• 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?
  • Stacey Warren - Expert
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  • katieb
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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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