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 kg and the mass of the Moon is 7.35*10^22 kg and the distance between them is 384400 km.
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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.
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