Need discussion about a heavy spring hanging down

[akak.ak88]

Homework Statement

Hang a heavy spring from it's top.
Let it hanging down naturally.
The total mass of the spring is M.
The spring constant of the spring is k.
Asking it's extension.



2. Homework Equations
Describing the potential energies( spring force energy or gravity)
with math or physics equation.
Describing the motion of the spring
with math or physics equation.
Or please tell me everything you know or interesting about this heavy spring.




3.My professor told the whole class that he is going to put any questions about this heavy spring in our midterm test, so that I need many helps to let me know this heavy spring more clearly. This spring is so important to me.

 

[LawrenceC]

One thing you know is that the distance the center of mass of the spring drops from its unstreached position represents a change in potential energy. When the spring comes to rest and quits oscillating, the aforementioned potential energy change must equal the stored potential energy in the spring.

When it comes to rest after oscillating, the stretch in the spring will be nonuniform. It'll stretch more where its fastened at the top than where it is attached to M. Potential energy is not stored equally throughout the spring. The storage of potential energy increases as you go upwards due to the additional stretch.

The weight expanding the spring coils varies linearly with the distance from the top due to the uniformity of the coils (assumed).

The system will oscillate up and down until the internal damping of the spring and air drag damp the motion out. The temperature of the spring will rise slightly due to the work being done on it by the constant bending of the material as the spring oscillates.

 

[akak.ak88]

Thank your accurate concepts, which help me think more clearly
about the energy and motion.
I still have the problem about writing the formula or making assumption.
Could you please show me some initial formula about
the aforementioned potential energy change must equal the stored potential energy in the spring and how to describe the up and downs with the formula.
Really thank for answering, it really give me a good direction to think more.