Question

What is the distance from the center of the Moon to the point between Earth and the Moon where the gravitational pulls of Earth and Moon are equal? The mass of Earth is 5.97 × 1024 kg, the mass of the Moon is 7.35 × 1022 kg, the center-to-center distance between Earth and the Moon is 3.84 × 108 m, and G = 6.67 × 10-11 N ∙ m2/kg2. Group of answer choices 4.69 × 107 m 4.69 × 106 m 3.83 × 106 m 3.45 × 108 m

Answer 1

3.83*10^7 m

Step-by-step explanation:

Assume that the distance at which gravitational force due to both Earth and moon is zero and it l is given by force force balance

F(moon) = F(earth)

F(moon) = GM(moon) / r²

F(earth) = GM(earth) / (d - r)²

If F(moon) = F(earth), then

GM(moon) / r² = GM(earth) / (d - r)²

7.35*10^22 / r² = 5.97*10^24 / (3.84*10^8 - r)²

now, we take the square root of both sides, we have

2.71*10^11 / r² = 24.4*10^11 / (3.84*10^8 - r) =>

2.71 / r² = 24.4 / (3.84*10^8 - r)

if we cross multiply, we have

24.4r = 1.04064*10^9 - 2.71r

24.4r + 2.71r = 1.04064*10^9

27.11r = 1.04064*10^9

r = 1.04064*10^9 / 27.11

r = 3.83*10^7 m