- #1

alicia113

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## Homework Statement

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

r1

the point from the center of the Earth to x and

r2

the point from x to the center of the moon

Therefore

(1)

r=r1+r2

Using this assumption of the object X we get..

Fg=Gm1xr21

where

m1

is the mass of the earth, and x is the mass of object X

Fg=Gm2xr22

where

m2

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:

(2)

Gm1xr21=Gm2xr22

Now, we obtain a ratio between

m1

and

m2

so we can express

m1

in terms of

m2

m1m2=81

(Note: I rounded to 81 just to make typing it up here easier...)

Therefore,

m1=81m2

Now, going back to equation (2), G and x will cancel out and we replace

r1

with

r−r2

(from equation (1)) leaving us with:

81m2(r−r2)2=m2r22

m2

will cancel out:

81(r−r2)2=1r22

81=(r−r2)2r22

i take the square root of both sides, and put the denominator on the left side

9r2=r−r2

10r2=r

r2=r/10

r2=3.84∗104

Since

r1=r−r2

r1=3.84∗105−3.84∗104

r1=3.456∗105