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This question arises when I am studying fictitious forces in an undergrad introduction to physics course.

Suppose I am standing in an elevator with an acceleration $a$ directed upward. From the ground viewpoint, with $N$ being the normal force on me, we have $N - mg = ma $ and so $N = mg + ma$. From my viewpoint, I feel two force: an upward force of $m(g+a)$ and a total downward force (gravity + fictitious force) of $m(g+a)$ too. If I were standing somewhere on a planet with gravitational acceleration $G=g+a$, then I feel these two forces too. That's why in the elevator I have the feeling of being on the planet with $G = g+a$. OK, so far so good.

Now let's say I'm freefalling. From my viewpoint, I feel two force: an upward fictitious force of $mg$ and a downward gravitational force $mg$. The situation is pretty the same as if I were standing on the ground. But clearly I wouldn't feel that normal when freefalling; in fact I feel weightless. Why?

Thanks for your help.

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Essentially a duplicate of – Qmechanic Sep 26 '13 at 11:41
@dj_mummy Yes, you are welcome :) – tom_a2 Sep 26 '13 at 12:32
@Qmechanic That's a helpful link, thanks! So one possible answer to my question is that the upward force from the ground acts on my feet more heavily. Yet when I freefall, the upward and downward force acts on my atoms equally, so they indeed cancel. Another explanation is that I am just moving in the curved space-time generated by gravity, so I feel essentially weighless. – tom_a2 Sep 26 '13 at 12:37
@tom_a2 I edited my answer. Hope it is even simpler to understand now. – dj_mummy Sep 26 '13 at 13:06
up vote 0 down vote accepted

I think the answers in the duplicate have covered most of the key points. I will just add to them

The feeling of weight, strain, stress etc. is due to the differential force acting on different points of the body. Even while floating in curved space-time you may feel strain, weight etc. because your body is not point size, the particles of your body will be accelerated in different directions relative to each other. To maintain the rigidity of your form, the body exerts a force to hold itself together. This force is weight,strain,stress etc. So irrespective of your state of motion from any frame of reference, if the particles of body are being forced in different directions relative to each other, your nervous system will register this as weight,stress,strain in the respective parts of the body.

The cause of relative acceleration between parts of your body is twofold- tidal acceleration due to the structure of space time, the action of non gravitational forces on the different particles in your body. So even in non-inertial frames without the effects of curved-space you might feel this weight,strain etc.

The only time you will feel absolutely weightless, is if the relative acceleration between all the particles in your body is equal to 0. Due to the fact that the Earth is curved and that you are not a point particle, there is a slight inward lateral strain even when you free fall in vacuum on Earth.

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If I stand on the ground, then the $mg$ grav force and the $mg$ normal force do cancel out too. Then why didn't I feel weightless? – tom_a2 Sep 26 '13 at 12:12

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