No. You do not feel your weight. You feel the force normal acting on you. This video shows why and demonstrates what you feel on an elevator.
Want Lecture Notes? This is an AP Physics 1 topic.
0:22 Showing that you do not feel your weight
1:10 What does the scale actually measure?
2:10 Elevator example
3:12 Determining your apparent weight on the elevator
4:23 An elevator in free fall!
5:42 Apparent weightlessness
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Thank you to Youssef Nasr for transcribing the English subtitles of this video.