= Mass change in a viscously damped spring oscillator
{title2=$\omega_d^2=k/m-b^2/(4m^2)$}
For fixed <spring constant> $k$ and linear <damping> coefficient $b$, replacing <mass> $m$ by $\lambda m$ changes the exponential envelope rate from $\gamma=b/(2m)$ to $\gamma/\lambda$. The damped <angular frequency> becomes $\omega_{d,\lambda}^2=(\omega_d^2+\gamma^2)/\lambda-\gamma^2/\lambda^2$. Thus envelope decay times scale exactly as <mass>, while the familiar inverse-square-root <frequency> scaling is a weak-damping approximation. A constant gravitational <force> shifts equilibrium without changing either rate.
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