Apparent Weight.

How does this apparent weight thing work? Why do people experience the normal force, but not the weight? I don't think I have the exact answers here, but rather a plausible argument.

First, let me model a person as the following:
Wait. That doesn't look anything like a person. I said it was a model, so its ok. This is a model of a person's spine. The model consists of 5 masses (the red balls) connected by springs (the white lines). This person is standing on the surface of the Earth. I have also represented the gravitational force on each part of the person's spine as yellow vectors. Note that they are all the same length. The gravitational force on each one is the same. The compressions of the springs are not the same. The lower springs are compressed more than the higher springs because they have to exert more force on the lower masses. Take the top mass (assume all masses are 1 kg). The gravitational force on this top mass is F_g = (1 kg)(9.8 N/kg) = 9.8 Newtons.




So, for this mass to be in equilibrium, the spring must also push up with 9.8 Newtons. If that spring connected the top mass and the next mass is the same spring, then this same spring should push DOWN on the next lower mass with 9.8 Newtons (Newton's 3rd law says forces come in pairs). So, the second lower spring would have to push up 18.6 Newtons. 9.8 Newtons to counter act the weight of the second mass, and another 9.8 Newtons to counter at the spring above it pushing down. Since that spring must provide more force, it will be compressed more (the more you stretch or compress a spring, the more force it exerts).

Key point: The gravitational force on each piece of this 'person' is the same, but lower parts are compressed more. In my model of a person, gravity is "sensed" by how body is compressed.

Suppose I now put my "person model" in an elevator that accelerates upward at 3 m/s². Here is what that would look like:

The red arrow represents the acceleration. Notice that the gravitational force is the same as it was before, but the springs are compressed more. In this case, you would FEEL heavier. You are NOT heavier. Notice also that the blue arrow representing the force the floor pushes on the person is greater.
What if the 'person' is in an elevator that is accelerating down? (at 3 m/s²)

Now there is less compression, smaller force of the floor pushing on the person but the SAME gravitational force. Here you would feel lighter (but you are not).
Next case, accelerating down at 9.8 m/s²:

Once again, gravity is the same. But here, there is NO force of the floor pushing on the person and the masses are equally spaced. This is because there is no need for the spring to exert a force on each mass since the following is true
Since the acceleration is the same as the gravitational field (g), no other force is needed to make it move this way. Here you would feel weightless, but again - NOT.

Ok, here is my final case. What if there was zero gravity (as would happen if you were very far from any massive objects like planets or stars). Is it possible to make it "feel" like you have weight? Here is such a case:

In this case, the masses seem to be compressed the same as they were with gravity, but no acceleration. But here there is no gravitational force. How would this happen? If the floor is accelerating upwards at 9.8 m/s² then the floor would have to push on the person exactly the same as when the person was at rest in gravity.

Apparent weight is related to the force that is pushing on a person from an external object (this is what you feel). You don't really feel gravity because it pulls equally on all parts of you.

Here is the vpython program I used to create this model (in case you want it)
If you don't know about vpython - go here http://www.vpython.org