Superposition Principle for Gravitational Field

The superposition principle. When $n$ different masses $m_1, m_2, \dots, m_n$ are all present, the total gravitational field they produce together at any point $P$ is simply the vector sum of the individual fields each one would produce alone:

If instead of discrete point masses we have a continuous distribution of total mass $M$, the same idea is applied using integration.

Worked example (two masses on the axes). Masses $m_1$ and $m_2$ sit on the $y$- and $x$-axes respectively, each a distance $a$ from the origin, and we want the field at the origin $P$. The field due to $m_1$ alone points along $-\hat{j}$ (toward $m_1$) with magnitude $Gm_1/a^2$; the field due to $m_2$ alone points along $-\hat{i}$ with magnitude $Gm_2/a^2$. Adding them as vectors,

and if $m_1=m_2=m$, this simplifies (using $\hat{i}+\hat{j}$ as a single diagonal direction) to $\vec{E}_{total}=-\dfrac{\sqrt{2}\,Gm}{a^2}$ pointing diagonally back toward the origin from the midpoint of $m_1$ and $m_2$ -- the resultant direction is fixed purely by the relative sizes of $m_1$ and $m_2$. …