Hanging Spring

Vertical mass-spring system: equilibrium

A light vertical spring with force constant $k$ is fixed to the ceiling. A block of mass $m$ is attached to the lower end and allowed to hang at rest.

Let the natural (unstretched) length of the spring correspond to $y = 0$, where $y$ is the downward extension of the spring from its natural length. Thus the block at rest hangs at $y = y_{\text{eq}}$ below $y=0$.

The situation is shown in the diagram:

Given: $k = 80\text{ N/m}$, $m = 0.52\text{kg}$, $g = 9.8\text{m/s}^2$.

Using force balance in the vertical direction at equilibrium, find the magnitude of the extension $y_{\text{eq}}$ (measured downward from the natural length) first symbolically in terms of $m$, $g$, and $k$, and then numerically.