Conceptual Force of Gravity Acceleration 2 vs 3 objects

[PhysicsNowApple]

Hello,

I've been thinking about the nature of Newton's Laws and had a question about two scenarios where I'm noticing a difference in the way the problem is framed and the outcome. I cannot sleep because it is wracking my mind.

Let me start by stating what I know to be true going into the problem.

1. Net forces cause accelerations
2. Newton's 2nd Law states that object experience accelerations that are proportional to the net force acting on the object and inversely proportional to the mass of the object.
3. The force of Gravity is constant over small distances.
4. Near Earth, objects of differing masses experience the same free fall accelerations regardless of mass because inertial mass and gravitational mass are nearly identical. (In a vacuum)Ok, let's start with scenario 1. In scenario 1, there are three objects to consider. All three objects are assumed to have all their mass concentrated in a particle in the center of them.
Object 1: The Earth
Object 2: A 10kg sphere that is 1 meter above the Earth's surface.
Object 3: A 100kg sphere that is 1 meter above the Earth's surface.

The two spheres are 1 meter from each other horizontally. Their gravitational interaction is not the focus of this problem.

The spheres are allowed to free fall. In this case, Object 3 experiences a larger gravitational force from the Earth than Object 2. Also, because of Object 3's greater mass and the fact that inertial mass and gravitational mass are equal, Object 3 is more resistant to acceleration so it experiences the same acceleration as Object 2.

In this case, the forces acting on Object 2 and 3 are not equal, yet they experience equal accelerations toward Earth. The forces in question are gravitational forces.

Let's try scenario 2 with only the 2 objects. I want to take those same 2 spheres, but now teleport them to the void in between galaxies. In this space, the gravitational influences of the "nearby" galaxies are negligible. Now the only forces acting on the objects are the equal and opposite action-reaction pair of their mutual gravitational attraction.

In this case, I believe that they should experience accelerations that are inversely proportional to their masses. I mean, that I believe the larger sphere will undergo a smaller acceleration than the smaller sphere due to its greater mass. Am i correct?

In this case, in contrast to case 1, the FORCES are equal, yet the accelerations they experience are not. Also, to clarify, I understand why the forces are equal in this case compared to the first but my question regards their accelerations.

Is my reasoning sound? Is there a discrepancy here? Or is this entirely fine?

Why wouldn't they experience equal accelerations in the second case considering that the forces in question are gravitational forces?Thank you for taking the time to read.

 

[CWatters]

Repeat case 1 with just the Earth and object 2. The acceleration of the Earth is different to object 2.

This is the same situation as case 2.

 

[PhysicsNowApple]

Repeat case 1 with just the Earth and object 2. The acceleration of the Earth is different to object 2.

This is the same situation as case 2.

Yes, that is clear to me. I framed that in the title to point out that i see the difference is between the framing of two or three objects. But that does not answer the question of why in the 2 object scenario we are ok with accelerations between Earth and one of the spheres being different while also simultaneously being ok with the two spheres having equal accelerations when it is a 3 object scenario.

 

[Ibix]

If I were you, I would write out the gravitational forces and Newton's second law for the components you are interested in explicitly in terms of masses $m_1$, $m_2$, and $m_3$ and cancel. In your first scenario you have different forces, but the same terms cancel in the two equations, leading to Galileo's famous observation. In the second case, you have the same force, but different terms cancel. You don't expect equal accelerations.

 

[Dale]

The two spheres are 1 meter from each other horizontally. Their gravitational interaction is not the focus of this problem.

Now the only forces acting on the objects are the equal and opposite action-reaction pair of their mutual gravitational attraction

Why wouldn't they experience equal accelerations in the second case considering that the forces in question are gravitational forces

In the first case you are explicitly ignoring their mutual attraction. In the second case you are explicitly focusing on their mutual attraction. Why should it be surprising that you get different results when in one part you focus on something that you deliberately neglect in the other part?

The same laws are followed in the same way in each case. If you make different approximations then you will get different results.