How to derive the time period equation for a spring mass system taking into account the mass of the spring without involving energy analysis? I want to know the way to derive the time period equation of a spring mass system accounting for the mass of the spring but not using the energy analysis method but by proceeding in the same way as we do by ignoring the mass of the spring. Please help. I did not find any texts at my level. Any links would suffice gratefully.
 A: I needed to solve the same question. Here is what i did:-
Suppose that we have a spring of mass $M$ with a mass $m$ attached to it, suspended against gravity. Assume that the mass of the spring is distributed evenly through its length. So, we can imagine our spring to be divided into infinitesimally small springs. Each small spring will have its own rate of oscillation depending upon the effective mass felt by it. Suppose the length of the spring lies along $x$ axis with origin at the bottom tip of the spring. 
Now, assume that we have completely compressed the spring such that it can no more be compressed against its solid self. Let the length of the spring in this state be $L$. Now, for a small spring element located at a distance $x$ from the bottom in this state, time period $T$ of oscillation will be given by-
$T=2\pi \sqrt{\frac{\frac{M.x}{L} +m}{k} } $
Here, $k$ is the spring constant of the spring element.
So, frequency $f$ of simple harmonic motion can be given by-
$f=\sqrt {\frac{k} {\frac{M.x}{L} +m} } $
We realize that every spring element will have different frequency of oscillation. To simplify and approximate, we can imagine the entire spring to have the frequency of the spring element located at the middle of the spring.
So, for the time period of the entire spring, we get-
$T=2\pi \sqrt{\frac{\frac{M}{2} +m}{k} } $
Problem solved...!
Note that this solution will be valid only for a small time duration after the spring is allowed to oscillation. For example, if we let the spring oscillate for several minutes, we will start observing discrepancy. But for small time duration since the beginning of the experiment, the results will be quite accurate.
Edit: I just checked my physics lab manual, which has following equation for $T$:
$T=2\pi \sqrt{\frac{\frac{M}{3} +m}{k} } $
So, my answer was close. My approximation turns out to be too crude, and instead of assuming the frequency of the whole system to be close to that of the spring element halfway, we can get a better approximation if we assume it to be one-third from bottom, for the sake of calculus...
The equation in the book is also approximate, one must remember.
