There is a point between the earth and the moon, where the gravitational force on a space ship due to earth and moon together is zero. Find the distance of that point from the earth Given the disatnce between earth and moon is `3.845 x 10^(8) m` and the moon has `1.2 %` of the mass of the earth.

Explanation

Let `M_(1), M_(2)` be the masses of earth and moon
respectively. Then `(M_(2))/(M_(1)) = (1.2)/(100) = 0.012`
Distance between earth and moon, `r = 3.845 xx 10^(8) m`. Let the force of space ship due to earth and moon becomes zero at a point, whose distance from earth is `r_(1)` and from moon be `r_(2)`. Then
`r = r_(1) + r_(2) = 3.845 xx 10^(8) m` ..(i)
At such a point, the gravitational force on space ship due to earth is equal to that due to moon. If `m` is the mass of space ship then
`(GM_(1)m)/(r_(1)^(2)) = (GM_(2)m)/(r_(2)^(2))` or `(r_(2)^(2))/(r_(1)^(2)) = (M_(2))/(M_(1)) = 0.012`
or `r_(2) = sqrt(0.012) r_(1) = 0.11 r_(1)`
From (i), `r_(1) + 0.11 r_(1) = 3.845 xx 10^(8) m`
or `1.11 r_(1) = 3.845 xx 10^(8) m`
or `r_(1) = 3.464 xx 10^(8) m`