r_{cm} = 4.67 10⁶ m. We see that none of the answers is correct, the closest is the third, but the center of mass is within the radius of the Earth.
The definition of the center of mass is
rcm= 1 / M ∑ rimi
where mi represents each body's location and M represents its total mass
We must choose a reference system for the calculation, and we will do it with the Earth's centre as its origin. The information we have is
Mass of the Earth Me = 5.98 1024 kg
Moon mass m = 7.36 1022 kg
Earth to Moon Distance r = 3.84 108 m
Let's apply this to our case
r_{cm} = 1 / (Me + m) (Me 0 + m R)
r_{cm} = 1 / (598 +7.36) 10²² (0 + 7.36 10²² 3.84 10⁸)
r_{cm} = 4.67 10⁶ m
We can see that this distance is less than the radius of the Earth
We see that none of the answers is correct, the closest is the third, but the center of mass is within the radius of the Earth.
The unique point at the centre of a spatial distribution of mass is known as the centre of mass, and it has the property that the sum of the weighted position vectors around it is zero. The centre of mass is the average location of a distribution of mass in space, to use an analogy from statistics.
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