= Damping measurement from free decay
An experimentally useful question is whether the disappearing <oscillation> follows <viscous damping>, and whether increasing the brake changes $b$ rather than $m$ or $k$. A published teaching protocol uses a cart on an <air track>, attached between two <springs>, with ceramic magnets providing <eddy-current damping> and an <ultrasonic motion sensor> recording <position> against <time>. Replace the magnets with equal <masses> in the baseline trial so the moving <mass> stays fixed. Obtain each <spring constant> from the slope of applied <force> versus extension, then use $k=k_1+k_2$. Time many baseline cycles to check $\omega_0\simeq\sqrt{k/m}$. Compare one-magnet and two-magnet free decays; the protocol calls for a fit of the full trace including its offset. <Driven Harmonic Oscillator>{ref}
The following additional controls make the inference testable. Level the <air track> and hold its air supply fixed. Check the <ultrasonic motion sensor> at several stationary flag positions against a ruler, including the full planned travel range; record the offset and scale correction. Check elapsed acquisition <time> against an independent clock, and choose a <sampling rate> that resolves many samples within the shortest expected <oscillation period>. Record the resting <equilibrium> before and after each run. Release the cart from a small measured <displacement> with minimal initial <velocity>, with the driving apparatus inactive. Repeat at several release sizes and magnet settings while keeping the <springs> and total <mass> unchanged. These are proposed calibration and repeatability checks, not reported measurements from the source.
For <underdamping>, fit the recorded <position> to $y(t)=y_{\mathrm{eq}}+Ae^{-\alpha t}\cos(\omega_{\mathrm d}t+\phi)$ by <nonlinear regression>. The readout to inspect is a sequence of alternating extrema about $y_{\mathrm{eq}}$ with shrinking <amplitudes>, and the deviations of the fitted curve from the recorded points. The model predicts
$$
\alpha=\frac{b}{2m}=\zeta\omega_0,\qquad
\alpha^2+\omega_{\mathrm d}^2=\frac{k}{m}.
$$
Compare the right-hand side, determined independently from <mass> and <spring constants>, with the fitted left-hand side. A good fit supports a constant-coefficient single <normal mode> over the observed range; it does not establish a universal microscopic <force> law. The published protocol denotes the decay rate by $b$ and the <velocity>-proportional coefficient by $c$; here they are $\alpha$ and $b$, respectively.
At <critical damping> or <overdamping>, fit the corresponding nonoscillatory response including its offset and both <initial conditions>, rather than forcing a damped cosine onto the data. Adjusting the brake can explore the transition, but the cited protocol does not establish that its one-magnet or two-magnet settings attain <critical damping>. A near-critical trace over a short interval can be hard to distinguish from neighbouring <damping ratios>.
Important limitations are finite sensor range, missed or reflected <acoustic wave> pulses, <measurement noise> near <equilibrium>, nonzero release <velocity>, and position-dependent magnetic braking. Changing the release size tests for <quadratic drag> or nonlinear <spring> response; inspecting deviations near turning points can expose <kinetic friction>. Slow drift of the offset can imitate a long <overdamped> tail. Report uncertainties in fitted rates and independently measured $m,k$; absence of a visible crossing only bounds the behaviour within the instrument's sensitivity and recording interval. No numerical experimental results are asserted here.
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