Question

At what distance from the Moon, expressed in lunar radii (and also in Earth radii), does the gravitational force due the Earth on a point mass, m, offset the gravitational force due to the Moon?

Answer 1

point mass = m

Let ths disatnce between earth and moon = L

let the object be at s diastance x from earth

GmMe/ x^2 = Gm Mm/ ( L-x)^2

Me/ x^2 = Mm/ ( L-x)^2

L = 384,400 x 10^3 m

Mm = 7.35 x 10²² kg

Me= 5.97x 10^24 kg.

5.97x 10^24 / x^2 = 7.35 x 10^{22 / (} 384,400 x 10^3 - x)^2

81.244 ( 384,400 x 10^3 - x)^2 = x^2

81. 244 ( 14776336 x 10^ 10 + x^2 - 768800x 10^2x ) = x^2

1200488641.984 + 80.244 x^2 - 62460397.2 x = 0

80.244 x^2 - 62460397.2 x + 1200488641.984 = 0

x = 778361.62 m = 7783.616 km apprx from earth