Galileo's thought experiment

[LAHLH]

Galileo is said to have reasoned that all masses must fall with the same acceleration despite their differing weights as follows.

Imagine having a heavy object say weighing 200N and a lighter mass say weighing 100N, then we may suppose that the heavier object fall will an acceleration 2a, and the lighter with acceleration a (as people may have been inclined to think before Galileo).

Now imagine connecting these two objects with a weightless tether/chain, what will happen? Well the heavier object will pull on the lighter object, to increase its acceleration, and conversely the lighter object would pull in the opposite direction on the heavier mass to decrease its acceleration. The result is that the composite object would fall with an intermediate acceleration between a and 2a.

But by our initial model, the composite object (having weight 300N), should fall with an acceleration faster than both the individual objects, of 3a. Thus we have been led into a contradiction and seen that the model has failed somewhere.

This is how Galileo is said to have from reason alone established that all objects would fall with the same acceleration on Earth independent of their weights.
-----------------------------------------------------------------------------------

My question is why can this not be applied in electrostatics to lead to a similar contradiction. For example let's suppose we assumed an object of "weight" (this is really F=qE) C falls at acceleration a, and a second "heavier" object of "weight" 2C, falls at acceleration 2a.

So to visualise this better perhaps imagine instead of a gravitational field an Electric field radially inward to the earth, so now E plays the role of g, and q that of gravitational mass (or "gravitational charge"), m.

Now we apply the same trick attaching a massless tether somehow, and deduce the composite object accelerates somewhere intermediate between a and 2a.

But by our initial reasoning the composite object of weight 3C, should fall at rate 3a.

Of course we know it really wouldn't because the inertial mass has also doubled assuming the first two charges have equal masses, so would Galileo's thought experiment lead to a misleading and false contradiction in this case, making us think that all charges must accelerate at the same rate.

Notice if the two objects had equal inertial masses m, and equal charges q, then indeed both of them would accelerate at rate a, AND the composite object with charge 2q and mass 2m, would also accelerate at mass a. Meaning Galileo's contradiction would have led to the correct result.

---------------------------------------------------------------------------------
This is all obvious to us since we know about the equivalence of gravitational and inertial mass, but Galileo would not have known this equivalence prior to his argument (otherwise why would he be making such an argument to deduce the rate at which objects fall, since knowing the equivalence of inertial and gravitational mass in advance would mean he already knew all masses would equally accelerate, making his thought experiment somewhat circular).

So my question really boils down to how did Galileo know inertial mass was equivalent to gravitational "charge" in advance of such a thought experiment which clearly relies on it?

To re-emphasise how it relies on it, imagine inertial and gravitational mass were not equivalent, then object one has weight $$W=m_{g} g$$, and let's say acceleration a, (and let's say inertial mass $$m_i$$), and object two has the weight $$3W=3m_{g} g$$, and assuming what people may have thought prior to Galileo therefore acceleration 3a. But now instead of what we know to be true now, that this implies inertial mass $$3m_i$$, let's assume inertial mass could be different to gravitational charge, and be $$2m_i$$.

Now performing Galileo's thought experiment, we would attach them via weightless chain, and see the heavier mass should increase the acceleration of the lighter and vice versa, until the composites acceleration is between a and 3a.
But by our acceleration is proportional to weight theorem the people prior to Galileo may have assumed we see the composite object (of weight 4W) should be expected to accelerate at 4a. Therefore we have Galileo's contradiction again, and according to him are forced into assuming all such objects must have the same acceleration.

But of course we know such an assumption would be incorrect, for in fact if object 1's acceleration is a, then object 2's is 1.5a $$3W/2m_i$$, and as for the composite object (of weight 4W, and $$m_i=m_i+2m_i=3m_i$$ ) its acceleration would be 1.33a

So the way I see it is Galileo's thought experiments only lead to the correct results, if one assumes in advance the equivalence of inertial mass and gravitational mass, otherwise of course things don't accelerate equally in uniform fields. But if Galileo new about such things before hand from some other means, then what's the point in such a thought experiment?

Perhaps he discovered this really from his actual experiments rolling things down inclines, and the thought experiment is just a flawed myth?

I think the best this thought experiment can tell you is that acceleration is not just dependent on force, but there must be some additive quantity (what we now call inertia) that each body possesses that acceleration also depends on.

 

[TurtleMeister]

My question is why can this not be applied in electrostatics to lead to a similar contradiction.

Because Coulomb's Law does not include mass?

This is all obvious to us since we know about the equivalence of gravitational and inertial mass, but Galileo would not have known this equivalence prior to his argument (otherwise why would he be making such an argument to deduce the rate at which objects fall, since knowing the equivalence of inertial and gravitational mass in advance would mean he already knew all masses would equally accelerate, making his thought experiment somewhat circular).

So my question really boils down to how did Galileo know inertial mass was equivalent to gravitational "charge" in advance of such a thought experiment which clearly relies on it?

Your question is contradictory. You first say that Galileo would not have known about the equivalence principle prior to his argument, and then you ask how did he know about the equivalence principle in advance of his thought experiment? I do not think that prior knowledge of the equivalence principle would be required for Galileo to conceive of his thought experiment. Anyone of his time could have dropped some rocks and observed their behavior.

 

[Pythagorean]

LAHLH:

1) Masses have another property besides how they're affected by gravitational pull. They also have inertia, which is their resistance to motion.

A particle with larger charge does not make it resist motion more, like mass does. Remember that:

a = F/m

so the bigger mass has a lower acceleration for the same force. You can not equivalently say:

a = F/q

This is the major difference between gravity and electrodynamic force (besides their not being negative mass, of course). That mass also influences the inertia of the object.

2) The gravitational constant g is actually an approximation that assumes a constant distance from the Earth. The real gravitational force is calculated using big G and the distant and magnitudes of the two masses in question.

$$F = G \frac{m_1 m_2}{r^2} = G \frac{m_E m}{r_E^2} = mg$$

so $$g = G\frac{m_E}{r_E^2}$$

where $$r_E$$ is the radius of the Earth and $$m_E$$ is the mass of the Earth.
In reality, the object is not always at the exact radius of Earth. It varies some small distance around it. But that distance is so small compared to the radius of Earth itself, that we can justify approximating it. In reality, there's some change in g as the object gets closer to Earth.

I'm not sure this (2) is actually relevant now that I think of it...

 

[LAHLH]

Because Coulomb's Law does not include mass?

You misunderstand my point I fear. The difference between the electrostatic force and gravitational force is its charge is not the same as inertial mass, therefore in the same electrostatic field E, different charges q and 2q,say, would accelerate differently. It just so happens that "gravitational charge", i.e. the mass you see in the equation $$F=m_{g}g$$, also is exactly the same as inertial mass and therefore ($$a=m_{g}g/m_{i}=g$$) all masses accelerate equally.

Your question is contradictory. You first say that Galileo would not have known about the equivalence principle prior to his argument, and then you ask how did he know about the equivalence principle in advance of his thought experiment?

You're quite right on this. I should really have asked can Galileo's thought experiment be useful for anything more than deducing the concept of inertia, and it is not a strong enough thought experiment to deduce all bodies in a field accelerate equally.

I do not think that prior knowledge of the equivalence principle would be required for Galileo to conceive of his thought experiment. Anyone of his time could have dropped some rocks and observed their behavior.

Yes, but if Galileo had "dropped rocks" already, or had rolled things down inclined planes as is now believed to the case, and observed they do accelerate equally, and had made this an assumption before his thought experiment, then the thought experiment would be completely useless and circular. He would be putting in the assumption that masses accelerate equally into a thought experiment whose very existence was supposed to deduce that fact.

 

[LAHLH]

LAHLH:

1) Masses have another property besides how they're affected by gravitational pull. They also have inertia, which is their resistance to motion.

A particle with larger charge does not make it resist motion more, like mass does. Remember that:

a = F/m

so the bigger mass has a lower acceleration for the same force. You can not equivalently say:

a = F/q

This is the major difference between gravity and electrodynamic force (besides their not being negative mass, of course). That mass also influences the inertia of the object.

I find it odd that you would try to tell me this, when it is a key feature in the arguments I make? Again my arguments were:

If Galileo before the thought experiment did not know that "gravitational charge", $$m_g$$ was equal to inertial mass $$m_i$$ (which is valid since, if he did know this fact then he could have just used $$a=m_{g}g/m_{i}=g$$ therefore deducing that all bodies accelerate equally prior to a thought experiment set out to deduce this fact!), then Galileo's thought experiment alone, would not be sufficient to deduce that all masses accelerate equally.

I used the EM field, simply to illustrate an example of such a field that has a "charge" not equal to inertial mass, just as indeed may have been the case for gravity if we weren't so fortunate to have it that $$m_{g}=m_i$$. Then indeed in EM things do not fall equally in a uniform field. Galileo not knowing prior to his actual experiment with inclined planes etc, could not have known that gravity was not a force like EM.
So if we performed his thought experiment in EM, tethering two charges together, as I did in my OP, we find (if we make the false assumption of the the olden times, that objects of greater "weight" or electrostatic force in this case, fall faster) that the composite body falls intermediately between the two independent objects (because of the pulling back by the less charged mass, and pulling down by the higher charged), when by our false ansatz, we expected the composite body to fall faster than both. Therefore we find the concept of inertia, but it would a massive leap (and a false leap here) to deduce because of this that all charges fall equally. So when Galileo makes this leap in the gravitational case he either:

1)knows $$m_g=m_i$$ rendering his thought experiment circular and futile in the first place
2) doesn't know $$m_g=m_i$$ and therefore is unjustified in making such a huge leap that all things accelerate equally, since just like in the case of an EM type field, they do not generally.

What I believe is Galileo's thought experiment really leads one into the conclusion: objects acceleration doesn't just depend on force (weight) there must be another additive quantity that it depends on (the thing we now call inertia) [this is quite significant because you could imagine people before Galileo thinking if an object has twice the weight (force) it will accelerate twice has fast etc] . What Galileo's thought experiment does not show is that this inertial mass equals gravitational mass, and it does not show that all bodies fall equally.

*NB I use charge quite generally sometimes, this could mean the usual q, in F=qE, or could mean the gravitational charge of $$F=m_{g}g$$, which I take care to distinguish from the inertial mass $$m_i$$ of F=m_{i}a [/tex].

Thanks

 

[TurtleMeister]

You misunderstand my point I fear. The difference between the electrostatic force and gravitational force is its charge is not the same as inertial mass, therefore in the same electrostatic field E, different charges q and 2q,say, would accelerate differently. It just so happens that "gravitational charge", i.e. the mass you see in the equation $$F=m_{g}g$$, also is exactly the same as inertial mass and therefore ($$a=m_{g}g/m_{i}=g$$) all masses accelerate equally.

I agree that there is no equivalence principle between electric charge and inertia. And that is the same reason you cannot draw an analogy between Galileo's experiment and electric charge and reach the same conclusion. Or do I still misunderstand?

I should really have asked can Galileo's thought experiment be useful for anything more than deducing the concept of inertia, and it is not a strong enough thought experiment to deduce all bodies in a field accelerate equally.

Well, I think it's a pretty strong argument for the case of gravity. It was not meant to apply to any other force.

Yes, but if Galileo had "dropped rocks" already, or had rolled things down inclined planes as is now believed to the case, and observed they do accelerate equally, and had made this an assumption before his thought experiment, then the thought experiment would be completely useless and circular. He would be putting in the assumption that masses accelerate equally into a thought experiment whose very existence was supposed to deduce that fact.

After a little further study I can see that Galileo's thought experiment was based more logic and not so much on previous experiments. It's purpose was to prove that Aristotle's view was wrong through contradiction or reductio ad absurdum.

 

[LAHLH]

I agree that there is no equivalence principle between electric charge and inertia. And that is the same reason you cannot draw an analogy between Galileo's experiment and electric charge and reach the same conclusion. Or do I still misunderstand?

I wasn't trying to draw an exact analogy between EM and gravity, only so much as that gravity would be somewhat like EM if inertial mass was not equal to gravitational mass. In retrospect my argument would have probably been clearer just saying "imagine a Galileo investigating a universe where $$m_{g}$$ is not equal to $$m_i$$..." and taking things from there. I just wanted to give a real life example of such a force where the analogy of gravitational mass (i.e. electrostatic charge), is obviously not equal to the inertial mass, meaning objects don't have uniform acceleration in it's field generally.

Well, I think it's a pretty strong argument for the case of gravity. It was not meant to apply to any other force.

So let me abandon the whole EM analogy, and proceed as I said I probably should have in the paragraph just above...Imagine a young Galileo sat pondering (before he had chance to do his inclined planes experiment and the like), but he lives now in a universe where $$m_{g} \neq m_i$$. His peers of the day, believe that the weightier (weight being the force the Earth is exerting on the object, $$W=m_{g}g$$, just to be absolutely definite here) an object is the greater that objects acceleration is.

Galileo thinks about two objects one of weight W and the other of weight 2W. His peers who think weight is the be all and end all, would say if object 1 accelerates at a, object two will then accelerate at 2a. Galileo then asks himself, what if I chain these two objects together, then surely the lighter (less weighty) object accelerating more slowly will drag the heavier object to a lower acceleration than 2a, and conversely, the lighter object will be dragged down more by the heavier mass, to an acceleration greater than a. Thus the composite mass must accelerate intermediately between a and 2a. But by the logic of my peers, they would have the composite mass, of weight 3W, accelerating at 3a!

He should now conclude that weight ($$W=m_{g}g$$) isn't the be all and end all when it comes to acceleration, and his peers are wrong. He should conclude that when adding bodies together there is another important additive quantity, that must slow the composite bodies acceleration to be somewhere between a and 2a. the inertial mass , $$m_i$$. (In this universe of course acceleration of composite body $$a_c=\frac{3W}{m_{i_1}+m_{i_2}}$$, which is only equal to a, in the very special case that inertial and gravitational mass have equivalence, i.e. when $$m_{i_1}=m_g$$ and $$m_{i_2}=2m_g$$ )

So the crux of my point really is that in just doing a thought experiment, Galileo could not be sure he wasn't in such a universe, therefore the leap to all things have equal acceleration, at the very end of his thought experiment does not follow. (You can't argue that Galileo had already done inclination experiments do deduce this equal acceleration of all bodies, since then if his thought experiment relies on that to supposedly show all things accelerate equally, then it is circular)

[I only used analogy to the EM field because it is mathematically very similar to a universe where $$m_g \neq m_i$$ just with $$m_g$$ replaced by electrostatic charge q. ]

In this universe where $$m_{g}\neq m_i$$ if Galileo at this point made the leap from this point in his thought experiment to the conclusion that all things accelerate equally, he would be committing a fallacy.

It's purpose was to prove that Aristotle's view was wrong through contradiction or reductio ad absurdum.

Well if that is it's purpose then it achieves it successfully,it successfully deduces the property of inertia, and it's important role in determining accelerations, it just doesn't demand things accelerate equally, as it is often popularised to show. That Galileo could not do by mind alone, only with the aid of experiment.

 

[TurtleMeister]

Ok, I understand now. I do not know if Galileo had any prior experimental evidence before he conceived of his thought experiment. Personally, I think it was probable that he did have some experimental evidence and just conceived of the thought experiment to add strength to his argument.

In this universe where $$m_{g}\neq m_i$$ if Galileo at this point made the leap from this point in his thought experiment to the conclusion that all things accelerate equally, he would be committing a fallacy.

What does $$m_g$$ refer to? Active gravitational mass or passive gravitational mass?

 

[LAHLH]

Ok, I understand now. I do not know if Galileo had any prior experimental evidence before he conceived of his thought experiment. Personally, I think it was probable that he did have some experimental evidence and just conceived of the thought experiment to add strength to his argument.

Yeah I think now, that that is probably the case.

What does $$m_g$$ refer to? Active gravitational mass or passive gravitational mass?

It is the mass in the equation $$Weight=mg$$, where g is the acceleration due to the Earth's gravity. So I guess it's passive mass, as it is a measure of how the body reacts to the Earth's uniform field, not the active mass of the body creating the field around it. I guess we could go even further and say that the active and passive gravitational masses don't need to be equal either in some strange universe. Although I don't think the active mass being different, would play a role in Galileo's thought experiment anyway? as we're are neglecting the two objects gravitational attraction to each other. We're only considering the two objects as moving under the Earth's uniform gravity, not setting up fields of their own, etc.

 

[Pythagorean]

His experimental evidence involved bowling balls rolling down a ramp and court drummers for time (as they didn't have stopwatches).

He was inspired in church, watching a massive chandalier swing like a pendulum, timing it with his heartbeat. I think his rationale was that all masses swing at the same rate as long as the line holding them is the same length.

 

[TurtleMeister]

It is the mass in the equation $$Weight=mg$$, where g is the acceleration due to the Earth's gravity. So I guess it's passive mass, as it is a measure of how the body reacts to the Earth's uniform field, not the active mass of the body creating the field around it.

Ok, I was unsure because in your previous posts you made reference to "gravitational charge" which is the equivalent of active gravitational mass. And some of your comments could only be interpreted as active gravitational mass.

I have always had a problem with the m_p distinction. It's hard for me to imagine a non equivalence between it and m_i. They seem like the same thing to me. Or two different manifestations of the same thing.

I guess we could go even further and say that the active and passive gravitational masses don't need to be equal either in some strange universe. Although I don't think the active mass being different, would play a role in Galileo's thought experiment anyway? as we're are neglecting the two objects gravitational attraction to each other. We're only considering the two objects as moving under the Earth's uniform gravity, not setting up fields of their own, etc.

Yes, you are correct. $$m_a \neq m_i$$ would not be detectable for ordinary sized objects.

 

[Andrew Mason]

So my question really boils down to how did Galileo know inertial mass was equivalent to gravitational "charge" in advance of such a thought experiment which clearly relies on it?

I don't think Galileo thought of gravity in terms of a force that was proportional to inertial mass. That was Newton. All he intended to show was that, ignoring resistance, all objects fell with the same acceleration.

Galileo simply observed: 1. that all objects tend to fall. 2. they all fall in the same direction - down and 3. identical masses fall at the same rate (they do not pull away from each other when falling).

From those observations, one could conclude that objects fall at the same rate regardless of mass. Otherwise, as you have pointed out, tethering two identical masses together would make them fall at a different rate than if they were separate.

One could not make these same observations about charge. Not all objects move in an electric field. Those that do, do not necessarily move in the same direction. Otherwise identical masses can move at different rates and in different directions in an electric field. Tethering two identical masses can affect their rate of motion.

AM

 

[Pythagorean]

He also found that they accelerated. The experiment Was something like this:

you get a small handheld ramp and put a small bumper at 1 ft, 4 ft, 9 ft, and so on and roll a ball down it. Assuming the bumpers don't take away a significant amount of motion, when you roll the ball down a ramp, it dings in uniform beat as it hits the bumpers.

 

[LAHLH]

Ok, I was unsure because in your previous posts you made reference to "gravitational charge" which is the equivalent of active gravitational mass. And some of your comments could only be interpreted as active gravitational mass.

ah, I see how that language could seem misleading, yes I definitely meant passive mass.

I have always had a problem with the m_p distinction. It's hard for me to imagine a non equivalence between it and m_i. They seem like the same thing to me. Or two different manifestations of the same thing.

It's certainly an interesting question why they are equal, I know Einstein based his GR around this equivalence principle, and that this insight was previously motivated by Mach. Dennis Sciama (Hawking's advisor) has an interesting related paper 'On the origin of inertia', explained here http://physics.fullerton.edu/~jimw/general/inertia/index.htm

 

[TurtleMeister]

It's certainly an interesting question why they are equal, I know Einstein based his GR around this equivalence principle, and that this insight was previously motivated by Mach. Dennis Sciama (Hawking's advisor) has an interesting related paper 'On the origin of inertia', explained here http://physics.fullerton.edu/~jimw/general/inertia/index.htm

Well, the thing that puzzles me is not why they are equal but why anyone would think that they could not be equal. I don't understand why m_p is even needed to describe the dynamics of gravity. Couldn't we do it with just m_a and m_i?

Thanks for the link. But it's a little beyond my level of comprehension.

 

[LAHLH]

I have the opposite problem, I could kind of imagine a universe where $$m_p \neq m_i$$, for me then gravity would be just like any other force like EM, and we could call $$q_g$$ and see an analogue to $$F=qE$$ etc. It does very much surprise me that $$m_i$$ is equal to $$m_p$$ (and $$m_a$$ for that matter), and I think this is a very deep fact that I don't fully understand myself yet.

This observation leads one to general relativity in some sense, the fact that gravity is a distinguished force in that it acts to accelerate all objects, and accelerate them equally, whereas other forces such as EM only act on electrically charged particles like electrons, so one can observe other particles that remain non accelerated.
This leads you to the idea of a preferred class of trajectories through spacetime, "freely falling frames", were freely falling, means non accelerated by any forces other than gravity. (All objects subject only to gravity, i.e. not EM etc, will appear non accelerating in these frames).
Notice you couldn't get such a set of frames, in EM, for a start two particles with opposite charges accelerate in different directions, and in general objects will have different charges and accelerate at different rates.
So this equivalence principle (known as the weak equivalence principle) is very important for GR's foundations.
Another way this is often stated is imagine you were in a tightly box/elevator with no windows, performing experiments to determine the motion of particles, because of the universality of gravity there is no way you distinguish uniform acceleration to gravity in this box. If the elevator was jerked up suddenly it would appear to you like a gravitational field downwards had been turned on etc. This isn't true of EM, since if an electric field is switched on, some particles like neutrons will sit unnaccelerated, but under the elevator being jerked up everything including the neutrons would be accelerated downward.
(See the start of ch2 Sean Carroll's Spacetime and Geometry for a better discussion of these ideas, that I have just nicked here, the book is free online somewhere)

So the fact inertial mass and "gravitational charge" or whatever you want to call it, are equal is very important indeed, but as of now I can't claim to understand why they are equal.

 

[LAHLH]

As an interesting experimental side note Eotvos performed experiments showing the equivalence of gravitational and inertial mass to $$10^-12$$ decimal places

 

[TurtleMeister]

I have the opposite problem, I could kind of imagine a universe where $$m_p \neq m_i$$, for me then gravity would be just like any other force like EM, and we could call $$q_g$$ and see an analogue to $$F=qE$$ etc.

So the q in $$F=qE$$ would be m_p? If so, then what would be the analogue to m_a?

As an interesting experimental side note Eotvos performed experiments showing the equivalence of gravitational and inertial mass to $$10^-12$$ decimal places

I think this http://www.npl.washington.edu/eotwash/publications/pdf/schlamminger08.pdf" at the University Of Washington is the best to date. There has been a lot of tests of the equivalence of m_p and m_i. But there has been very little interest for testing the equivalence of m_a. The best laboratory test to date was the Kreuzer experiment of 1966. And that was only good to 5 decimal places. This is one of my areas of interest.

 

[LAHLH]

So the q in $$F=qE$$ would be m_p? If so, then what would be the analogue to m_a?

Well, active mass of a body can be thought of as mass that generates gravity (whereas passive is mass that which responds to gravity). So if you look at $$F=G\frac{m_a}{r^2}$$, this is the gravitational force field per unit test mass set up by the body (Note if $$m_a$$ here is active mass of Earth and r is radius of Earth this number is g) . It is analogous to the electric field in EM $$F\frac{1}{4\pi\epsilon_0} {q_a}{r^2}$$.

Wouldn't it be a very weird universe if passive and active mass could be different?
For example take two bodies with equal active mass $$m_a$$, they will set up equal gravitational fields for the other to move in (or equivalently equal potentials) $$\phi(r)$$. But let's say one had passive mass $$m_p$$ and the other $$2m_p$$, then each would "feel" this field differently, the one with greater mass, would feel a force twice as big toward the other body, as that body feels toward it ($$F_1=-m_p \grad(\phi)$$ and $$F_2=-2m_p \grad(\phi)$$ ).

So the whole idea of Newton's third law would seem to break down, as you don't have equal and opposite forces any more, when two bodies like this interact gravitationally.

The whole idea of putting a certain mass in when calculating the potential or force field a body creates, and a different mass for how the body reacts to gravity seems very unnatural to me. It seems to go against the symmetry of the equation: $$F=G\frac{m_1 m_2}{r^2}$$

 

[TurtleMeister]

Well, active mass of a body can be thought of as mass that generates gravity (whereas passive is mass that which responds to gravity). So if you look at $$F=G\frac{m_a}{r^2}$$, this is the gravitational force field per unit test mass set up by the body (Note if $$m_a$$ here is active mass of Earth and r is radius of Earth this number is g) . It is analogous to the electric field in EM $$F\frac{1}{4\pi\epsilon_0} {q_a}{r^2}$$.

Ok, I see what you're getting at. You're thinking that if $$m_p \neq m_i$$ then the force of gravity is no longer linked to it's inertial mass, just like electric charge. However, if this is what you're thinking then you must also say that $$m_a \neq m_i$$. So let's assume that's true. Then the force of gravity would be the same as electric charge (substituting G for k):

$$F = \frac{Gq_1q_2}{r^2}$$

So the (active charge?) of q1 would act on the (passive charge?) of q2 and the (active charge?) of q2 would act on the (passive charge?) of q1. Do I have it right?

Wouldn't it be a very weird universe if passive and active mass could be different?
For example take two bodies with equal active mass $$m_a$$, they will set up equal gravitational fields for the other to move in (or equivalently equal potentials) $$\phi(r)$$. But let's say one had passive mass $$m_p$$ and the other $$2m_p$$, then each would "feel" this field differently, the one with greater mass, would feel a force twice as big toward the other body, as that body feels toward it ($$F_1=-m_p \grad(\phi)$$ and $$F_2=-2m_p \grad(\phi)$$ ).

Shouldn't that be:

$$F_1 = m^{p}_1(\phi_2)$$ and $$F_2 = -2m^{p}_2(\phi_1)$$

Since the active masses are equal $$\phi_1 = \phi_2$$

So the whole idea of Newton's third law would seem to break down, as you don't have equal and opposite forces any more, when two bodies like this interact gravitationally.

Yes, I agree. Using the current methodology F₁ must equal -F₂ to prevent a third law violation. But why should we assume that Newton's third law would break down? If it did then we could live in a world of perpetual motion machines and unlimited free energy. So to avoid that mess we must conclude that either a non equivalence between m_i and m_p/m_a is impossible or something is wrong with the methodology.

The whole idea of putting a certain mass in when calculating the potential or force field a body creates, and a different mass for how the body reacts to gravity seems very unnatural to me. It seems to go against the symmetry of the equation: $$F=G\frac{m_1 m_2}{r^2}$$

I agree.