Is the Earth's gravitational pull constant?

In summary: So if I had two balls of equal mass and placed them a fixed distance apart, then the force between them would be the same, regardless of what else was happening in the universe. But if I picked up one of the balls and started moving it around, the force between it and the second ball would decrease as the distance between them increased.Welcome to PF.
  • #1
samm
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TL;DR Summary
Is earth pulling ever body with same amount of force?
If yes, then the accelerations would be different for different masses(inertias)? but acceleration due to gravity (g) is constant for all bodies.
If earth pulls different masses with different forces, then 'g' could be constant, but then, gravitational force depends on distance.
As an object falls, the distance between earth and the body decreases, so, force must increase and for a particular mass, the acceleration should also increase.
In the first part, I'm asking about acceleration of freefalling objects with different masses.
In the second part I'm asking about acceleration of one object with decreasing distance.
Please explain where am I getting it wrong.
Thank You!
 
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  • #2
The gravitational force between objects of mass ##m## and ##M## whose centers are distance ##r## apart is $$F=\frac{GMm}{r^2}$$where ##G## is Newton's gravitational constant. You can see that the force is different on objects of different mass ##m## and at different distances.

However, a mass ##m## that has a force ##F## applied to it accelerates at ##a=F/m##. That means that in this case, acceleration due to gravity is $$a=\frac 1m\frac{GMm}{r^2}=\frac{GM}{r^2}$$So the gravitational acceleration only depends on the center-to-center distance between the objects.

So any source that says gravitational force is constant is wrong. Any source that says gravitational acceleration is constant is missing a key qualifier - it's only true at fixed altitude. You can detect variation over a few tens of meters of altitude with equipment easily in reach of a DIY enthusiast.

[Note: The ##F=GMm/r^2## formula is strictly only true outside spherically symmetric masses. Formulae for other situations can be derived, but the fully general formula is a (sometimes messy) integral.]
 
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  • #3
The Earth pulls different masses with different amounts of force according to Newton's law of gravitation ##F = G\frac{m_1m_2}{r^2}##. Gravitational force is proportional to both the mass of the Earth and the mass of the other object. You are correct that the force and acceleration of a given body do increase as it falls, as can be seen from the inverse square relationship to the distance ##r## in the above formula. However, in the case of the Earth, ##r## is the distance of the object's center of mass from the Earth's center of mass (Earth's core), which must be at least equal to the radius of the Earth (over 6000 km). The distance which most objects fall in ordinary life is so tiny in comparison that any change in acceleration over the course of a fall is negligible. It's a good approximation to say that falling objects all experience the same gravitational acceleration ##g##, equal to the gravitational acceleration at the Earth's surface.
 
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  • #4
Welcome to PF.

samm said:
Is earth pulling ever body with same amount of force?
No. The acceleration due to gravity, g, is roughly constant, but the force is dependent on the mass of the object being accelerated.
Force = mass * g .

The acceleration due to the Earth's gravity depends on your height above the Earth's surface, and your distance from the centre of the Earth. There are also small differences depending on topography and geology.

samm said:
As an object falls, the distance between earth and the body decreases, so, force must increase and for a particular mass, the acceleration should also increase.
You had that backwards. As an object falls it gets closer to the Earth, so acceleration, g, increases, and the force then increases.
 
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samm said:
In the second part I'm asking about acceleration of one object with decreasing distance.
Please explain where am I getting it wrong.
Saying that the free fall acceleration ##g## is constant is an approximation.
 
  • #7
In fact, everything is an approximation.
One of the most entertaining facts about the constancy with which things accelerate towards Earth is that no matter how big nor small, it's always the same. (see the green line below)
Even the black hole at the center of our galaxy would accelerate at the same speed as a feather towards Earth.
The only booger is that the Earth would be accelerating towards the black hole quite a bit faster than towards the feather. (the red line)

Acceleration of Earth and Objects. 2023-07-28 at 14.54.54.png

[edit: minor but obvious error in the graph]
 
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  • #8
When talking about this subject, terminology tends to be mixed up quite a lot, and the colloquial use of words tends to vary wildly from their strict physics-mathematical definitions.

If someone says "gravitational pull is constant" you really have to ask for clarification of what exactly they mean by that, because it's a very vague sentence. It could be interpreted as correct or as incorrect, depending on what the person is actually referring to.

Certainly for two given unchanging masses, at a certain given distance, their gravitational pull is constant and doesn't change eg. depending on time or where in the universe they are in. That's why the big G is called "the gravitational constant": As far as we know, it doesn't change. It's one of the big physical constants. It is assumed (with good reason) that G doesn't change over time and is the same everywhere in the universe. That's why saying that "gravitational pull is constant" could be interpreted as a correct statement.

However, if "gravitational pull is constant" is interpreted in another way, it can be considered incorrect. If it's referring to the gravitational force between the two objects (at least if we are talking about Newtonian mechanics), as the gravitational formula clearly shows, this force depends on the masses and their distance, and it can certainly change if those things change. Even if the expression is referring to the acceleration caused by gravity, well, that depends on the distance (although, incidentally, doesn't depend on the masses), so that too can change over time, depending on how the objects move in relation to each other. Certainly acceleration on Earth's orbit is slightly less than on the surface, because of the difference in distance to the Earth's center.

It's an ambiguous expression, and thus it cannot all of its own, without further clarifications, be considered "true" or "false" because it depends on what the person means by it.
 

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