Do Newton's Laws Predict Equal Forces Between Two Bodies of Similar Mass?

[hmsmatthew]

I am struggling to understand Newtons law:

F = ma = GMm/r^2

a = GM/r^2

In the above equation, the small m's cancel out to give a constant acceleration due to gravity from the perspective of big M. I consider a planet and a small moon (little m) a certain distance away from the planet, (assuming small moon is not in motion or in orbit i.e. it will just fall towards the planet).

Normaly we would say the moon would fall towards the Earth with the acceleration defined above, in the same way as an object on the planet, independant of the mass of the object.

But from the small moons perspective we can use the above equation to say it will attract the planet towards it with a certain acceleration derived in the same way (a smaller acceleration).

Is the correct interpretation of this to say the accelerations cancel? or add together ? Both values for the acceleration of the moon towards the planet cannot be true at the same time.

What if the moon is the same size as the planet? then does the moon fall towards the planet whilst the planet towards the moon adding the accelerations?

If the moon is more massive than the planet then surely the planet would fall to the moon.

If i could "hold" the planets apart and they had the same mass/diameter would there be a mid point between them where i could place an object which would be stationary because the forces towards each planet are of equal strength ?

I am very confused about this and do not know how to explain things mathematically

Any help would be greatly appreciated,

Matt

 

[The legend]

Is the correct interpretation of this to say the accelerations cancel? or add together ? Both values for the acceleration of the moon towards the planet cannot be true at the same time.

Matt

None, actually. It's true that the moon too exerts the same force , but it exerts that force ON EARTH. And Earth exerts force ON MOON. So you cannot add or subtract the accelerations or forces acting since they are on different bodies.

And yes, both celestial bodies "fall" toward's each other. But since the Earth is soooo large compared to the moon, it moves very very little.

Hope you get the answer for the other questions from this explanation.

 

[Doc Al]

Normaly we would say the moon would fall towards the Earth with the acceleration defined above, in the same way as an object on the planet, independant of the mass of the object.

But from the small moons perspective we can use the above equation to say it will attract the planet towards it with a certain acceleration derived in the same way (a smaller acceleration).

Both objects exert force on each other and thus accelerate. When one is small and the other huge, we generally ignore the much smaller acceleration of the larger object. Since the moon's mass is about 1/80 that of the earth, it's acceleration would be correspondingly smaller.

Is the correct interpretation of this to say the accelerations cancel? or add together ? Both values for the acceleration of the moon towards the planet cannot be true at the same time.

The accelerations--in this simple case--are best thought of as being with respect to some inertial frame. Each would accelerate towards the other, the relative acceleration would be the sum of the individual accelerations.

What if the moon is the same size as the planet? then does the moon fall towards the planet whilst the planet towards the moon adding the accelerations?

Sure.

If the moon is more massive than the planet then surely the planet would fall to the moon.

They accelerate toward each other.

If i could "hold" the planets apart and they had the same mass/diameter would there be a mid point between them where i could place an object which would be stationary because the forces towards each planet are of equal strength ?

Sure. Imagine an object between the two planets. Use Newton's law of gravity to express the force on the object from each planet, which will be a function of distance. Solve for the point where those forces are equal. If the planets have equal mass, that point will (obviously) be equally distant from each.

 

[TurtleMeister]

Once you understand the frame of reference then things will be much clearer. Newton's laws, such as a = GM/r^2, require that the frame of reference be inertial. In the case of the two body problem the inertial frame of reference is the inertial center of mass of the two bodies. For example, if both bodies have the same mass then the center of mass will be at the mid point between them. If body 1 is more massive than body 2 then the center of mass will be closer to body 1. If both bodies are initially stationary, they will accelerate toward that point (the center of mass). And they will meet at that point. Obviously if the center of mass is closer to body 1 then body 1 will have the lower acceleration and body 2 will have the higher acceleration.

The relative acceleration is NOT based on the inertial center of mass. In that case the frame of reference is either one of the two bodies. It is the absolute sum of the accelerations of the two bodies toward the center of mass. Or it can be calculated from the mass of the two bodies: A_rel = G(m1+m2)/r^2

 

[paisiello]

I am struggling to understand Newtons law:

If i could "hold" the planets apart and they had the same mass/diameter would there be a mid point between them where i could place an object which would be stationary because the forces towards each planet are of equal strength ?

Matt

Yes, this is one of the Lagrange points. Any two body system (e.g. the earth-moon system) has five such points where the object would be seen as stationary relative to the other objects.