# Extension in Massive Springs under gravity

Suppose we have a spring which has a uniform cross sectional area $$A$$, and has a uniform linear mass density $$J$$, and has a total unstretched length $$L$$. Let the spring also obey Hooke's Law.

Further, suppose we hang this spring in a room from the ceiling and hold the spring in unstretched state. Then we slowly bring the spring to its equilibrium state under gravity (if I just release the spring from initial state, I think the Motion of the spring will be really complex, so I assume we bring it to a state of equilibrium very slowly)

Finally, we are to find the extension in the spring.

I'm at a loss as to how to approach this question. I've never encountered massive Springs in my course of study and am at a loss of how to approach problems related to them.

I'm looking for a hint to approach the problem.

Any Help would be appreciated, Thank You.

The trick is to evaluate the load situation of a slice of lenght $$\Delta h$$. As it is at rest, the downward force at the bottom of the slice $$(F_h)$$, plus the weight of the slice must be equal to the upward force at the top of the slice $$(F_{h+\Delta h})$$.
After converting to tensions, applying Hooke's law, and going to limit where $$\Delta h \to 0$$ we get a differential equation. Using the appropriate boundary conditions, it is possible to find the displacement as a function of $$h$$.