The mass of the moon is 7.35x10^22 kg. At some point between Earth and the Moon, the force of Earth's gravitational attraction on an object is canceled by the Moon's force of gravitational attraction. If the distance between the Earth and the moon (center to center) is 3.84x10^5 km, calculate where this will occur, relative to the Earth.
The Attempt at a Solution
ok now this is from another thread, and i completely understand how she got everyting but the 81! could someone please explain to me how she got the 81.. thts the only part I am lost in..
We assume an object is at the L1 point (the point where the gravitational fields of the two objects cancels out). We will call this object X
r is the radius between the Earth and the Moon
We will call
the point from the center of the Earth to x and
the point from x to the center of the moon
Using this assumption of the object X we get..
is the mass of the earth, and x is the mass of object X
is the mass of the moon, and x is the mass of object X
Since we know that, at the point where the object X is situated, the gravitiational pull from both the Earth and the Moon will be equal, we can equate the above two equations getting the following:
Now, we obtain a ratio between
so we can express
in terms of
(Note: I rounded to 81 just to make typing it up here easier...)
Now, going back to equation (2), G and x will cancel out and we replace
(from equation (1)) leaving us with:
will cancel out:
i take the square root of both sides, and put the denominator on the left side