Answer:
RE = 3.456*10⁸ m
Explanation:
Given
G = 6.67*10⁻¹¹ N*m²/Kg²
ME = 5.97*10²⁴ Kg
MM = 7.35*10²² Kg
R = 3.84*10⁸ m
The forces acting on a mass m from the Earth (mass ME) and the moon (mass MM) are:
FE = G*m*ME/RE² and FM = G*m*MM/RM²
where RE and RM are distances from the mass m to the Earth and the Moon.
From the given conditions:
FE = FM and
RE + RM = R =3.84*10⁸ m (the distance between the Earth and the Moon).
Then:
ME/RE² = MM/RM² ⇒ (R-RE)²/RE² = MM/ME
⇒ (R/RE) - 1 = √(MM/ME)
⇒ RE = R / (1 + √(MM/ME)) =
⇒ RE = 3.84*10⁸ m / (1 + √(7.35*10²² Kg / 5.97*10²⁴ Kg)) = 3.456*10⁸ m.
