Acceleration and Newtons gravitation

[bassplayer142]

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.

 

[Hootenanny]

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?

 

[DavidWhitbeck]

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.

 

[bassplayer142]

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.

 

[D H]

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.

 

[DavidWhitbeck]

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.

 

[Shooting Star]

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?

 

[D H]

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.

 

[DavidWhitbeck]

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.

 

[bassplayer142]

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.

 

[DavidWhitbeck]

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.