Question

The masses of the Earth and Moon are \({\bf{5}}{\bf{.98 \times 1}}{{\bf{0}}^{{\bf{24}}}}\;{\bf{kg}}\) and \({\bf{7}}{\bf{.35 \times 1}}{{\bf{0}}^{{\bf{22}}}}\;{\bf{kg}}\), respectively, and their centers are separated by \({\bf{3}}{\bf{.84 \times 1}}{{\bf{0}}^{\bf{8}}}\;{\bf{m}}\).

(a) Where is the CM of the Earth–Moon system located?

(b) 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?

Short answer

(a) The center of mass of the Earth-Moon system is\(4.66 \times {10^6}\;{\rm{m}}\)away from the Earth.

(b) The Moon will rotate around the Earth, and the Earth will rotate about the Sun.

Step-by-step solution

Motion of the Moon around the Sun

The movement of the Moon around the Sun is the addition of two motions:

(i) the movement of the Moon about the Earth-Moon CM, and

(ii) the movement of the Earth-Moon CM about the Sun.

Given information

The mass of the Earth is\({M_{\rm{E}}} = 5.98 \times {10^{24}}\;{\rm{kg}}\).

The mass of the Moon is\({M_{\rm{M}}} = 7.35 \times {10^{22}}\;{\rm{kg}}\).

The separation between the center of the Earth and the Moon is\({x_{\rm{M}}} = 3.84 \times {10^8}\;{\rm{m}}\).

Calculate the center of mass of the Earth-Moon system

(a)

Consider the center of the Earth as the origin. Then the distance of the center of the Earth from the CM is\({x_{\rm{E}}} = 0\).

The center of mass of the Earth-Moon system can be calculated as shown below:

\(\begin{array}{c}{x_{{\rm{cm}}}} = \frac{{{M_{\rm{E}}}{x_{\rm{E}}} + {M_{\rm{M}}}{x_{\rm{M}}}}}{{{M_{\rm{E}}} + {M_{\rm{M}}}}}\\{x_{{\rm{cm}}}} = \frac{{\left( {5.98 \times {{10}^{24}}\;{\rm{kg}}} \right)\left( 0 \right) + \left( {7.35 \times {{10}^{22}}\;{\rm{kg}}} \right)\left( {3.84 \times {{10}^8}\;{\rm{m}}} \right)}}{{\left( {5.98 \times {{10}^{24}}\;{\rm{kg}}} \right) + \left( {7.35 \times {{10}^{22}}\;{\rm{kg}}} \right)}}\\{x_{{\rm{CM}}}} = 4.66 \times {10^6}\;{\rm{m}}\end{array}\)

Thus, the center of mass of the Earth-Moon system is \(4.66 \times {10^6}\;{\rm{m}}\) away from the Earth.

Motion of the Earth and Moon system

(b)

As the CM lies very close to the Earth, the Moon rotates around the Earth while the Earth is rotating around the Sun. Here, the Earth is considered as the center of mass of the Earth-Moon system.