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I have still not yet learn about the centre of mass of a system but I really do need to solve this question that keeps bugging me. I know that internal forces cannot accelerate the centre of mass of a system, but assuming that I have a block-earth system and the block is dropping from a point and considering that gravitational force is an internal force for this system, won't the centre of mass accelerate since the block is accelerating while the earth is not?

Btw, it would be great if someone can explain the concept of centre of mass of a system to me in a much simple way.

Thanks

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The statement "Internal forces cannot accelerate the center of mass of a system" you told is exactly correct. We can choose an appropriate system so that you can find external, internal forces and accelerations easily. Here let us choose the Earth and block as a single system, there are no externals forces acting on this system as gravitational forces are internal forces. So the center of mass of the earth-block system doesn't accelerate.

As you stated that block is accelerating while the Earth doesn't is wrong. Because the block is accelerating towards the Earth at the same time the Earth is also accelerating towards the block. But the amount of acceleration of Earth is negligible compared to block. Also, the mass of the block is minute compared to Earth. Finally, mass times acceleration of both the bodies are the same except in opposite directions. So the center of mass of this Earth-block system doesn't accelerate.

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Actually, from Newton's Third Law of motion, the Earth is attracted towards the block with exactly the same magnitude of force, with which it attracts the block. But we don't see it accelerating simply because its mass is (numerically) much, much greater than the gravitational force. You can also calculate this acceleration, from Newton's law of universal gravitation. We have: $$F=\frac {GMm}{R^2}$$ Hence, $a_{Earth}=\frac {F}{M}$ which comes out to be negligibly small in most cases, hence unnoticeable.

When the block is falling towards the Earth, the CM of the Earth-block system stays right where it was at the start, because it has neither any velocity nor acceleration (disregarding any influence from other bodies, such as the Sun).

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