Looks like a MIT 8:01 problem to me :)
It is too much for me at present but this link might help a little:
https://books.google.com/books?id=PTj2eEbT_jQC&pg=PA67&dq=elongation+of+long+hanging+rod+due+to+own+weight&hl=en&sa=X&ei=7TGfVI_7NcufgwSx-YKYCg&ved=0CGwQ6AEwCQ#v=onepage&q=elongation%20of%20long%20hanging%20rod%20due%20to%20own%20weight&f=false
When relaxed, an elastic cord has length L, cross section area A, mass M, and Young modulus Y.
An object of mass m is hung from the ceiling using the cord. The system reaches a steady state.
What is the longitudinal mass distribution lambda(z) of the stretched cord as a function of distance z from the object?
Hint: consider the stress-strain relation only for an infinitesimal segment of the cord. You may assume that the cross section remains A.
As usual, don't forget to explore the limits of your result.
3 answers
try deleting the s on https
http://books.google.com/books?id=PTj2eEbT_jQC&pg=PA67&dq=elongation+of+long+hanging+rod+due+to+own+weight&hl=en&sa=X&ei=7TGfVI_7NcufgwSx-YKYCg&ved=0CGwQ6AEwCQ#v=onepage&q=elongation%20of%20long%20hanging%20rod%20due%20to%20own%20weight&f=false
http://books.google.com/books?id=PTj2eEbT_jQC&pg=PA67&dq=elongation+of+long+hanging+rod+due+to+own+weight&hl=en&sa=X&ei=7TGfVI_7NcufgwSx-YKYCg&ved=0CGwQ6AEwCQ#v=onepage&q=elongation%20of%20long%20hanging%20rod%20due%20to%20own%20weight&f=false
Note equation 1.78 seems to be missing an = sign just left of the integral sign