of Galileo's choice of FoR and conclusions.., even though he did not have access to modern clocks, or any understanding of GR.

Thanks. I think, speaking as a mere chemist with some grasp of Newtonian mechanics only, it would be helpful to the argument to agree what a purely Newtonian view would conclude and then to add onto that how the relativistic view might differ. I think that on a purely Newtonian view RJB is right. I cannot express a view on what happens if GR is taken into account and would welcome someone's explanation as to how that would alter the result.

 

[Neddy Bate]

A black hole does not come under the caveat of m1 <<< m2, unless it's a microscopic black hole, even then, a black hole most certainly does not have a uniform gravitational field.

True. But I take this as evidence that RJBeery's thought experiment was not intended to be limited to either of those caveats. His use of the words "technically wrong" in the thread title also suggests that he wanted to include every possible 'technicality' that might allow a circumstance where Galileo could be wrong. You argue that he should not be allowed to do that, because it takes Galileo out of context. That is a good point, but it seems to me that RJB intended to take him out of context.

 

[Trippy]

True. But I take this as evidence that RJBeery's thought experiment was not intended to be limited to either of those caveats.

If RJBerry's thought experiment is not intended to be limited to either of those caveats then RJBeery's thought experiment bares NO RELEVANCE what-so-ever to the statements Galileo made or Galileo's thought experiments because both of those caveats were explicit or implicit in the original phraseology.

His use of the words "technically wrong" in the thread title also suggests that he wanted to include every possible 'technicality' that might allow a circumstance where Galileo could be wrong.

Technically RJBerry's entire argument is a straw-man argument because it's based off something that superficially resembles Galileo's rather than anything Galileo actually had to say.

You argue that he should not be allowed to do that, because it takes Galileo out of context. That is a good point, but it seems to me that RJB intended to take him out of context.

I've made three points.

The first point is not so much that he should not be allowed to do that, but more that any consideration of whether or not a claim is accurate needs to be done within the domain of applicability applied to the original claim in the first place.

If I claim that under the circumstances where A is true, and B is true then C applies, demonstrating that C is false where A and/or B is false doesn't disprove C, which is what RJBeery is (at least) implicity claiming) at worst, it shows that C is incomplete.

If RJBeery genuinely believed that Galileo's statement was ever meant to describe the closing velocity between the earth and a one solar mass black hole then he hasn't understood Galileo's statement in the first place.

The other two points that I made were that the distinctions being talked about, when applied to a golf ball and a bowling ball are physically and practically meaningless.

To be honest, I've been debating closing or moving this thread since Friday - especially given that RJBerry so far seems to have ignored my points (as well as other peoples points).

 

[Trippy]

I wasn't referring to your comments. It's my bad since I don't answer separate quotes. Which is a problem for me sometimes. Otherwise your posts have been exemplary in my estimation.

Thank you. :*)

 

[Aqueous Id]

You're making the same mistake that Galileo [supposedly] did; the one that I identified in the OP over 100 posts ago. You can make the Earth a reference frame by definition but that is a convenience, not a technically and infinitely accurate presumption.

In every kinematics problem, you must first choose the reference frame. There is no "right' or "wrong" frame, other than a choice made out of ignorance. Assuming you have the skills to validate the frame as appropriate, then there is hope. Other than that, there is SciForums, where you can get free help for your freshman physics questions.

You tell me to pick a frame of reference. I did.

YOu choose an appropriate frame in order to get the correct answer, not because I brought it up.

Then you tell me to do it again. I DID.

I was asking you to locate it physically. Where is it relative to Galileo?

Then you say that we could choose the Earth if we wanted,

In this case you just about have to.

but many people have already explained many times that additional adjustments to your calculations would be required if you wanted to do this

I'm not aware of what other people have said. I'm responding to your posts as I get the alerts. But "adjustments" is nonsense. The desired model either exists, or it does not. If it exists, then all of the calculations are one organic chunk of work. Things like adjustments are worthless.

The simple Newtonian analysis acts as if the Earth did not move as I attempted to illustrate around 60 posts ago with the following picture:

Your diagram looks silly. You should try to draw the free body diagram as you would do if it were a test question.



------------------------------------

Aqueous Id is still arguing that fall time depends on your choice of coordinates,

Actually, no, I haven't said anything of the sort. At present I am trying to help RJ organize his thinking as a skilled person would do.

The positions and velocities of single objects depend on one's choice of coordinates, so different observers can see the ball moving at different speeds.

Ok.

But relative position,
a.k.a. the distance between two objects, mathematically $$x_{rel}= X-x$$, is independent of coordinate choice in Newtonian physics.

No, once you locate the x-axis that contains X and x, you have assigned a reference frame and lost that independence you seek.

In your example, for instance, all observers should agree on the velocity of the ball relative to the referee, even as they disagree on the absolute velocity of the ball. Because time and \(x_{rel}\) are both independent of coordinate choice,

No, for example if the ship turns 45°, the observer on the dock will not see the ball pass in front of the referee at the same time the referee sees the ball cross the net.

it is also true that all time-derivatives of \(x_{rel}\) are independent of coordinate choice.

No, see above.

My formula gives relative acceleration as a function only of relative position, so it holds true in all coordinate systems.

Your formulas were written by someone who evidently never even took algebra.

You lost me here. What part of my language/procedure is invented?

Your use of proprietary speech, your inability to reformulate a freshman physics problem, dragging in a circuitous reference to the Hamiltonian, and your messed up math--together spell "invention".

Hamiltonian dynamics are a standard method in classical mechanics;

That just illustrates that you have little or no familiarity with the typical texts for Physics I.

they're probably the single most useful thing I learned from my Classical Mechanics course as an undergrad.

Did you really take Classical Mechanics? Impossible, methinks, unless you are a ghost from the 17th c. But Lo, methinks you never matriculated at all.

Relative coordinates are a long-standing trick,

Trickery noted.

especially for two-body problems,

I see you are an adept kind of shrewd.

and I use them at work routinely.

A little honesty would go a long way here. You ought to state up front that you don't actually apply physics skills in your daily work.

The rest of what I did was literally just applying Newton's law of gravity and the kinetic energy formula to a falling ball.

I think it's a little late to be promoting your math.

Nor am I. Maybe "reference frame" was a poor choice of words; I meant Newtonian reference frame, i.e. choice of coordinates.

Alas, too late, poor Yorick.

Yes, it was by definition an Earth-centered frame. But in this experiment, the Earth accelerates upward,

Conclusion before the fact.

so it is not inertial.

Wrong. The frame is inertial iff the g-field is homogeneous and isotropic throughout.

Therefore, any frame centered on the Earth throughout the experiment must also not be inertial.

You have that backwards.

You can't make an accelerating frame inertial by definition, because Newton's laws are modified in a non-inertial frame.

No, you make it inertial by choosing it such that the field is not changing, which is why your/RJ's reference frame is non-inertial.

When he treated the Earth-centered frame as inertial, Galileo was making a very good but strictly imprecise approximation.

That alone tells readers you are not who you purport to be.

My solution, which actually is in an inertial frame, does not suffer the same issue.

No, your origin cuts across the Earth's g-field, therefore it is not properly called 'inertial'.

In RJ's illustration, the wall isn't falling to the Earth. It's supposed to act as an anchor point to hold the Earth in place.

This tells readers you never had to draw free-body diagrams, methinks, in case they didn't notice your math was not merely an update per 'the' Hamiltonian.

This is to highlight the contrast with the actual situation, in which the Earth can freely accelerate toward the ball as the ball falls.

It's to highlight that the horse simply won't drink. Go figure.

 

[Neddy Bate]

Trippy,

I concede that Galileo probably never claimed that a celestial object with the mass of the sun would fall to earth at the same rate as a bowling ball, (with both falling from the same height). But it seems there are some in this thread who would claim that is true. In that sense, Fednis, RJB, and others have a better grasp of the physics being discussed here than those other folks do. It probably would have been wiser for RJB to leave Galileo out of it, and just present his case.

 

[RJBeery]

If RJBerry's thought experiment is not intended to be limited to either of those caveats then RJBeery's thought experiment bares NO RELEVANCE what-so-ever to the statements Galileo made or Galileo's thought experiments because both of those caveats were explicit or implicit in the original phraseology.

If you feel I've not given you attention it's because, to this point, your comments are inane. Did you miss this post of mine?

I think we can all agree that this has no practical use for dropping small objects to the Earth. This is a theoretical exercise, and the point of the OP is to show that, TECHNICALLY, Galileo was mistaken. He could also have made the claim that light travels with an infinite speed and would have been forgiven because, for all practical purposes (particularly at that point in history), he would have been correct.

This thread is clearly being pedantic and technical. If anyone is offended because we're talking about absurdly minuscule differences in drop times then you don't understand the point of the exercise.

Trippy, you're trying to defend the MAN when I am clearly not attacking him. The issue was settled when the good professor chimed in. Watching you and the others do your linguistic gymnastics trying to rescue your ego in the face of defeat is certainly entertaining but it's getting old.

Hell, it wouldn't surprise me if you moved this thread to pseudo-science just to erase it from the pages of history.

 

[RJBeery]

There is some truth to part of what you have been tring to present but it requires that the mass to be dropped, or originate somewhere other than as a part of the earth's orginal mass.

I think I can safely speak for RJ when I say we both agree with this. Since you dropped your assertion that the moon would fall at the same rate as a baseball, I've had no disagreement with you. But yours was not the only competing argument in this thread; Aqueous Id is still arguing that fall time depends on your choice of coordinates, and RajeshTrivedi is still arguing that the ball shares the Earth's upward acceleration.

Yes, I agree on this, although I had to convince myself mathematically because my intuition failed me here. I've reduced the "time to fall equation" from the OP to the following (and renaming it "time to connect equation" since it involves two free-floating bodies):

t = sqrt((2d * r^2)/G * M))

Look at the simplicity!
d = distance between objects
r = distance between objects' respective centers of mass
G = gravitational constant
M = total system mass

The amazing thing to note here is that it doesn't matter how you divide the mass. A pea dropping to the Earth from 10 meters up will take the same time as splitting the Earth in two equal pieces and separating them by 10 meters.

You can click here to see Wolfram Alpha calculate the 10 meter fall to be 1.427 seconds...regardless of the mass distribution. Amazing to me

 

[Motor Daddy]

The amazing thing to note here is that it doesn't matter how you divide the mass. A pea dropping to the Earth from 10 meters up will take the same time as splitting the Earth in two equal pieces and separating them by 10 meters.

You can click here to see Wolfram Alpha calculate the 10 meter fall to be 1.427 seconds...regardless of the mass distribution. Amazing to me

What's even more amazing is that while your numbers predict it's just a matter of time (which according to you can be calculated) before Mercury crashes into the sun, and then some time later Venus, then Earth, Mars...all just a matter of time before they all crash into the sun.

On the other hand, Earth is getting further away from the sun in reality, so it's just a matter of time before the earth just keeps getting farther away, not to be confused with crashing into...

 

[RJBeery]

What's even more amazing is that while your numbers predict it's just a matter of time (which according to you can be calculated) before Mercury crashes into the sun, and then some time later Venus, then Earth, Mars...all just a matter of time before they all crash into the sun.

On the other hand, Earth is getting further away from the sun in reality, so it's just a matter of time before the earth just keeps getting farther away, not to be confused with crashing into...

Yes, MD, any simplistic analysis of gravity without including orbits and other factors will conclude in crashes. You think this adds to the discussion in some pertinent way?

 

[Fednis48]

You included three quotes attributed to me that were not from any of my posts. Probably an error in copying the initial quote marker. Maybe you could edit the posts to indicate the quotes true origin/link.

Whoops! Sorry about that. The quotes were from Aqueous Id. By the time you pointed it out, the edit button had expired, but it looks like both you and Aqueous figured it out, so no harm done. Unrelatedly, I'd like to echo exchemist in saying there's sometimes value in working through the Newtonian case and then discussing how the relativistic case might differ; if relativistic corrections somehow exactly cancel out this effect, I would be very surprised, and I would love to see that result.

To be honest, I've been debating closing or moving this thread since Friday - especially given that RJBerry so far seems to have ignored my points (as well as other peoples points).

Please don't. You seem awfully exasperated that we're debating an effect too small to be of any practical meaning, but as a theorist, I believe that delving into technicalities can be a good way to gain deeper understanding of the underlying theory. NeddyBate, exchemist, and I have all explicitly stated we've learned something from this discussion, and I have to assume others have as well. If nothing else, rest assured that no one in this thread is saying Galileo made any serious mistakes, or trying to denigrate his landmark contribution to physics.

No, once you locate the x-axis that contains X and x, you have assigned a reference frame and lost that independence you seek.

Ok, we really need to sort this out. I'll be as precise as I can. In Newtonian mechanics, the quantity \(|\vec{x_{rel}}|\equiv |\vec{X}-\vec{x}|\)is independent of choice of coordinates. In a problem like this where all motion is along a single axis, we can define the scalars X and x as the two objects' positions along the axis of motion. This gives \(x_{rel}\equiv X-x\) as the coordinate-independent quantity. If you disagree with this, we have more significant problems than the technicality this thread is about.

No, for example if the ship turns 45°, the observer on the dock will not see the ball pass in front of the referee at the same time the referee sees the ball cross the net.

No, see above.

Your formulas were written by someone who evidently never even took algebra.

Are you seriously getting on my case for forgetting to specify that my position coordinates are measured along the Earth<->ball axis? Fine. My formula is valid for all coordinate systems in which the x-axis is parallel to the Earth<->ball separation. This doesn't change my conclusion at all, since choice of axis orientation is independent of choice of origin.

Your use of proprietary speech, your inability to reformulate a freshman physics problem, dragging in a circuitous reference to the Hamiltonian, and your messed up math--together spell "invention".

That just illustrates that you have little or no familiarity with the typical texts for Physics I.

Did you really take Classical Mechanics? Impossible, methinks, unless you are a ghost from the 17th c. But Lo, methinks you never matriculated at all.

This is getting surreal. Classical Mechanics is a core course for a physics degree at every university I know of; it basically covers the dynamics of systems in the non-relativistic, non-quantum limit. Hamiltonian mechanics are a staple technique for such systems, and it's beyond bizarre that you're accusing me of "inventing" things when you could just go to the Wikipedia page on Hamiltonian dynamics and verify all my techniques/terminology for yourself.

A little honesty would go a long way here. You ought to state up front that you don't actually apply physics skills in your daily work.

Don't presume. My current research involves the dynamics of pairs of interacting atoms in the absence of external forces; I literally do calculations in relative coordinates for money at least 3 times a week.

Wrong. The frame is inertial iff the g-field is homogeneous and isotropic throughout.
...
No, your origin cuts across the Earth's g-field, therefore it is not properly called 'inertial'.

Where did you get that definition? That's not what "inertial" means. An inertial frame is one that isn't accelerating. In our problem, there is no force on the center of mass of the two objects, so the center of mass is not accelerating. That means the center-of-mass frame is an inertial frame. The Earth is accelerating in this frame, but all inertial frames are in a constant state of rectilinear motion with respect to one another, so any frame co-moving with the Earth frame cannot be inertial. To say that such a frame is inertial because it has a constant g-field is just nonsense.[/tex]

 

[Motor Daddy]

Yes, MD, any simplistic analysis of gravity without including orbits and other factors will conclude in crashes. You think this adds to the discussion in some pertinent way?

Well what the heck, does your gravity just pertain to bowling balls, chains, and golf balls (oh my)?

Does it not also pertain to Suns, and planets, and moons (oh my)?

 

[OnlyMe]

Whoops! Sorry about that. The quotes were from Aqueous Id. By the time you pointed it out, the edit button had expired, but it looks like both you and Aqueous figured it out, so no harm done. Unrelatedly, I'd like to echo exchemist in saying there's sometimes value in working through the Newtonian case and then discussing how the relativistic case might differ; if relativistic corrections somehow exactly cancel out this effect, I would be very surprised, and I would love to see that result.

Yes, the edit funtion is rather short fused. I'd also like to see the Like button returned and quote links reach back further than the top of the page. But what the heck a work in progress...

Moving on - though my initial responses were knee jerk reaction... The point was that the way the OP is layed out there are only two objects to drop that have any real world implications. Including a SMBH was..., well.., and an unidentified fouth mass has no signifcance until a mass value has been attributed to it. That leaves the golf ball and bowling ball and there would be no difference, between Newtonian physics and GR, in the results of any drop test from 10 meters with either of those. As has been pointed out the center of mass for the two body problem and the center of mass for the earth itself are not measureably different... and the distance the earth would move during the drop is even less significant to either. The relativistic results would be the same as Galileo's conclusions. In fact that was what I was trying to point out by repeatedly linking the Standford press release, The force of gravity is the same for atoms and baseballs {1999} http://news.stanford.edu/pr/99/atomgravity990825.html.

Only when the dropped mass becomes large enough that the two body center of mass is significantly different than the earth's center of mass..., does the point that RJ introduced become an issue. I believe that much has been agreed on, but I often misunderstand so, I reasert it as my understanding of the discussion at this point.

In real world terms the earth's rotation and orbital velocity.., even the presence and relative location of the moon durin the drop, could have an equal or even more significant affect on the results than any difference between Galileo's, the Newtonian or relativistic case, of the drop of a golf ball or bowling ball from 10 meters. Any variation would be far less than expected systemic error... Unmeasurable.

BTW I added Farsight's clocks to try and emphasize the FoR issue. There is no realistic difference in the center of mass of the earth and the two body problem when you are talking about golf balls and bowling balls.

I still hold that there is no even technically wrong issue with Galileo's conclusions. They have been repeated many times with greater and greater accuracy and always returned results that only refined his work. Nothing that has overturned it... On this issue anyway.

 

[RJBeery]

I still hold that there is no even technically wrong issue with Galileo's conclusions. They have been repeated many times with greater and greater accuracy and always returned results that only refined his work. Nothing that has overturned it... On this issue anyway.

OnlyMe, perhaps you could state specifically what you believe Galileo's conclusions were. Depending on your wording, we could all be agreement on this issue, or I'm about to quote THREE college professors who say that you're wrong.

 

[Trippy]

Trippy, you're trying to defend the MAN when I am clearly not attacking him.

No I'm not, I have specifically and explicitly addressed your assertions, and pointed out how they are wrong. You dismiss them as being inane in order to avoid addressing them. That's you're issue, not mine.

Which of my three points do you disagree with?
1. The statement is not technically wrong because you're considering situations it was never intended to be applied to
2. The result is physically meaningless anyway.
3. The result is realistically meaningless anyway.

Perhaps instead of dodging you could show some integrity and address these points. I mean even on the first point alone...

The issue was settled when the good professor chimed in.

Which part, the part where they agreed with my demonstration that the motion is somewhere between virtually meaningless and completely meaningless (a tiny fraction of the radius of a proton).

Watching you and the others do your linguistic gymnastics trying to rescue your ego in the face of defeat is certainly entertaining but it's getting old.

Now you're just being rude, which is ironic really - especially given that you agreed with me at first.

Hell, it wouldn't surprise me if you moved this thread to pseudo-science just to erase it from the pages of history.

Right, so you're emotionally invested in this discussion now, so now what?

The only thing I've asked you to do is address my three points, so far you've done nothing but be rude to me.