Answer to Question #249468 in Physics for Jubin

Question

A spacecraft is going from the earth to the moon. Find such a position from the earth over there The gravitational force is zero. Given
Earth mass = 6.0 × 10^24 kg. Moon mass = 74 × 10^22 kg; The distance between the center of the earth and the center of the moon = 3.8 x 10^8m

Answer 1

Lets set the origin at the Earth. Then the force at distance $r$ from the Earth is given as follows:

$$F_1 = G\dfrac{M_Em}{r^2}$$

where $G = 6.67\times 10^{-11}\dfrac{m^3}{kg\cdot s^2}$ is the gravitational constant, $M_E$ is the mass of the Earth, $m$ is the mass of the ship.

The force from the Moon at the same point is given as follows:

$$F_2 = G\dfrac{M_Mm}{(d-r)^2}$$

where $d$ is the distance between the center of the Earth and the center of the Moon. The total gravitational force is 0 if:

$$F_1=F_2\\$$ $$G\dfrac{M_Em}{r^2} = G\dfrac{M_Mm}{(d-r)^2}\\ \dfrac{M_E}{r^2} = \dfrac{M_M}{(d-r)^2}$$

Expressing $r$, find:

$$(M_E-M_M)r^2 - 2dM_Er + M_Ed^2 = 0$$

Substituting the values and solving the quadratic equation, obtain:

$$r\approx 2.8\times 10^8m$$

Answer. $2.8\times 10^8m$.