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In this problem, you will calculate the location of the center of mass for the Earth-Moon system, and then you will calculate the center of mass of the Earth-Moon-Sun system. The mass of the Moon is 7.35×1022 kg , the mass of the Earth is 6.00×1024 kg , and the mass of the sun is 2.00×1030 kg . The distance between the Moon and the Earth is 3.80×105 km . The distance between the Earth and the Sun is 1.50×108 km .

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Assuming that the earth is the center of all coordinates. The center of mass of the earth will be located at the coordinate [tex]x_1 = 0[/tex]

To calculate the center of mass of the Earth-moon we require to proceed to the calculations given by,

[tex]X_{e-m} = \frac{m_1x_1+m_2x_2}{m_1+m_2}[/tex]

Where

[tex]m_1[/tex]= Mass of earth

[tex]x_1[/tex] = Distance to center of mass for earth

[tex]m_2[/tex] = mass of moon

[tex]x_2[/tex]= Distance from moon to earth (to center of mass)

Replacing the values we have

[tex]X_{e-m} = \frac{(6*10^{24})(0)+(7.35*10^{22})(3.8*10^5)}{6*10^{24}+0.0735*10^{24}}[/tex]

[tex]X_{e-m} = \frac{27.93*10^{27}}{6.07*10^{24}}[/tex]

[tex]X_{e-m} = 4.599*10^3km[/tex]

We extrapolate this procedure to the sun, but now including the three bodies, in this way

[tex]X_{e-m-s}=\frac{(m_1+m_2)X_{e-m}+m_3x_3}{m_1+m_2+m_3}[/tex]

[tex]X_{e-m-s}=\frac{(6.07*10^{24})(4.599*10^3)+(2*10^{30})(1.5*10^8)}{6.07*10^{24}+2*10^{30}}[/tex]

[tex]X_{e-m-s}= 1.5*10^8km[/tex]

So we have the two center of mass:

System Earth-Moon equal to [tex]4.599*10^3[/tex]km

System Earth-Moon-Sun equal to [tex]1.5*10^8[/tex]km

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