# Gravitational Attraction

tascja

## 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 Earth and the Mon (centre to centre) is 3.84x10^5 km, calculate where this will occur, relative to Earth.

## Homework Equations

Fg = (G)(m1)(m2) / r^2

## The Attempt at a Solution

FgEARTH = (G)(m1)(mE) / r^2
FgMOON = (G)(m1)(mM) / r^2
dM + dE = 3.84x10^5 km = 3.84x10^8 m

(G)(m1)(mE) / dE^2 = (G)(m1)(mM) / dM^2

(5.98x10^24) / dE^2 = (7.35x10^22) / (3.84x10^8 - dE)^2

(7.35x10^22) / (5.98x10^24) = [(3.84x10^8 - dE)^2](dE^2)

0.01229 = (1.5x10^17 - 3.84x10^8dE + dE^2)(dE^2)

I don't know where to go from here? can i move the 0.01229 to the other side and at the same time move the dE^2 to the other side (so that it gets canceled out)??

Homework Helper
I don't know what you are doing. If Fearth=Fmoon then G*m1*mE/rE^2=G*m1*mM/rM^2 and rM+rE=r, where rE is the distance from earth, rM is the distance from the moon and r is the total distance=3.84x10^5 km. That's two equations in two unknowns, rM and rE.

tascja
thats exactly what i wrote in mine its dE instead of rE. and then you isolate for rE:
dE = 3.84x10^8 m - dM

and then you can solve by substitution and get what i have at the top, and now i still have the same question

tascja
okay so i cross multiplied the ratios and then used the Quadratic Formula to solve for dE and i get a number of 5x10^43. Does that sound right?

Staff Emeritus
Homework Helper
(5.98x10^24) / dE^2 = (7.35x10^22) / (3.84x10^8 - dE)^2

(7.35x10^22) / (5.98x10^24) = [(3.84x10^8 - dE)^2](dE^2)

Can you fix the error that occurs between these two steps?

tascja
yea i multiplied wrong:
so i cross multiplied like this instead:

(5.98x10^24) / dE^2 = (7.35x10^22) / (3.84x10^8 - dE)^2

(7.35x10^22)(dE^2) = (5.98x10^24)(1.5x10^17 - 3.84x10^8dE + dE^2)

0 = 5.9x10^24dE^2 - 2.3x10^34dE + 8.97x10^42

Staff Emeritus
Homework Helper
You are on the right track, but here are more hints:

The quadratic formula gives two solutions, not just one.
You might double-check carefully the numbers you just posted in those equations.
Perhaps it would help first to divide through by 10^22

tascja
yea the other answer for the quadratic formula gave me a negative number, and since I am looking for distance, negative numbers are meaningless, but ill try your suggestion of dividing by 10^22

tascja
okay so i realized that some of my answers were off, but now i keep getting a negative under the square root sign??

kamerling
yea i multiplied wrong:
so i cross multiplied like this instead:

(5.98x10^24) / dE^2 = (7.35x10^22) / (3.84x10^8 - dE)^2

(7.35x10^22)(dE^2) = (5.98x10^24)(1.5x10^17 - 3.84x10^8dE + dE^2)

0 = 5.9x10^24dE^2 - 2.3x10^34dE + 8.97x10^42

you made a mistake in multiplying out (3.84x10^8 - dE)^2
I recommend finding the general formula for dE as a function of m_e m_m and R.

you use the quadratic formula here with b^2 very close to 4*a*c so any mistake will result in
an outcome that is many orders of magnitude off.

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tascja
okay so i restarted the question over (was making way too may mistakes lol) and i got as final answers dE = 7.78x10^10 and 1.7x10^8. Now because it is looking for the distance from the Earth do i just use 7.78x10^10m ??

kamerling
It's still not right. The point where the forces cancel must be between the Earth and the moon, so 7.78x10^10m can't be right. the other point is closer to the Earth than to the moon. This is really a case where it's better to find the general formula before you start pushing all those powers of 10 around.

tascja
o okay i see what you mean, sorry i thought you meant there was a specific formula that i didnt know... so for what your saying then:

(mE)(dM^2) = (mM)(dE^2)

and then plug in? for:

(5.98x10^24)(3.84x10^8 - dE)^2 = (7.35x10^22)(dE^2)

and then id foil and multiple the left side and just distribute on the right side?

Homework Helper
o okay i see what you mean, sorry i thought you meant there was a specific formula that i didnt know... so for what your saying then:

(mE)(dM^2) = (mM)(dE^2)

and then plug in? for:

(5.98x10^24)(3.84x10^8 - dE)^2 = (7.35x10^22)(dE^2)

and then id foil and multiple the left side and just distribute on the right side?

Don't be in such a hurry to plug in. Replace dM with R-dE, where R is the total distance from Earth to moon. Now you have a quadratic equation in dE. Try solving it without putting numbers in, like kamerling suggests. It's easy to make mistakes with all of the numbers floating around. They all look alike.

tascja
okay: no numbers at all would be:

(mE)(r^2) - 2(mE)(R)(dE) + [(mE)(dE^2) - (mM)(dE^2)] = 0

then from here there really is no way to solve except by quadratic formula is there?

kamerling
okay: no numbers at all would be:

(mE)(r^2) - 2(mE)(R)(dE) + [(mE)(dE^2) - (mM)(dE^2)] = 0

then from here there really is no way to solve except by quadratic formula is there?

This is right if r is the same as R. There is quite a lot you can cancel after using the

tascja
yes r is the same as R i just forgot to capitalize...

and i don't really see what can cancel:
** let the numbers inside {} be like under the square root sign
= -b +/- {b^2 -4ac} / 2a

= -[-2(mE)(R)(dE)] +/- { [-2(mE)(R)(dE)]^2 - 4[(mE)(dE^2) - (mM)(dE^2)](mE)(r^2)} / 2[(mE)(dE^2) - (mM)(dE^2)]

kamerling
and i don't really see what can cancel:
** let the numbers inside {} be like under the square root sign
= -b +/- {b^2 -4ac} / 2a

= -[-2(mE)(R)(dE)] +/- { [-2(mE)(R)(dE)]^2 - 4[(mE)(dE^2) - (mM)(dE^2)](mE)(r^2)} / 2[(mE)(dE^2) - (mM)(dE^2)]

you're solving for dE so you can't get that in the result of the quadratic formula.

what are a, b and c in $$\frac { -b \pm \sqrt {b^2 - 4 a c}} { 2 a }$$

Staff Emeritus
Homework Helper
yes r is the same as R i just forgot to capitalize...

and i don't really see what can cancel:
** let the numbers inside {} be like under the square root sign
= -b +/- {b^2 -4ac} / 2a

= -[-2(mE)(R)(dE)] +/- { [-2(mE)(R)(dE)]^2 - 4[(mE)(dE^2) - (mM)(dE^2)](mE)(r^2)} / 2[(mE)(dE^2) - (mM)(dE^2)]

Work on the terms inside the square root sign next. Something cancels.

Staff Emeritus
Gold Member
I'd suggest going through some of your notes or a textbook on algebra again. Most of the problems you're having will disappear and it will set you up for the rest of your studies.

tascja
okay so simplifying under the square root sign will give:

3(mE^2)(R^2)(dE^2) + (mE)(mM)(R^2)(dE^2)

Staff Emeritus
Homework Helper
You're dropping the "4" that appears in the expression.

{ [-2(mE)(R)(dE)]^2 - 4[(mE)(dE^2) - (mM)(dE^2)](mE)(r^2)}

tascja
i see so under the square root its actually only:

4(mE)(mM)(R^2)(dE^2)

Staff Emeritus
Homework Helper
Almost, but as Kammerling said dE should not appear, since dE is the variable you're trying to solve for. Sorry, I missed that point myself up to now.
So it's really 4(mE)(mM)(R^2) inside the square root.

From you're post #17 in this thread, we now "sort of" have
= -[-2(mE)(R)(dE)] +/- { 4(mE)(mM)(R^2) } / 2[(mE)(dE^2) - (mM)(dE^2)]

And I say "sort of" because, as mentioned before, the dE's should simply be omitted.

tascja
yea i kinda realized that too so i rewrote the equation and its:

= 2(mE)(R) +/- {4(mE)(mM)(R^2)} / 2mE - 2mM

and i plugged everythign in and i still get a funny number: 3.8x10^8 which if you look at the original question that is the distance between Earth and the Moon (centre to centre) so it really doesn't make sense as an answer??

Staff Emeritus
Homework Helper
You're not far off. Something went awry when you plugged in the numbers, but you are close.

tascja
man big numbers frustrate me, i didnt even round so that id get a better answer but okay so here are my numbers:
mE: 5.98x10^24 kg
mM: 7.35x10^22 kg
R: 3.84x10^8 m

= 2(5.98x10^24)(3.84x10^8) +/- { 4(7.35x10^22)(5.98x10^24)(3.84x10^8)} /
2(5.98x10^24) - 2(7.35x10^22)

= 4.6x10^33 +/- {6.8x10^56} / 1.2x10^25

= 4.6x10^33 / 1.2x10^25

= 3.8x10^8

**i rounded the numbers i wrote here just to write less, but i plugged in complete numbers into the calculator

tascja
okay scrap that lol, i tried it a completely different way:

dE = {mE}(R) / {mM}+{mE}

= 3.457x10^8 m

** which sounds more reasonable as an answer

kamerling
okay scrap that lol, i tried it a completely different way:

dE = {mE}(R) / {mM}+{mE}

= 3.457x10^8 m

** which sounds more reasonable as an answer

According to your equation dE = 3.79*10^8 which is much too close to the moon.

$$\frac {2 m_e R \pm \sqrt {4 m_e m_m R^2}} { 2m_e - 2m_m}$$

this is really correct. You can still cancel a 2 and get R^2 out from under the square root sign. I hope you do not do the algebra in ascii, but write it out on paper.

A calculator that does variables is very useful. It's much harder to make mistakes if
you can enter:

>>> me = 5.98e24
>>> mm = 7.35e22
>>> R = 3.84e8
>>> me*R/(mm+me)
379337573.30469972

(this is in python)

Staff Emeritus
Homework Helper
= 2(5.98x10^24)(3.84x10^8) +/- { 4(7.35x10^22)(5.98x10^24)(3.84x10^8)} /
2(5.98x10^24) - 2(7.35x10^22)

= 4.6x10^33 +/- {6.8x10^56} / 1.2x10^25

Here you forgot to square R inside the square root { } expression. It should be
(3.84x10^8)^2

Anyway, you got it to work out (perhaps by reading the other thread with this same problem).

p.s. Note to kammerling: he is using { } to signify square rooots.

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