(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

My question is in regards to a relatively simple problem:

A massless, ideal spring with a spring constant k hangs from the ceiling; it is originally neither compressed nor stretched. We attach a block of mass m to the spring. In terms of the mass of the block, the spring constant k, and the acceleration due to gravity, g, how far does the spring stretch from it's equilibrium position?

While this question is not terribly complicated, I am confused because I feel that I should be able to do it two ways and get the same answer: calculations using 1) force (Hooke's Law) and 2) Gravitational Potential Energy/Elastic Potential Energy should presumably give the same result for the displacement from equilibrium (x) of the spring. But they don't (at least when I do it). I will demonstrate what I have done below.

2. Relevant equations

In order to do this Problem using net force and Hooke's Law we only need two equations:

F(net) = ma

F(spring) = -kx

In order to do this Problem using conservation of energy we only need three:

Total Energy(initial) = Total Energy (final)

U(spring) = (1/2)kx^2

U(gravity) = mgh

3. The attempt at a solution

Via net force equations:

F(net) = ma = 0 because the block is suspended from the spring. This means that

F(spring) - F(gravity) = 0

so F(spring) = F(gravity)

so kx = mg

so x = mg/k

Via Energy equations:

Energy(initial) = mgh(initial) = Energy(final) = U(spring) + mgh(final)

Rearranging we get:

mg(h(initial)-h(final)) = (1/2)kx^2

h(initial)-h(final) = x (the displacement) so:

mgx = (1/2)kx^2

Solving this we get:

x= 2mg/k

This is clearly different than the original answer.

So my first question is: Why are these two calculations different? It seems like they should give me the same value for x.

And my second question is: how can I add an addtional term into my energy equation to get the same answer as the force method? This should be fixable.

I know this is a long post, but any clarification on this topic would be greatly appreciated. Thanks!

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# Inconsistent calculations with Hooke's Law/Elastic Potential Energy

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