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If you place a weight on a spring and it is an equilibrium, then you have this equation:

$mg = kx$

you would solve for k and get:

$k = mg/x$

but, if we used conservation of energy, assuming that when the mass is on the spring it is at x = 0, we would get:

$1/2kx^2 = mgx $ we would get $k = 2mg/x$

What assumption am I making that is false? How would the k value be different with two correct statements?

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You need to account for the KE when the mass is dropped and it moves downward. Unless the spring doesn't move, you can't use this energy equation. And if it doesn't move I don't think you could consider it a linear spring.

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