Why doesn't the Earth accelerate towards us?

Question

According to Newton's third law of motion that states that every action has an equal and opposite reaction. So, if the Earth exerts a gravitational pull on us (people) then even we should exert a force equal and opposite (in terms of direction) on the Earth. It is intuitive to think that this force is really small to get the Earth to move. But, if we take a look at the second law of motion that states that F = ma we see that however small the force, there will be some amount of acceleration. Therefore, even though we exert a very small gravitational force on the Earth it should be enough to get the Earth to move even though the acceleration is a very small amount. But what I said clearly does not happen. So there must be some flaw with my reasoning. What is that flaw?

Answer 1

The earth is pulling you down and you are pulling up on the earth. If your weight is ~60 kg, the force is roughly 600N. Since you are only 60kg, you experience 10 m/s^2. The force on the earth is the same 600N but over a mass of 6 x 10^24 kg, so it experiences an acceleration of 1 x 10^-22 or 0.0000000000000000000001 m/s^2. That is pretty undetectable. If there is a flaw in your reasoning, it would be that you don't appreciate how massive the earth actually is and how you cannot detect things when it's to the -22 power.

I think to further complicate things for why you don't "see" it happening is that there are masses distributed all around the planet so even though you may pull up on the earth somewhere in North America, there is some mass pulling the earth the other way in Australia that could cancel it out.

So even if the all the mass on earth gets concentrated to a single point on the earth's surface, you may not "see" much of a pull on the earth if the sum of all the masses is small compared to 6 x 10^24 kg.

Answer 2

There is really not a flaw in your reasoning. If I was in free fall and had a mass of 60 kg, I would have a force of 588 N and accelerate at 9.8 m/s2.

The Earth has a mass of around 6 x 10^23 kg so if it experiences the same reaction force it will have an acceleration of 9.8 x 10^-22 m/s2. To put this in perspective if we were in free fall for our whole lifetime of around 70 years, the Earth would have only moved 0.0024 meters (2.4 mm).