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While learning about motion of centre of mass, I came across this statement online,

It is also more accurate to say that the Earth and Moon together revolve about their common center of mass.

I am not able to understand this. Can anyone please explain this statement?

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    $\begingroup$ Do you know what "center of mass" means? If so can you see that conservation of linear momentum requires that a free rotation of the pair must be about that point? Give us some help here in understanding where you are stuck. $\endgroup$
    – dmckee --- ex-moderator kitten
    Commented Sep 12, 2014 at 2:15
  • $\begingroup$ Get somebody to help you with an experiment: take each others hands and start swinging around in circles like in a folk dance (polka?). Where is the center of your movement? $\endgroup$
    – CuriousOne
    Commented Sep 12, 2014 at 2:28
  • $\begingroup$ I know abt " com " . But I don't know abt this , "conservation of linear momentum requires that a free rotation of a pair must be abt that point ." I think I am not able to understand because of unknowing this ... $\endgroup$
    – Saravanan Ramesh
    Commented Sep 12, 2014 at 2:49

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Suppose the Earth was stationary, and the Moon revolved around the centre of the Earth:

Earth Moon

If $v$ is the orbital velocity of the Moon then at point $A$ the linear momentum of the Moon is $mv$. Half an orbit later, at point $B$, the velocity of the Moon is $-v$, because it's in the opposite direction, so the momentum of the Moon is $-mv$.

If the Earth is stationary then its momentum is zero, and that means the total momentum of the Earth Moon system is not conserved, because it's changing from $mv$ to $-mv$ every half an orbit. But if we take the Earth-Moon system as an isolated system then its momentum must be conserved so we have a contradiction. That means the Earth can't be stationary.

Obviously what happens is that the Earth moves as well as the Moon:

Earth Moon system

At any point in time the momentum of the Earth is equal and opposite to the momentum of the Moon, so the total momentum (in the centre of mass frame) is zero.

Centre of mass

And that's why conservation of momentum requires both the Earth and the Moon to revolve around the centre of mass.

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  • $\begingroup$ If you take the Earth stationary there is no need to impose conservation of momentum of the orbiting body. This way to derive a correct result is somehow forced. It is better to show where the COM is and takes the obvious consequence. Isn't? $\endgroup$
    – Alchimista
    Commented Feb 7, 2018 at 10:04

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