Acceleration and Newtons gravitation

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Discussion Overview

The discussion revolves around the implications of Newton's laws of motion and gravitation, particularly in the context of two masses interacting gravitationally. Participants explore the concept of acceleration in a system where both masses exert forces on each other, questioning the assumption that one mass can be considered stationary. The scope includes theoretical reasoning and conceptual clarification regarding gravitational interactions and acceleration.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant suggests that the acceleration of a satellite towards a larger mass is only accurate if the larger mass is stationary, proposing that the total acceleration should consider both masses' accelerations.
  • Another participant questions the assumption that the larger mass is stationary and asks for clarification on the reference frame for measuring acceleration.
  • It is noted that Newton's third law implies that the larger mass experiences an equal and opposite force, thus it must also accelerate.
  • A participant acknowledges that while the larger mass isn't stationary, neglecting its acceleration can be justified in cases where its mass is significantly larger than that of the smaller mass.
  • Discussion includes the use of the center-of-mass frame, which allows for a different perspective on the motion of the two bodies without assuming one is stationary.
  • There is a suggestion that the difference in acceleration becomes significant when the masses are comparable, challenging the idea that the acceleration difference is always negligible.
  • Clarifications are made regarding the context of the discussion, with one participant emphasizing that they are referring to two masses at rest that begin to accelerate, rather than orbiting bodies.

Areas of Agreement / Disagreement

Participants express differing views on the assumptions regarding the stationary nature of the larger mass and the implications of neglecting its acceleration. There is no clear consensus, as some participants argue for the validity of considering both accelerations, while others maintain that in practical scenarios, the larger mass's acceleration can often be ignored.

Contextual Notes

Participants highlight that the discussion does not resolve the complexities involved in gravitational interactions, particularly in systems where masses are comparable. The assumptions made regarding the stationary nature of the larger mass and the reference frame for acceleration measurements remain points of contention.

bassplayer142
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I was thinking about Newtons law and come across something interesting. F=ma=GMm/r^2
therefore a=GM/r^2. This is the acceleration of the satellite towards the larger mass. But this acceleration is only accurate assuming that the larger mass is stationary. But f=Ma=GMm/r^2, so there is an acceleration of the larger mass towards the smaller.

Wouldn't the total acceleration or real acceleration be a addition of the two separate accelerations. I know that this is a very small and probably negligible acceleration difference and this has probably already been discussed. Thanks in advance.
 
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bassplayer142 said:
But this acceleration is only accurate assuming that the larger mass is stationary.
What leads you to believe that this is the case? Furthermore, what are you measuring the acceleration relative to?
 
Newton's 3rd Law: the larger mass experiences a force of the same magnitude but opposite direction.

Newton's 2nd Law: F=Ma

What does that imply? That the larger mass accelerates. Newton's Law of Gravitation does not assume that the larger mass is stationary, and in fact if it was, his own laws of motion would be violated.
 
I know that the larger mass isn't stationary. What I'm saying is that if you had two objects in a closed system they would both be accelerating towards each other. Just working out one of the equations above would be incorrect. I'm neglecting relativity too. I would be measuring relative to point where the larger mass was at the beginning.
 
bassplayer142 said:
Just working out one of the equations above would be incorrect.
Applying the laws of physics incorrectly will yield invalid results. Which is exactly why physicists and astronomers don't do that. Newton's law of gravity applies to both bodies.

That said, while it is technically incorrect to ignore the acceleration of the both bodies, in practice one can ignore the acceleration of the larger body if the mass of the smaller body is many orders of magnitude smaller than that of the larger body. For example, artificial satellites in orbit around the Earth.
 
If you use the center-of-mass frame then it will be stationary and the difference between the positions will satisfy the Newton's 2nd Law equation without having to make any assumption about one of them being stationary. And then it's just a differential equation you can solve.
 
bassplayer142 said:
Wouldn't the total acceleration or real acceleration be a addition of the two separate accelerations. I know that this is a very small and probably negligible acceleration difference and this has probably already been discussed.

But why didn't you consider the case where the masses are comparable or equal? The difference wouldn't be negligible then.

The answer has been given by the others, but what did you conclude?
 
DavidWhitbeck said:
If you use the center-of-mass frame then it will be stationary and the difference between the positions will satisfy the Newton's 2nd Law equation without having to make any assumption about one of them being stationary. And then it's just a differential equation you can solve.
A pair of objects orbit their common center of mass. Neither object is stationary. Solving for this motion directly is a bit daunting. The center of mass point of view let's one go back to a body-centered point of view. The motion here is a bit easier to deduce, and from that one can return to the inertial center of mass frame.
 
Yes by "it will be stationary" I meant the center-of-mass. Solving for the motion is not that bad, because it's a classic result whose derivation is reproduced in calculus textbooks frequently.
 
  • #10
I'm not quite referring to two objects orbiting each other but rather two objects in space. Sorry I didn't make that clear before. I understand how you would use the center of mass of a system but I am talking about two masses at rest near each other that start accelerating. It's ok though because my question was answered, thanks.
 
  • #11
bassplayer142 said:
I'm not quite referring to two objects orbiting each other but rather two objects in space. Sorry I didn't make that clear before. I understand how you would use the center of mass of a system but I am talking about two masses at rest near each other that start accelerating. It's ok though because my question was answered, thanks.

It doesn't matter, the center-of-mass is still stationary because the gravitational forces are internal to the system. I am making no assumption about the trajectories being orbits.
 

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