91Ó°ÊÓ

The given data can be listed below as,

• The mass of the earth is$6×{10}^{2}4 kg$ .
• The mass of the moon is$7×{10}^{2}2 kg$ .
• The center-to-center distance of the earth and the moon is$4×{10}^8 m$ .

The law of the center of mass states that if a rigid object is pushed at its center of mass, then the object will always continue to move.

The equation of the position of the center of mass gives the distance of the center of mass of the Earth-Moon system.

From the law of the center of mass, the distance of the earth’s center to the Earth-Moon system is expressed as:

$r_{dist}=\frac{M_E{r}_E}{M_m+M_E}+\frac{M_m{r}_m}{M_m+M_E}$

Here, $r_{dist}$is the distance of the earth’s center to the Earth-Moon system, $M_m and M_E$are the mass of the moon and the earth respectively, $r_m and r_E$are the center to center distance of the earth and the moon and center to center distance of the earth that is 0.

Substituting the values in the above equation, we get-

$$\begin{gathered} r_{\text{dist}}=\frac{\left(7×{10}^{22}\mathrm{kg}\right)×\left(4×{10}^8\mathrm{m}\right)}{\left(6×{10}^{24}\mathrm{kg}\right)+\left(7×{10}^{22}\mathrm{kg}\right)} \\ {r}_{\text{dist}}=4666666.667 m \end{gathered}$$

Thus, the distance of the center of the earth and the center of the mass of the Earth-moon system is $4666666.667 m.$