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- anonymous

There is a point on the line joining the centers of the Earth and the moon where their combined field strength is zero. Is this point closer to the Earth or the Moon?

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- anonymous

- schrodinger

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- JamesJ

JamesJ: On your question, what do you think intuitively is true?
JamesJ: If you're not sure, ask yourself this:
JamesJ: what would happen if the moon and earth had the same mass?
JamesJ: Now what if the mass of the moon were slightly less; where would that point be then?
physicsme: towards the moon?
JamesJ: Yes. And where would it be if their masses were equal?
physicsme: Earth
physicsme: ?
JamesJ: Where would the point of zero gravitational force be if the moon and earth had equal mass?
physicsme: in between
JamesJ: but where in between. Suppose these were the only two masses in the universe. Where would the point of zero gravity be if their masses were equal.
physicsme: center
JamesJ: right. Exactly in the middle. So now, if the mass of the moon goes down from equality with earth, you said a minute ago the point would move towards ...
physicsme: the moon
JamesJ: right. Hence the answer to your question is?
physicsme: The lovely moon :)
gogind: Hi, sorry to interrupt. How do I write vectors in latex?
gogind: like A^hat
JamesJ: yes
JamesJ: \[ \hat{a} \]
gogind: ah, thank you
physicsme: and how would we calculate the distance of the point from Earth?
physicsme: *from the center of the earth
JamesJ: To calculate it exactly, you'll need the mass of the earth and moon, as well as the distance between their centers.
physicsme: yeah
physicsme: and then how would we solve it?
JamesJ: Write E for mass of the earth, M for the moon. Suppose the distance between their centers is D. Then the point where the two forces of gravity cancel is where for a test mass m
JamesJ: F_earth = F_moon. Namely,
JamesJ: GEm/r^2 = GMm/(D-r)^2
JamesJ: where we're measuring distance r from the earth. Notice now that G and m cancel, so we're left with
JamesJ: E/r^2 = M/(D-r)^2 or
JamesJ: (D-r)^2/r^2 = M/E
JamesJ: Notice that as M/E < 1, it must be that
physicsme: yeah, thanks
JamesJ: (D-r)/r < 1, which means the point is indeed closer to the moon.
JamesJ: If M = E, then M/E = 1 and
JamesJ: (D-r)/r = 1 => r = D/2, the midpoint.
physicsme: got it

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