SHM Spring System is Independent of Gravity?

[GameJammer]

For a mass-spring system,
Period, T = 2pi * root(m/k)

So using hookes law,
F = kx
But if the spring is being stretched by a mass due to gravity,
mg = kx
So,
k = mg/x

But then this means,
Period, T = 2pi * root(mx / mg)
or,
T = 2pi * root(x / g)

Where have I gone wrong? I've been told countless times that a spring-mass system's period is independent of g, but it seems my proof states otherwise.

Thanks

 

[TSny]

Hlello, GameJammer. Welcome to PF!

So using hookes law,
F = kx
But if the spring is being stretched by a mass due to gravity,
mg = kx
So,
k = mg/x

Just to make sure, what is the precise meaning of x in this equation? Is it an arbitrary value of x or some specific value of x?

 

[GameJammer]

x is the extension of the spring, it's part of Hookes law

 

[TSny]

Yes. But it's the extension of the spring under what condition?

 

[GameJammer]

It's the extension of the string due to force F

 

[TSny]

Does x in your equation k = mg/x represent how much the spring is stretched if you just hang the mass on the spring and let it sit in equilibrium? Or does it represent something else?