Qualitative query regarding Newton's Law of Universal Gravitation

[dukeofsphere]

1. Take the equation F=[G(m₁)(m₂)]/r²

2. Assume a single mass of 20 (units don't matter), which will be divided in the following ways:
2.1. System 1: m₁=20, m₂=0
2.2. System 2: m₁=19, m₂=1
2.3. System 3: m₁=18, m₂=2
2.4. System 4: m₁=17, m₂=3
.
.
.
2.11.System 11: m₁=10, m₂=10

Now...

3. If we keep both G and r constant, then the force is going to depend on the allocation of mass by the factor (m₁)(m₂)

4. The factors for the systems would then be:
4.1. System 1: (m₁)(m₂)=20 (or zero?)
4.2. System 2: (m₁)(m₂)=19
4.3. System 3: (m₁)(m₂)=36
4.4. System 4: (m₁)(m₂)=51
4.5. System 5: (m₁)(m₂)=64
.
.
.
4.11.System 11: (m₁)(m₂)=100

5. My question is this: why does the force of gravity change when the distribution of mass of the overall system doesn't change? This might seem obvious to someone else, it may be mathematically obvious why, but I still don't understand intuitively why this happens. I would have thought that if you have 20 units of mass in one system and 20 units of mass in another, all else being equal, the force in the two systems would be the same.

What am I missing here, conceptually?

 

[Pythagorean]

20*0 = 0

I'm not sure what you mean with your question:

"I would have thought that if you have 20 units of mass in one system and 20 units of mass in another, all else being equal, the force in the two systems would be the same."

Each system has two masses in it. A single mass alone in the universe doesn't exert any gravitational force at all. Force is something that happens between a system of two or more masses.

So both masses in each system are important to the question of force, thus

F = G(m1*m2)/r^2

Notice that it would make no difference if you switched mass1 and mass 2, reinforcing the concept that force exists between the two, not in one of them.

 

[dukeofsphere]

20*0 = 0

I'm not sure what you mean with your question:

"I would have thought that if you have 20 units of mass in one system and 20 units of mass in another, all else being equal, the force in the two systems would be the same."

Each system has two masses in it. A single mass alone in the universe doesn't exert any gravitational force at all. Force is something that happens between a system of two or more masses.

So both masses in each system are important to the question of force, thus

F = G(m1*m2)/r^2

Notice that it would make no difference if you switched mass1 and mass 2, reinforcing the concept that force exists between the two, not in one of them.

Okay, for example let's say m₁=19.5 and m₂=0.5. Total mass of the system is 20. The force is 9.75(G)/r²

Now, instead of 19.5/0.5, let's say 10/10. Mass of the total system is still 20. But the force is now 100(G)/r²

Maybe my problem is that it doesn't really make sense to talk about the mass of the "overall" system?

But I still don't understand, if there is still the same amount of stuff in space in both systems, why the gravitational force is so different.

 

[dukeofsphere]

And, also, I should have said that I'm not thinking about this in strict classical terms, but in more of a "geodesic/geometric" kind of a way.

For example, take the Sun. if there were no planets around and it was just the Sun sitting there, its mass would still distort space-time; there would still be a field. I take this to mean that if there were a secondary mass-particle, the force would then present itself in the classical sense.

Now, let's say the Sun is split in two. Now we have two stars. Is it now true that the force between the two - or the distortion in space-time caused by the two new masses - is much greater than if the Sun had not split?

 

[Feldoh]

Maybe my problem is that it doesn't really make sense to talk about the mass of the "overall" system?

But I still don't understand, if there is still the same amount of stuff in space in both systems, why the gravitational force is so different.

Your problem is that you're not considering the spatial part of Newton's Law of Gravitation. It does makes sense to talk about the overall mass of a system -- BUT we do not just talk about the sum of the mass, we talk about something that is spatially dependent called the center of mass. The center of mass is just the average of all the mass included in a system at the averaged coordinate position.

Purely from an algebraic standpoint we can see that if you change the ratio m₁m₂ while keeping the distance of the two particles the same that the force on each particle will be different.

 

[Pythagorean]

Okay, for example let's say m₁=19.5 and m₂=0.5. Total mass of the system is 20. The force is 9.75(G)/r²

Now, instead of 19.5/0.5, let's say 10/10. Mass of the total system is still 20. But the force is now 100(G)/r²

Maybe my problem is that it doesn't really make sense to talk about the mass of the "overall" system?

But I still don't understand, if there is still the same amount of stuff in space in both systems, why the gravitational force is so different.

quite simply, the sum of two masses has nothing to do with the force between the two masses.

What might help some of your confusion is our discussion is based on point particles. If you want to talk about separating the sun, you have to start talking about mass distribution (which, as Feldoh pointed out, is a spatial consideration of the geometric arrangement of a bunch of tiny point particles) then the problem gets much more complicated, especially since you won't have spherical mass sources anymore.

If you separate the two halves of the sun some distance, the energy (to do the work to separate the halves against their gravitational pull) would have to come from somewhere else besides the gravitational fields of each half off the sun (an explosion in the sun?) but either way, energy is conserved, energy won't come out of nowhere.

 

[dukeofsphere]

quite simply, the sum of two masses has nothing to do with the force between the two masses.

What might help some of your confusion is our discussion is based on point particles. If you want to talk about separating the sun, you have to start talking about mass distribution (which, as Feldoh pointed out, is a spatial consideration of the geometric arrangement of a bunch of tiny point particles) then the problem gets much more complicated, especially since you won't have spherical mass sources anymore.

Yeah, best stick to point particles; I still need to think about the center of gravity concept, and how it relates to what I'm trying to get across.

I'm also getting confused with respect to the motion of the two particles. In my mental experiment, I'm considering two particles which are not revolving around a common center of mass; I'm thinking of two particles heading straight for each other on a collision course. I guess if you were to take a snapshot of the collision-bound particles in transit, it would look just like the orbiting scenario, in terms of the force-vectors and center of mass. [STRIKE]Or would it? I'm not so sure now.[/STRIKE]

Also, if the mass-points are revolving, then the gravitational force is constant over time (assuming perfectly circular orbit). But if they're heading at each other, the distance changes, and so the force changes with it with respect to time.

Gonna sleep on it.

 

[Feldoh]

Also, if the mass-points are revolving, then the gravitational force is constant over time (assuming perfectly circular orbit). But if they're heading at each other, the distance changes, and so the force changes with it with respect to time.

Gonna sleep on it.

You are correct. In the case of two particles orbiting around in circular orbits you might want to look up the two-body problem. It might help to explain this case. Basically the idea is that we consider one of the particles stationary and that the other is revolving around the stationary particle. This is, in principle, how the sun/earth orbit system works -- typically we consider the sun to be stationary while on earth.

You are right again in the second case. The force will constantly be changing as each particle moves straight towards the other.