Somewhere between the Earth and the Moon is a point where the gravitational attraction of the Earth is canceled by the gravitational pull of the Moon.The mass of the Moon is 1/81 that of the Earth.How far from the center of the Earth is this point?
A)8/9 the way to the Moon
B)9/10 the way to the Moon
C)3/4 the way to the Moon
D)80/81 the way to the Moon

Question

Quizplus · Accepted Answer

The point where the gravitational forces of the Earth and the Moon cancel each other out is known as the Lagrange point L1. The formula to find the distance $r$ from the Earth to L1 in terms of $R$, the distance from the Earth to the Moon, and the masses of the Earth ($M_e$) and the Moon ($M_m$), is $r = R \left(\frac{M_m}{M_m + M_e}\right)^{1/3}$. Given that $M_m = \frac{1}{81}M_e$, substituting this into the formula gives $r = R \left(\frac{1}{1 + 81}\right)^{1/3} = R \left(\frac{1}{82}\right)^{1/3}$. This fraction is approximately 1/9 of the distance from the Earth to the Moon, meaning the point is 1 - 1/9 = 8/9 or approximately 9/10 of the way to the Moon from the Earth, not exactly but closer to 9/10 than to any other options provided.

Somewhere Between the Earth and the Moon Is a Point