Question

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 .

Answer 1

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

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

$X_{e-m} = \frac{m_1x_1+m_2x_2}{m_1+m_2}$

Where

$m_1$= Mass of earth

$x_1$ = Distance to center of mass for earth

$m_2$ = mass of moon

$x_2$= Distance from moon to earth (to center of mass)

Replacing the values we have

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

$X_{e-m} = \frac{27.93*10^{27}}{6.07*10^{24}}$

$X_{e-m} = 4.599*10^3km$

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

$X_{e-m-s}=\frac{(m_1+m_2)X_{e-m}+m_3x_3}{m_1+m_2+m_3}$

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

$X_{e-m-s}= 1.5*10^8km$

So we have the two center of mass:

System Earth-Moon equal to $4.599*10^3$km

System Earth-Moon-Sun equal to $1.5*10^8$km