alka8q7

2021-11-18

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.

Ryan Willis

Beginner2021-11-19Added 15 answers

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 $({x}_{i},{y}_{i})$ 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${m}_{e}=5.98\times {10}^{24}\text{}kg$ , the mass of Moon is ${m}_{m}=7.36\times {10}^{22}\text{}kg$ , and the distnace between them is $R=3.84\times {10}^{8}$ 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:

${x}_{cm}=\frac{0+(7.36\times {10}^{22}\text{}kg)(3.84\times {10}^{8}\text{}m)}{7.36\times {10}^{22}+5.98\times {10}^{24}\text{}kg}=4.67\times {10}^{6}m$

$x}_{cm}=4.67\times {10}^{6$ m

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

Given

From the back cover of the book, the mass of earth is

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

Since our system consists of Earrth and Moon only, we can write equation (1) for our system as follows:

Now , we enter the given values, so we get:

Therefore, the center of mass of the Earth-Moon system is at

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