This is a part of the "Cambridge Physics Problems" thread, an exercise which I have taken up myself. Please read
here for a short introduction on the types of questions featured in threads of the same name. Please search for "Cambridge Physics Problems" for threads featuring past questions.
I appreciate all of your help on this. If possible, please show me the mathematical workings of the question (since this is what the question emphasises).
Thank you!
Question:
A cable of mass m and area of cross-section A has length L when it is laid out on the ground and measured. It is then suspended vertically on one end.
By considering a small length
δx of the suspended cable, show that the total energy stored as a result of the extension of the cable under its own weight is
6AELm2g2, where E is the Young modulus of the material of the cable. Assume that Hooke's law applies.
Attempt:
I am assuming that at different distances from the point of hanging, the wire carries different values of stress (and hence different values of strain). So in this case, the "discs" at different sections of the wire will produce different values of extension, and I right?
Stress in a section of the cable at a distance x below the point of suspension is expressed as
LA(L−x)mg.
Thus strain in a section of the cable at a distance x below the point of suspension is expressed as
LAE(L−x)mg.