Damped Simple Harmonic Motion — Question 10

Question 10

A detection threshold counts peaks but does not end the motion

Same-side peaks satisfy $A_n=A_0e^{-n\delta}$, starting at $n=0$. Let $A_0=10.0\,\mathrm{mm}$, $\delta=.20$ and $T_d=.50\,\mathrm{s}$. A detector records a peak only when its magnitude is at least $1.0\,\mathrm{mm}$.

1. Find the last recorded peak index, its time, and the total number of recorded peaks including the initial one.

2. Find the continuous exponential-envelope threshold time and explain why it differs from the last recorded peak time. Does the ideal linear oscillator stop there?

Question 10 - Solution

A detection threshold counts peaks but does not end the motion

(a) Derivation and calculation

$$A_n\ge1.00\,\mathrm{mm}\iff10e^{-.20n}\ge1\iff n\le\frac{\ln10}{.20}=11.5129.$$$$\boxed{n_{\mathrm{last}}=11},\qquad\boxed{t_{\mathrm{last}}=11(.500)=5.50\,\mathrm s}.$$$$\boxed{N_{\mathrm{recorded}}=11-0+1=12}.$$

(b) Result and interpretation

$$A_{11}=1.10803\,\mathrm{mm},\qquad A_{12}=.90718\,\mathrm{mm}.$$$$\beta=\delta/T_d=.400\,\mathrm{s^{-1}},\quad\boxed{t_{\mathrm{envelope}}=\frac{\ln10}{.400}=5.75646\,\mathrm s}.$$The interpolated peak envelope crosses threshold between recorded peak times. The ideal motion decays asymptotically and does not stop at the detector threshold.

Physical check

The inclusive condition records the initial peak and would also record a peak exactly on the threshold.