So here's the question:

Calculate the approximate distance between the centre of the Earth and the centre of the moon. You may use the mass of the Earth as $6\times10^{24} kg$ and $G = 6.67\times10^{-11} Nm^2kg^{-2}.$

I can't work this out, at least. This is what I've done.

$F=\frac{GMm}{r^2}=ma$ where $a=g$, so $r=\sqrt{\frac{Gm}{g}}$ but I feel like $a\neq g$ as it is far away?

  • $\begingroup$ The sign on the exponent of 10^11 in G should be negative. $\endgroup$
    – S. McGrew
    Apr 17 '18 at 13:52
  • $\begingroup$ The key to solving this problem is knowing how long a (sidereal) month is. $\endgroup$
    – JEB
    Apr 17 '18 at 13:57
  • $\begingroup$ So one month is the period of the moon? $\endgroup$ Apr 17 '18 at 14:04
  • $\begingroup$ @DanD'silva Yes. 1 Sidereal month. $\endgroup$
    – JEB
    Apr 17 '18 at 15:17

Looks like a homework problem. Just knowing the value of G and the mass of the Earth is not enough. You need to know at least another value, such as the orbital period of the Moon. For example, if you know that the Moon takes 28 days to complete its orbit around the Earth, you can calculate the centripetal acceleration as a function of r. The acceleration due to gravity equals the centripetal acceleration. Take your equation for r, substitute the centripetal acceleration (function of r) for g, and solve for r. This is only very approximate, because in fact the Earth and moon orbit around their common center of mass, so the radius of the moon's orbit is smaller than the distance from Earth to Moon. If you know the mass of the Moon you can calculate whee the common center of mass is, and make the appropriate correction.


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