Uniform-Earth chord oscillator (${\omega }_0^2=4\pi G\rho /3$)

Inside a homogeneous spherical mass, gravity is proportional to position. Projection onto a fixed straight chord gives $\ddot{y}=-{\omega }_0^{2}y$, independently of its perpendicular offset from the sphere centre. Release from one surface endpoint at rest reaches the opposite endpoint after $\pi /{\omega }_0$. The transverse force is supplied by the tunnel constraint. This idealization neglects removed tunnel mass, rotation, density variation and dissipative forces.

Adding a velocity-proportional force $-b\dot{y}$ produces a damped harmonic oscillator. The fastest non-oscillatory release from rest is critical damping, with the repeated-root solution in the title. Reaching fraction $\epsilon$ of the initial displacement requires $(1+s)e^{-s}=\epsilon$, $s={\omega }_{0}t$. At one percent, $s=6.638352\dots$, rather than $\log 100$. Exact arrival at equilibrium takes infinite time; the tolerance reference must be stated.