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If I connect one end of a big spring to the ground and pull up the other end... How much weight I will be lifting in that moment? Does it increase as I'm pulling up or it remains a constant thing for that spring?

OR: If I find the quantity of the force that I'm applying to the spring... Can I know how much weight that much force would lift from the ground?

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Weight in physics is usually defined as the force on some object of mass $m$ due to gravity.

How much weight I will be lifting in that moment?

Since "weight" is a force, this question does not make much sense to me. You cannot lift a "force".

If I connect one end of a big spring to the ground and pull up the other end.

You will extend the spring by some length $x$. Normally springs are easy to extend at the start and get increasingly more difficult to extend the longer they get. In many cases (and for small extensions) the force needed to extend a spring is proportional to the extension (Hooke's Law):

$$F=kx$$

This is only the force needed to essentially deform the material of the spring (i.e. to extend it). As you can see from the expression, the force increases as you extend the spring further.

If it is a big heavy spring, in addition to this force you will also need to overcome the gravitational force for the parts of the spring that you lift.

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