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If we assume the head the heaviest part of the body, (which may not be a warranted assumption at all,) then no matter what orientation its in (lying down or upright) it will still remain the heaviest part of the body. It may even become slightly heavier, when lying or upside down, since a bit more blood may be forced into that area. Whatever body part or area is indeed heaviest, will remain heaviest.
However, we're making a few assumption here, firstly, that the head is the heaviest to begin with, that the force of gravity is constant, and that we're observing within an everyday, typical frame of reference. If you had a device to determine the force of gravity next to your head, you would find that your head (or any part of your body) would weigh more lying down than standing up. That's because weight is a function of gravity; the weight of someting equals its mass times the force of the gravitational field in which that mass finds itself, or:
and the force of gravity strengthens a very tiny bit from the top of your head to the bottom of your feet. The mass of your head, or any other body part does not change.
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