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How far in miles does the Moon have to be for the Earth not to feel its effects?

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    $\begingroup$ Technically, the answer would be "infinity". On a more practical level, the moon will always have some gravitational effect on the Earth, no matter how far away it is. The real question is how small you want those effects to be. $\endgroup$
    – Phiteros
    Mar 20 '17 at 3:57
  • $\begingroup$ I don't believe that to be true. There is a distance where the moon will have no effect on the Earth in the sense of gravitational effects. I need an actually number please. $\endgroup$
    – Joe
    Mar 20 '17 at 3:59
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    $\begingroup$ It doesn't matter what you believe to be true. What matters is what the math says. And the math says that you would need to be infinitely far away to feel zero force. $\endgroup$
    – Phiteros
    Mar 20 '17 at 4:13
  • $\begingroup$ Joe, the force of gravity drops as $1/r^2$, and so never goes to zero. But if you want to pick a specific effect, for example tidal force, you can ask how far the moon has to be to drop by half, or 90%. I can't guarantee the answer will be simple, but right now your question's answer is really "infinity" as @Phiteros says. $\endgroup$
    – uhoh
    Mar 20 '17 at 7:21
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    $\begingroup$ Please stop posting the same question over and over and over. $\endgroup$
    – HDE 226868
    Mar 20 '17 at 16:50
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We can use Newton's law of gravity to calculate the force the Earth feels from the moon:

$$ F_g = \frac{GM_{Earth}M_{Moon}}{r^2} $$

Where G is the universal gravitational constant, $6.62*10^{-11}$ $m^3 kg^{-1} s^{-2}$, $M_{Earth}$ and $M_{Moon}$ are the masses of the Earth and Moon in kilograms, and $r$ is the distance between their center s in meters. We can rearrange this equation to solve for a distance like so:

$$ r = \sqrt{\frac{GM_{Earth}M_{Moon}}{F_g}} $$

Now we can input any force we like to see how far away the Moon would have to be in order for us to feel that force. If we want the force of gravity to go to zero, we can take the limit as $F_g\rightarrow 0$. When we do this, we see that for the Earth to feel zero gravitational force from the Moon, it would have to be infinitely far away.

However, infinity is not a very useful number; why don't we look for something a little more practical? Let's say that we want the Earth to feel exactly 0.01 Newton of gravitational force from the Moon. That's about the weight of 1 gram of matter at the Earth's surface. In that case, we will have $r=5.411*10^{19}$ meters or $3.4*10^{14}$ miles, about $1.4*10^{11}$ times further away.

Keep in mind, though, that this is still not zero force. It's small enough that we probably wouldn't be able to detect it, but it is nonzero.

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  • $\begingroup$ What about a distance where Earth would start spinning violently on its axis? $\endgroup$
    – Joe
    Mar 20 '17 at 4:16
  • $\begingroup$ I am actually doing this for science extra credit. My teacher said that we would die if the moon got too far away. He wants us to figure out when this will happen. $\endgroup$
    – Joe
    Mar 20 '17 at 4:23
  • $\begingroup$ Now that's a lot more complicated. If you do some basic research online, you should be able to find estimates though. $\endgroup$
    – Phiteros
    Mar 20 '17 at 4:26
  • $\begingroup$ Can you help? I tried and could't find anything. $\endgroup$
    – Joe
    Mar 20 '17 at 4:27
  • $\begingroup$ This Wikipedia article should answer some of your questions. $\endgroup$
    – Phiteros
    Mar 20 '17 at 4:38

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