A rocket is launched from the Earth towards the Moon, following a straight trajectory, that joins the centers of the two bodies. Since the mass of the Earth M_E is approximately 81 times the mass of the Moon M_M, determine the point in the trajectory where the magnitude of the gravitational fields due to the Earth and the Moon canceling each other out. Consider the Earth-Moon system isolated from the rest of the Universe, the system is stationary and with the total mass of each body concentrated at its center.


Problem data:

• Earth Mass:    M_E;
• Moon Mass:    M_M;
• Ratio between the masses of the Earth and the Moon:    M_E = 81M_M .

Problem diagram:

As the problem considers the masses of the Earth and the Moon concentrated in their centers, the problem reduces to two points representing the Earth and the Moon with a distance d between them and the rocket a point, of mass m, at a distance x from the Earth. (Figure 1).


We assume the positive direction oriented from Earth to Moon.
The gravitational force of attraction, \( {\vec{F}}_{rE} \), acts between the Earth and the rocket, and between the Moon and the rocket, the gravitational force of attraction, \( {\vec{F}}_{rM} \).

Solution

The force of gravitational attraction is given by Newton's Law of Universal Gravitation Applying expression (I) to the Earth-rocket system, where r = x Applying again expression (I) to the Moon-rocket system, where r = d−x For the magnitude of the gravitational fields due to the Earth and the Moon to cancel each other out, we must impose the following condition substituting the expressions (II) and (III) into the expression above canceling the Universal Gravitational Constant G and the rocket mass m from both sides of the equation, and substituting the ratio between the masses of the Earth and the Moon given in the problem