Question

The mass of the earth is $\mathbf{6}\mathbf{\times }{\mathbf{10}}^{\mathbf{2}}\mathbf{4}\mathbf{\hspace{0.33em}}\mathbf{kg}$, the mass of the moon is$\mathbf{7}\mathbf{\times }{\mathbf{10}}^{\mathbf{22}}\mathbf{\hspace{0.33em}}\mathbf{kg}\mathbf{,}$, and the center-to-center distance is $\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{8}}\mathbf{\hspace{0.33em}}\mathbf{m}$. How far from the center of the earth is the center of the mass of the Earth-Moon system? Note that the Earth’s radius is$\mathbf{6}\mathbf{.}\mathbf{4}\mathbf{\times }{\mathbf{10}}^{\mathbf{6}}\mathbf{\hspace{0.33em}}\mathbf{m}$.

Short answer

The distance of the center of the earth and the center of the mass of the Earth-moon system is$4666666.667\hspace{0.33em}m$ .

Step-by-step solution

Identification of the given data

The given data can be listed below as,

• The mass of the earth is$6\times {10}^{2}4\hspace{0.33em}kg$ .
• The mass of the moon is$7\times {10}^{2}2\hspace{0.33em}kg$ .
• The center-to-center distance of the earth and the moon is$4\times {10}^8\hspace{0.33em}m$ .

Significance of the law of the center of mass of the Earth-moon system

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.

Determination of the distance of the center of the earth to 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, $M_m\hspace{0.33em}and\hspace{0.33em}{M}_E$are the mass of the moon and the earth respectively, $r_m\hspace{0.33em}and\hspace{0.33em}{r}_E$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-

$$\begin{gathered} r_{\text{dist}}=\frac{\left(7\times {10}^{22}\mathrm{kg}\right)\times \left(4\times {10}^8\mathrm{m}\right)}{\left(6\times {10}^{24}\mathrm{kg}\right)+\left(7\times {10}^{22}\mathrm{kg}\right)} \\ {r}_{\text{dist}}=4666666.667\hspace{0.33em}m \end{gathered}$$

Thus, the distance of the center of the earth and the center of the mass of the Earth-moon system is $4666666.667\hspace{0.33em}m.$