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According to Wikipedia "The gravitational potential $V$ at a distance $x$ from a point mass of mass $M$ can be defined as the work $W$ that needs to be done by an external agent to bring a unit mass in from infinity to that point: $$V(\vec{x}) = \frac{1}{m}\int^x _{\infty} \vec{F}\cdot d\vec{x} = \frac{1}{m}\int^x _{\infty}G\frac{Mm}{x^2}dx$$

where $G$ is the gravitational constant, and $\vec{F}$ is the gravitational force."

Question 1: What is the meaning of "from infinity" here? We could integrate "from infinity" in a variety of ways, yet it seems we must integrate in the direction away from the point mass in question, right? Since otherwise we would get a different integral.

Question 2: How is the Gravitational Potential defined when there are multiple point masses in space?

Qmechanic
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Sam
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1 Answers1

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Question (1):

"Infinity" here refers to a distance that is at an infinite radial distance away from the source mass $M$. Essentially at $x = \infty$.

Note that this defintion is incomplete. The definite integral you have written is equal to the gravitational potential difference between the two points (one at a radial distance $x$ and the other at a radial distance of $\infty$). However, it is usually a convention to assign a gravitational potential of zero to points that are at an infinte radial distance away from the source mass $M$, so the difference reduces to the gravitatonal potential of the point at a radial distance $x$ from the source mass.

Question (2):

The potential associated with a distribution of point masses at a point is the scalar sum of the individual potentials at that point due to the masses constituting the system. This is often called the superposition of gravitational potential.

It is: $$V(x) = -\sum_{i = 1}^{n} G \frac{m_i}{|x-x_i|}$$ for a system of n-particles having masses $m_1, m_2, ..., m_n$ at distances $x_1, x_2...,x_n$ from the origin respectively at a point which is at a distance $x$ from the origin. This, of course also takes into consideration that the potential at $x = \infty$ is zero.

Hope this helps.

Cross
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