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We have been talking about the center of mass of the earth and the moon. We agreed from calculation and data that it falls inside the earth and not somewhere in space.

My friend had a very convincing argument.

I wanted to know if its accurate.

The argument is as follows:

Condition 1: Earth and moon are very large objects and therefore gravitational forces plays the dominant role.

Condition 2: From history we know the moon has been following around the earth.

Condition 3: As the intuitive understanding of barycenter suggests that it is the point where all mass is assumed to be concentrated. So, we guessed incorrectly that if the center of mass moves everything follows and since moon follows the earth we guessed the center of mass must have been inside earth.

So, Q.E.D We can somehow suggests convincingly that the barycenter falls inside the earth.

We feel the deduction is wrong but someone please continue the argument intuitively that is accurate.

Jyotishraj Thoudam
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2 Answers2

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The equation for the position of the centre-of-mass, CoM:

$$x_{cm}=\frac{\sum xm}{\sum m}\quad\overset{\text{for 2 bodies}}=\quad \frac{x_1m_1+x_2m_2}{m_1+m_2}$$

In words: Choose an origin (a coordinate system), and sum all the objects' position times their masses, and divide with the total mass.

Such a linear procedure is luckily easy to grasp intuitively. If you have two bodies:

  • Equal masses: the CoM will be just as far from each; in in the middle.
  • One with double the mass: the CoM will be double as far from the light one, as it is from the heavy one. It could fx be 2000 km from the light body, and 1000 from the heavy one. In other words, it will be 1/3 from the heavy one and 2/3 from the other.
  • One with 3 times the mass: the CoM will be three times as far from the light one as from the heavy one. It could be 3000 km from the light on and 1000 from the heavy. In other words, 1/4 from the heavy and 3/4 from the light body.
  • One with 4 times the mass: the CoM will be 4 times as far from the light one as from the heavy one. In other words, 1/5 from the heavy and 4/5 from the light one.

This pattern continues, and can be an intuitive way of quickly figuring out, where the CoM is positioned, and thus which point they both orbit. The Earth weighs about 80 times more than the Moon, so:

  • the CoM of the Moon-Earth system will be 80 times as far from the Moon as from the Earth - and we are talking from center to center. This is around $~1/80$ (about $~1.2$ %) of the total distance from the Earth's center to the Moon.

If the Earth's center on a good day is 400.000 km from the Moon's center, then 1/80 of that is roughly 5000 km. The Earth is $~6400$ km in radius, so indeed, the combined CoM of the Earth-Moon system lies inside the Earth. Your deduction is correct though the three conditions you set up seem a bit "mooshy" and not so clear - I can't see how you actually ended with this correct result from that, but nevertheless, is ended correctly.

This quick calculation seems backed up by Wikipedia.

Steeven
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Whenever two bodies are gravitationally bound, they orbit their center of mass. Sometimes it happens that one body is much more massive than the other: in that case, the COM is closer to the larger body, so it moves less, but it still moves.

Your argument seems to be something along the lines of "since the Earth barely moves, the COM of the system is inside the Earth". But it's all a question of degree: all you can assert is that the COM is close to the Earth, but exactly how close depends on the details. Jupiter goes around the Sun, and yet their center of mass is outside the Sun, because Jupiter is so far away.

Javier
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