Can the logarithmic decrement be found from extension of spring?

Question

Consider a spring-mass system in which a mass hangs freely from a spring fixed to a ceiling. Can the logarithmic decrement be found simply from the extension of the spring? The only parameters known are the initial and final spring lengths.

Answer 1

No, it can't.

The unstretched spring extension is the equilibrium solution to the unforced equation $$m\ddot x+\gamma\dot x=-k(x-x_0),$$ whereas the stretched extension is the equilibrium solution to the forced equation $$m\ddot x+\gamma\dot x=-k(x-x_0)-mg.$$ The equilibrium positions are completely insensitive to the damping constant $\gamma$, since you set $\dot x=0$ to find them, so systems with different $\gamma$s will have the same extensions. However, $\gamma$ directly affects the logarithmic decrement, so systems with different logarithmic decrements will have the same extensions.

What you need, therefore, is some sort of time-dependent measurement, which will yield a quantity that directly involves $\gamma$. In general, the logarithmic decrement is exactly such a quantity - it is easy to measure, it is explicitly dependent on time, and it cleanly gives you $\gamma$ in the forms that you care about. You can get around this by measuring other things, but they must all involve motion in some way.