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I evidently have not been paying attention, merely noting that John's contributions have been less frequent lately.
...assuming that the potential energy is given by 1/2 kx^2, ... is incorrect because the stretching isn't linear. They effectively rederive the standard "add 1/3 of the mass of the spring" result...
My own solution to the problem is available at
http://www.csupomona.edu/~ajm/special/Dec08TPT.pdf
...
A. JOHN "Slo" MALLINCKRODT
Professor Emeritus of Physics