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How can I find the radius of the circle (average) centered at the center of the earth where the gravitational force of the earth cancels the gravitational force of the other celestial bodies? Will the velocity which is required to reach the circumference of the circle be the escape velocity of earth? I think because the net acceleration on the object outside the circle will be directed away from the earth.. so that the body has truly escaped the earths gravity..?

Sorry for my non scientific language im not even a layman.

N.S.JOHN
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  • The gravitational force due to other bodies is very feeble, the force is zero at the center of earth – Courage Dec 26 '15 at 15:31
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    @TheGhostOfPerdition: I think the OP means a circle far away from the surface of the Earth, not deep inside. If so, the Sun gravity will certainly be noticeable. Maybe this XKCD strip may help. – rodrigo Dec 26 '15 at 15:32
  • There will not be a circular locus of points due to the gravitational force of the Sun, as well as the other planets, and their changing positions. Even outside the solar system (which is dominated by the Sun's gravity), the forces due to stars are in a variety of directions. – Bill N Dec 27 '15 at 02:20

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Your question is a bit vague. There are a lot of objects moving around the solar system, and they are not always in the same place.

That said, there are two objects that affect the Earth in a more or less constant way: the Moon and the Sun.

The gravity of two massive bodies rotating each other, provided that one of them is quite bigger than the other (such as the Earth-Moon or the Sun-Earth systems), cancel at several points: those are the Lagrangian points.

The nearest one, the Earth-Moon L1 is about 326000 km high.

About your question:

Will the velocity which is required to reach the circumference of the circle be the escape velocity of earth?

No, the escape velocity of Earth is (by definition) the velocity needed to reach an infinite distance from the surface of the Earth. To reach the Earth-Moon L1 you need less speed because:

  1. It is nearer than infinity (obviously).
  2. You have help from the gravity of the Moon.

That said, there are another point where the Earth and Moon gravity cancel out, deep inside the Earth. Since the gravity as you go deep inside the planet decreases (proportional to the distance to the center) while the Moon gravity changes little (350000 km vs 356000 km is not a big deal), there is a point half-way (I didn't do the calculations) where the gravity of both the moon and the Earth would cancel out.

rodrigo
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  • Won't you need infinite velocity to reach infinite distance as there is acceleration of the earth which we consider always? – N.S.JOHN Dec 26 '15 at 15:47
  • @N.S.JOHN: No, because the gravity acceleration decreases rapidly with the square of the distance. The acceleration decreases much faster than the velocity. That's precisely why the concept of escape velocity exists. – rodrigo Dec 26 '15 at 15:49
  • But it never becomes zero. – N.S.JOHN Dec 26 '15 at 15:49
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    Imagine the sum $ 1/1^2 + 1/2^2 + 1/3^2 ... $. You could think that it will grow indefinitely, but it actually converges to a well known value. An infinite sum of infinitely small values is not necessarily infinite. – rodrigo Dec 26 '15 at 15:51
  • What do u mean by acceleration decreases much faster than velocity? Acceleration is change in velocity! – N.S.JOHN Dec 26 '15 at 15:51
  • @N.S.JOHN: Velocity decreases with time because there is an acceleration in the opposite direction. Acceleration decreases in time because the distance increases, and gravity is weaker at long distances. But gravity decreases with the square of the distance, while velocity decreases linearly with the acceleration. – rodrigo Dec 26 '15 at 15:53