At what distance between the Earth and the Moon is the gravitational force due to the Earth same as that of the Moon.The mass of Earth is 6*10^24...

The gravitational force of attraction between two bodies
of mass m1 and m2 is given by F = G*m1*m2/ r^2, where G is a constant and r is the
distance between the two bodies.

Here we have to determine
the point between the Earth and the Moon, where the gravitational force of attraction
due to the Earth and that due the Moon is equal.

Let the
point be at a distance D from the Earth. As the distance between the Earth and the Moon
is given as 384400 km, the distance of the point from the Moon is 384400 -
D.

Let a body of mass m be placed at this point. The
gravitational force of attraction due to the Earth is G*Me*m/D^2, and the gravitational
force of attraction due to the Moon would be G*Mm*m/ (384400 -
D)^2

G*Me*m/D^2 = G*Mm*m/(384400 -
D)^2

=> Me/D^2 = Mm / (384400 -
D)^2

=> (384400 - D)^2 / D^2 = Mm /
Me

=> 7.35*10^22 /
6*10^24

=> 1.225/
100

(384400 - D) / D =
0.1106

=> 384400*0.1106 - 0.1106*D =
0.1106*D

=> 0.22135D =
384400*0.1106

=> D =
192061

Therefore the distance of the point
from the Earth where the gravitational force of attraction due to the Moon as well as
the Earth is equal is 192061 km.