Earth and moon where does gravity cancel out

[lozzajp]

Homework Statement

where does the gravity cancel out between the Earth and moon?

Homework Equations

s=3.84405x10^8m
earth m = 5.98x10^24kg
moon m =7.35x10^22kg
G = 6.67x10^-11N m^2/kg^2
mass of object between = 1kg for simplicity
F = G x m / r^2
r = ?

F Earth = F moon
G x Earth / r^2 = G x moon / (s-r)^2

The Attempt at a Solution

I get so far with that and I've ended up with:
earth = (moon x r^2) / (s-r)^2

not sure if its right and how do i isolate r?

 

[aftershock]

Let's call the distance away from Earth r, and the total distance between the Earth and moon R.

A mass in between the Earth and moon will experience a gravitational force from both.

F_earth = GM_em/r²

F_moon = GM_mm/(R-r)^2

For the net force to be zero the two above forces must equal each other.

 

[lozzajp]

i understand that but i don't know how to solve for r (R = s in my part)

 

[Mesmerr]

You need to use simple algebra to isolate r.

start with multiplying both sides by the rights side denominator. then you need to multiply out Earth and divide everything by r². By doing this you should be able to get a single r² term. then just isolate r.

 

[aftershock]

i understand that but i don't know how to solve for r (R = s in my part)

I worked it out, and it's definitely doable but the algebra is a little more intense than it probably needs to be. I thought of an easier way to do it. Let R be the total distance between the moon and the earth.

The distance away you are from the Earth must be some percentage of total R. So if you're halfway of the total distance then .5R , if you're a quarter then .25R, etc. So let's say you're distance from Earth is aR where a is some number <1

The distance from the moon must then be 1-a

So we can say this.

GM/aR = Gm/(1-a)R here I called M the mass of Earth and m the mass of the moon.

Now you can just solve for a, that is significantly easier. Multiply a by the total distance and that's how far from Earth you need to be (well the center of the earth)

 

[lozzajp]

thats a good way to do it! thanks for that.

do post the intense algebra? :)

 

[aftershock]

thats a good way to do it! thanks for that.

do post the intense algebra? :)

You cancel G and cross multiply to get

(R-r)²/r² = m/M

Foil out R-r

(R²-2Rr+r²)/r² = m/M

You cross multiply the m/M to bring it to the left and r² to bring it to the right.

Distribute the new M/m term on the left. Now subtract r² so it's back on the left. Give r² a common denominator wit the rest of the terms. Now you can factor out r²

The end result will be a constant plus r times some constant plus r² times some constant = 0

So you can plug that into the quadratic formula.

 

[haruspex]

Oops.

 

[aftershock]

I guess that's what the question is after, but it isn't strictly correct. If you were at that place you would not find that the two gravities cancelled. That's because once inside the Earth there's no net force from from the parts of the Earth that are further from the centre than you are. But I hope that's not to be taken into account here.

I'm not sure what you mean? What does being inside the Earth have to do with this? You're in space located in between the moon and the earth.

 

[haruspex]

I'm not sure what you mean? What does being inside the Earth have to do with this? You're in space located in between the moon and the earth.

Oops, was thinking of a different question entirely. Sorry for the noise.

 

[lozzajp]

You cancel G and cross multiply to get

(R-r)²/r² = m/M

Foil out R-r

(R²-2Rr+r²)/r² = m/M

You cross multiply the m/M to bring it to the left and r² to bring it to the right.

Distribute the new M/m term on the left. Now subtract r² so it's back on the left. Give r² a common denominator wit the rest of the terms. Now you can factor out r²

The end result will be a constant plus r times some constant plus r² times some constant = 0

So you can plug that into the quadratic formula.

So it is impossible without quadratic formula.. guess I have some reading to do thanks for all your help!