Question:

The masses of the Earth and Moon are $5.98 \times 10^{24} \textrm{ kg}$ and $7.35 \times 10^{22} \textrm{ kg}$, respectively, and their centers are separated by $3.84 \times 10^8 \textrm{ m}$.

- Where is the CM of the Earth–Moon system located?
- What can you say about the motion of the Earth–Moon system about the Sun, and of the Earth and Moon separately about the Sun?

Source: Giancoli, Douglas C., Physics: Principles with Applications, 7th Edition, 2014.

Quick Answer:

- $4.7\times 10^6 \textrm{ m from the center of the Earth}$
- see video

### Transcript for this Giancoli solution

This is Giancoli Answers with Mr. Dychko. Let's choose the origin of our coordinate system to be the center of the Earth and we have the distance from the center of the Earth to the center of the moon is 3.84 times 10 to the 8 meters and so we'll find the center of mass of this Earth-moon system. So the*x*-coordinate of the center of mass, well, first of all just say the

*y*-coordinate of center of mass is along this line connecting the Earth and the moon because there's an equal amount of mass above and below evenly distributed so by symmetry, you can say the center of mass

*y*-coordinate is 0 and so the

*x*-coordinate is gonna be mass of the Earth times its distance to the origin which is zero so this term is actually zero and plus the mass of the moon times its distance from the origin which is the distance between the Earth and the moon center's that's gonna be multiplied by the mass of the moon— 7.35 times 10 to the 22 kilograms— divided by the total mass of the Earth and the moon and that gives about 4.7 times 10 to the 6 meters from the centre of the Earth is the centre of mass. So it's gonna be somewhere around here, I guess, pretty close to the Earth; this whole distance is on the scale of 10 to the 8 whereas this distance here is times 10 to the 6 from the centre of the Earth so this full distance is more than a 100 times greater than this distance here so, you know, it should be maybe even really close to the surface. Then the motion's is the next question so we have the Sun here in the middle of our solar system and the centre of mass of the Earth-moon is gonna go around in the elliptical orbit that we are familiar with from our pictures of the solar system and then the moon itself or the Earth itself will be rotating around the centre of mass of the Earth-moon system as it travels in this orbit around the Sun. So we are gonna have this high frequency revolution around the center of mass of the Earth-moon system at the same time as this low frequency once per year revolution of the Earth-moon centre of mass around the Sun.