What we humans experience as weight is not actually the force due to gravity. What we feel as weight is actually the force of the ground (or chair or whatever) pushing upwards against us to counteract the force due to gravity. A particle in a container in free-fall experiences weightlessness. This is because there is no reaction to the particles weight from the container, as it is being pulled down with the same force. The acceleration of the container equals the acceration of the particle, which equals the acceleration caused by gravity. So weightlessness is when the acceleration of a body is equal to the acceleration caused by gravity.
A body orbiting the earth has the force of gravity pulling on it. This force is inversely proportional to the square of the distance of the body from the centre of the earth:
(where f = force, r = distance from body to centre of earth, m = mass of body, a = acceleration of body, v = velocity of body, and k is a proportionality constant)
f = k/r2
By Newton's second law, acceleration equals the force divded by the mass of a body:
a = f/m
So the acceleration caused by gravity for this particle is equal to:
a = k/r2m
Although the body orbiting the earth travels at a nearly constant speed, it does not have a constant velocity. Velocity is a vector quantity depending on both speed and direction. A change in direction implies a change in velocity, which in turn implies acceleration. This centripetal (center-seeking) acceleration, caused by gravitational attraction of the body to the earth towards the center of the earth, equals:
a = v2/r
For weightlessness to occur, this acceleration has to equal the acceleration caused by gravity:
v2/r = k/r2m
v2 = k/rm
This shows that the weightlessness depends on the velocity the body is travelling, the radius of the orbit and the mass of the body.
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