# How far from the center of the earth is the center of mass of the earth + moon s

How far from the center of the earth is the center of mass of the earth + moon system? Data for the earth and moon can be found inside the back cover of the book.
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Ryan Willis
The center of mass of a system consists of a set of obect numbered i=1,2,3... where i has mass ${m}_{i}$ and is located at position $\left({x}_{i},{y}_{i}\right)$ is given by:
${x}_{cm}=\frac{1}{M}\sum _{i}{m}_{i}{x}_{i}$ (1)
${y}_{cm}=\frac{1}{m}\sum _{i}{m}_{i}{y}_{i}$ (2)
Given
From the back cover of the book, the mass of earth is , the mass of Moon is , and the distnace between them is $R=3.84×{10}^{8}$ m
Calculations
We can conisder Earth and Moon to be along a straight line, and we ca take this straight line to be the x-axis.
Also we take the origin to be at the center of Earth, so Earths position is ${x}_{e}=0$ and Moon's position is ${x}_{m}=R$
Since our system consists of Earrth and Moon only, we can write equation (1) for our system as follows:
${x}_{c}m=\frac{{m}_{e}{x}_{e}+{m}_{m}{x}_{m}}{{m}_{e}+{m}_{m}}$
Now , we enter the given values, so we get:

${x}_{cm}=4.67×{10}^{6}$ m
Therefore, the center of mass of the Earth-Moon system is at $4.67×{10}^{6}$ m from the center of Earth which is less than Earth's radius $6.37×1{-}^{6}$m