Question #43533

This question tests the student's ability to transition from rigid-body approximations to continuous-medium mechanics in the presence of a spatial gradient.

The primary trap lies in assuming either a constant average density or a constant wave speed.

Students often blindly apply the standard formulas $\Delta L = \frac{\rho g L^2}{2Y}$ and $v = \sqrt{\frac{Y}{\rho}}$, failing to recognize that both the static stress distribution and the dynamic wave propagation speed are local properties that must be integrated over the variable density profile $\rho(y)$.

By asking for the ratio $\frac{\Delta L}{T^2}$, the problem forces a rigorous setup of two distinct differential equations: one for static equilibrium ($d\sigma/dy = \rho(y)g$) and one for kinematics ($dt = dy/v(y)$), effectively filtering out rote memorization.