Where is the point where the Earth's gravity is cancelled by the moon?

[chantelkohl]

Homework Statement

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 away 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

Homework Equations

F=GMm/r^2

The Attempt at a Solution

GM/r^2=G(1/81)M/r^2-d

 

[Pengwuino]

I don't really get what that 'd' is doing hangen out there all by itself. If you look at your attempt, you need to have the 'r' on the other side denoted as something different, say for example r', where r' is the distance to the moon. What you're looking (in my opinion) is a ratio of the distances from the Earth to that certain point where the forces or equal and the distance from the moon. Start by multiplying both sides by $$r'^2$$ where r' is the distance to the moon.

 

[kerimek]

^2

To solve for d, we can use the fact that the gravitational force of the Earth and the moon at this point should be equal and opposite, cancelling each other out. This can be expressed as:

Fg = Fm
GMm/d^2 = G(1/81)M/(r-d)^2

Simplifying and rearranging, we get:

d = (80/81)r

Therefore, the point where the gravitational attraction of the Earth is cancelled by the moon is located at a distance of 80/81 the way to the moon from the center of the Earth. This is option (d) in the given choices.