Thought Experiment - Center of Gravity

[bigsplit]

It's been so long since I've posted, I am not even sure I can write this scenario correctly.

Imagine the center of gravity (CG) between the Sun and the Earth. It could be any two bodies really with some beef to them. The CG is moving as the earth orbits the sun. Hit pause and place an apple at the CG. Hit play. What happens to the apple?

[Grey]

It gets vaporized, because the center of gravity of the Earth and Sun is inside the Sun. In a general case, you need to know more about the system to know what will happen. It's important to realize that the center of gravity of a system is generally not a gravitationally neutral point. For two objects of different mass, the center of gravity will be closer to the more massive object, and a small test object will be pulled that way.

[Jens]

Hit pause and place an apple at the CG. Hit play. What happens to the apple?

Another complicating factor: the earth and sun are not stationary relative to one another (in other words, they move apart, then move closer again, then move apart, etc.). When you place the object, what velocity do you impart to it? I suppose you are imagining giving it the same velocity as the COG, but just wanted to make sure.

[bigsplit]

It gets vaporized, because the center of gravity of the Earth and Sun is inside the Sun. In a general case, you need to know more about the system to know what will happen. It's important to realize that the center of gravity of a system is generally not a gravitationally neutral point. For two objects of different mass, the center of gravity will be closer to the more massive object, and a small test object will be pulled that way.

I would think that the center of gravity would be much closer to earth, maybe even inside the earth. Please explain.

[bigsplit]

What would happen if the velocity were the same. What would happen if it were "stationary".

[Ken G]

I would think that the center of gravity would be much closer to earth, maybe even inside the earth. Please explain.

Is the COG of you and the Earth closer to you, or closer to the center of the Earth? How about the COG of you and a dust particle? What if there are ten dust particles, where is the COG then? The COG concept would sure be hard to use if it followed around the smallest object! Maybe you don't realize the the Sun's gravity is much stronger than the Earth's, that its gravity rules the whole solar system.

What would happen if the velocity were the same. What would happen if it were "stationary".

If you only have the Sun and the Earth, with no other objects, then the natural inertial reference frame is the one where the COG doesn't move at all, no orbit, nothing. Then the two questions you asked here are the same question. And it has been answered-- the force would be much stronger toward the Sun, both because the Sun has much more mass, and because the Sun is much closer. The Earth's gravity would not have any important effect at all, it would be just what the Sun does to things.

[bigsplit]

Is the COG of you and the Earth closer to you, or closer to the center of the Earth? How about the COG of you and a dust particle? What if there are ten dust particles, where is the COG then? The COG concept would sure be hard to use if it followed around the smallest object! Maybe you don't realize the the Sun's gravity is much stronger than the Earth's, that its gravity rules the whole solar system.
.

OK - I know what you are saying. Told you I would struggle with the question. I am talking about the point in space between the earth and the sun where mass would be pulled equally by both bodies. What is that point called?

[mkline55]

Let's see if I understand the problem. First, you want the point where the apple is pulled equally in opposite directions by the gravity of the Sun and the gravity of the Earth. I don't have a name for that, but it is just a math problem, and it is closest to the Earth. Second, you inserted the apple with no motion of its own relative to the Earth and Sun. Then you allowed the Earth to continue its orbit. I would assume that the Earth's gravitational pull on the apple decreases because the Earth is moving relative to the apple. Then the apple would begin acceleration toward the Sun, but not exactly toward the center, since it is still also influenced by the Earth's gravity. That's also a math problem, but a little beyond my reckoning.

[bigsplit]

Let's see if I understand the problem. First, you want the point where the apple is pulled equally in opposite directions by the gravity of the Sun and the gravity of the Earth. I don't have a name for that, but it is just a math problem, and it is closest to the Earth. Second, you inserted the apple with no motion of its own relative to the Earth and Sun. Then you allowed the Earth to continue its orbit. I would assume that the Earth's gravitational pull on the apple decreases because the Earth is moving relative to the apple. Then the apple would begin acceleration toward the Sun, but not exactly toward the center, since it is still also influenced by the Earth's gravity. That's also a math problem, but a little beyond my reckoning.

There is no name for it? Your explaination is what I was guessing, but I didn’t know if that point would “catch it and it would follow the path of that point.

[grant hutchison]

I've seen it called the "gravitational neutral point" or "gravitational null point", but they're not terms of art in celestial mechanics, because that point isn't of any real relevance.

If you give your apple a sideways velocity so that it keeps pace with revolution of the Earth around the Sun, you find it still won't stay in that position. It needs to be a little closer to the sun, so that the sun's gravity (reduced by the opposing pull of the Earth) is just enough to hold it in an orbit with the same period of revolution as the Earth. That's one of the Lagrange points - which are stable (or in this case, metastable) locations where you can place objects so that they corotate with a two-body system. Notation varies, but it's often numbered L1.

Grant Hutchison

[DaveC426913]

OK - I know what you are saying. Told you I would struggle with the question. I am talking about the point in space between the earth and the sun where mass would be pulled equally by both bodies. What is that point called?

Ah! You are talking about Earth's Lagrange points, specifically L1.

https://en.wikipedia.org/wiki/Lagrangian_point

The apple will remain in place relative to the Earth/Sun system.

[bigsplit]

I've seen it called the "gravitational neutral point" or "gravitational null point", but they're not terms of art in celestial mechanics, because that point isn't of any real relevance.

If you give your apple a sideways velocity so that it keeps pace with revolution of the Earth around the Sun, you find it still won't stay in that position. It needs to be a little closer to the sun, so that the sun's gravity (reduced by the opposing pull of the Earth) is just enough to hold it in an orbit with the same period of revolution as the Earth. That's one of the Lagrange points - which are stable (or in this case, metastable) locations where you can place objects so that they corotate with a two-body system. Notation varies, but it's often numbered L1.

Grant Hutchison

Thanks Grant.

[grant hutchison]

Ah! You are talking about Earth's Lagrange points, specifically L1.

https://en.wikipedia.org/wiki/Lagrangian_point

The apple will remain in place relative to the Earth/Sun system.

The "gravitational null point" is actually not the same thing as the L1 point. The L1 point corotates with the Earth-Sun system, so an object at L1 needs some Sun-ward acceleration to keep it in place - that is, it's on the sunward side of the gravitational null point.

Grant Hutchison

[DaveC426913]

The "gravitational null point" is actually not the same thing as the L1 point. The L1 point corotates with the Earth-Sun system, so an object at L1 needs some Sun-ward acceleration to keep it in place - that is, it's on the sunward side of the gravitational null point.

Sure, but now it's down to an interpretation of what the OP considers "placing" it there.

[George]

There is no name for it? Your explaination is what I was guessing, but I didn’t know if that point would “catch it and it would follow the path of that point.

Expanding on what Ken stated, imagine a see-saw to address the c.g. Consider how far a normal person would have to extend out on the see-saw where the other person weighs 330,000x more than the normal person and is sitting just one foot from the fulcrum?

[Ken G]

It's actually a pretty common mistake to think that the center of gravity of an object is a point that does not experience any net gravity from the object itself. This is only true for an object with a high degree of symmetry, like a football or a barbell. For a hockey stick, the atom located at the center of gravity does experience a net force of gravity from the hockey stick. This does not mean the hockey stick, isolated in free space, would have its center of gravity start to accelerate-- the hockey stick is held together by internal forces, which would compensate the gravity on the center of mass point, such that the atom at that point would not experience any net force. But it would experience a net internal stress force, to balance the net gravity force that it would also experience. If the hockey stick were pulverized into powder, the atom at the center of gravity would start to accelerate, changing the shape of the hockey stick. The center of gravity would of course not move, but the atoms at the center of gravity would be changing.

[bigsplit]

It's actually a pretty common mistake to think that the center of gravity of an object is a point that does not experience any net gravity from the object itself. This is only true for an object with a high degree of symmetry, like a football or a barbell. For a hockey stick, the atom located at the center of gravity does experience a net force of gravity from the hockey stick. This does not mean the hockey stick, isolated in free space, would have its center of gravity start to accelerate-- the hockey stick is held together by internal forces, which would compensate the gravity on the center of mass point, such that the atom at that point would not experience any net force. But it would experience a net internal stress force, to balance the net gravity force that it would also experience. If the hockey stick were pulverized into powder, the atom at the center of gravity would start to accelerate, changing the shape of the hockey stick. The center of gravity would of course not move, but the atoms at the center of gravity would be changing.

Thanks for all the input. I learned a lot. So if I may, I have a follow up. What would happen if you replace the apple with a clock at the null point in terms of time dialation?

[DaveC426913]

Thanks for all the input. I learned a lot. So if I may, I have a follow up. What would happen if you replace the apple with a clock at the null point in terms of time dialation?

It would tick away at 1 second per second, as all clocks do.

The question is: where are you standing when you measure it?

[bigsplit]

It would tick away at 1 second per second, as all clocks do.

The question is: where are you standing when you measure it?

I understand. Where we are now would be fine.

[Ken G]

Thanks for all the input. I learned a lot. So if I may, I have a follow up. What would happen if you replace the apple with a clock at the null point in terms of time dialation?

Gravitational time dilation depends not on the force of gravity, but on the gravitational potential. So the null point is of no particular significance for that, it's generally somewhere down in the potential well and will have gravitational dilation. If you like to think in terms of altitude in a contour map, or digging pits in the sand, a null point is a flat spot somewhere down inside the pit, but the depth in the pit is what determines the dilation.

[grapes]

Gravitational time dilation depends not on the force of gravity, but on the gravitational potential. So the null point is of no particular significance for that, it's generally somewhere down in the potential well and will have gravitational dilation. If you like to think in terms of altitude in a contour map, or digging pits in the sand, a null point is a flat spot somewhere down inside the pit, but the depth in the pit is what determines the dilation.

Or in this case, a lot farther closer to the sun than the earth

Did I just say "farther closer"??

[mkline55]

Or in this case, a lot farther closer to the sun than the earth

Did I just say "farther closer"??

Maybe you meant "least most closest" or "most least closest".

[George]

Gravitational time dilation depends not on the force of gravity, but on the gravitational potential. So the null point is of no particular significance for that, it's generally somewhere down in the potential well and will have gravitational dilation. If you like to think in terms of altitude in a contour map, or digging pits in the sand, a null point is a flat spot somewhere down inside the pit, but the depth in the pit is what determines the dilation.

Sand pits are kinda fun. So one pit (Sun's grav well) is very deep but very wide such that the slopes are gentle (until one goes farther [and gets] closer to the center) and the other pit is small (Earth's grave well) but with steeper slopes. The flat pit bottom would be the surface of each body. So, between the two, the high point would be L1 and, from pit bottom, we would see the L1 clock tick its fastest compared to ours. So is my wit on pits a fit or not worth spit?

[Ken G]

A question for the philosophers of the ages.

[DaveC426913]

I understand. Where we are now would be fine.

Everything in the solar system is to some extent, in the gravity well of the sun. We are in a deeper well than the L1 point.
So, we would observe it ticking ever so slightly faster.

Experiments were done to demonstrate this, comparing clocks in orbit* with clocks on the ground, and bore it out.

*taking into account the confounding dilation due to SR

[Ken G]

Everything in the solar system is to some extent, in the gravity well of the sun. We are in a deeper well than the L1 point.
So, we would observe it ticking ever so slightly faster.

Slower. (ETA: I see now that "it" means the clock at the L1 point, so the statement was correct.)

[George]

Slower.

Are you saying Earthers would see an L1 clock tick slower due to GR (ignoring SR)? But when brought side by side the L1 clock would be ahead of the Earther's (discounting SR)?

[Ken G]

Are you saying Earthers would see an L1 clock tick slower due to GR (ignoring SR)? But when brought side by side the L1 clock would be ahead of the Earther's (discounting SR)?

I think I misread the English, by "it" Dave meant "at the L1 point," I read it as "our clock." My mistake.

[bigsplit]

I think I misread the English, by "it" Dave meant "at the L1 point," I read it as "our clock." My mistake.

Thanks for all the answers. The gravitational null point is very interesting.