Weight is defined in physics as the force acting upon an object due to its mass. Or in simpler terms, weight is a force that is equal to an objects mass times the acceleration due to gravity, F=mg. Weightlessness is not an absence of that force, in fact it is the complete opposite. Weightlessness is when the only force acting upon an object is the force of gravity or its weight.
The force that we perceived as our weight is actually the normal force of the floor pushing against our feet counteracting your weight. When you feel weightless it is actually because of a canceling out of a normal force so it is essentially not acting. This is achieved by having a centripetal force act in the opposite direction of the normal force.
When in orbit, which is an example of being weightless, the force of gravity is always directed toward the Earth, the normal force is always opposite and the gravitational force. Then the centripetal force is always directed toward the center of the circle as represented in the image below.
So when an object is weightless it is the centripetal force cancels out the normal force and the only force acting on you is the gravitational force.
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