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I am learning newton's third law, and i got to this conclusion, i wanted to know if it's correct (within the boundaries of Newtonian mechanics)

Say I'm pushing a cupboard with my body, and I apply a force of 10 newtons. That means that the cupboard is applying a force of 10 newtons to me, but I'm not moving because the earth (through friction) is applying 10 newtons of force to me in the opposite direction. That means I am applying a force of 10 newtons on the earth.

That means that I'm changing the acceleration of the planet by applying a force of 10 newtons to a cupboard, and the change in acceleration is given by

$F = M_{mass-of-earth} * a$

I plug in the known values

$10 = 5,972 ×10^24 kg * a$

I solve for a

$a = 1.67448091 × 10^{-24}$

Is this reasoning correct? I feel that if I confirm this I will have understood what I felt I was missing in all my free body diagram exercises where forces are applied by arbitrary magical invisible entities.

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You forgot about the free body diagram for the cupboard! It is pushing on the earth too. So you can't apply a net force to the earth by pushing between the floor and the cupboard which are both fastened to the earth.

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  • $\begingroup$ i was missing that!, the force that stops the cupboard from moving comes from friction too, so by newtons third law the planet is getting an opposite force of 10n on the other direction and both forces applied to the planet sum up to cero. I feel a bit silly but now i feel i really get what's going on. thanks!! $\endgroup$
    – Joaquin Brandan
    Apr 25, 2016 at 2:50
  • $\begingroup$ Glad I could help! $\endgroup$
    – Duncan Harris
    Apr 25, 2016 at 2:54
  • $\begingroup$ what would happen if i actually make the cupboard accelerate?, i would beat the force of friction between the cupboard and the earth (say a force of 4 newtons of friction) but i would still be applying a force of 10n with my feet to make the cupboard accelerate with a force of 6 newtons. am i still missing something? $\endgroup$
    – Joaquin Brandan
    Apr 25, 2016 at 2:56
  • $\begingroup$ In that case you would accelerate the earth slightly while pushing the cabinet, but at some point the cupboard presumably stops moving again, at which point the cupboard accelerates the earth the other way. No net momentum is imparted to the earth by the process unless you manage to throw the cabinet into a stable orbit. Which is pretty hard, in case you haven't tried... $\endgroup$
    – Duncan Harris
    Apr 25, 2016 at 3:00

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