PHY.10:5 - Archetypal Grounding

PHY.10:5.1 - Change speed to separate two drag accounts

A specimen moves through the same fluid at an imposed positive speed v. In the proposed operating range, account L takes the opposing force magnitude to be F=b v; account Q takes it to be F=c v². The coefficients are positive and constant under their respective accounts.

At v=1 m/s, an earlier calibrated force comparison places the true drag between 0.95 and 1.05 N. Thus L permits b between 0.95 and 1.05 N·s/m, while Q permits c between 0.95 and 1.05 N·s²/m². Both explain that observation.

A repeat at the same speed leaves this disagreement intact. Instead impose v=2 m/s while retaining specimen geometry and the fluid conditions on which the coefficients depend. L predicts a force between 1.9 and 2.1 N; Q predicts between 3.8 and 4.2 N. With an additional bounded force-readout error of ±0.1 N, the possible indications lie in [1.8,2.2] N and [3.7,4.3] N. They are disjoint.

The drive and readout must sustain that comparison. Under Q the required mechanical power can reach 8.4 W, since P=Fv. A 5 W drive cannot establish the proposed steady speed for every admitted Q response. A 10 W drive at that speed can meet this particular power demand, while its other operating limits remain relevant.

Suppose the comparison returns 2.05 N. It is compatible with L and incompatible with Q under the stated ranges and conditions. A return of 2.9 N instead disagrees with both. It prompts examination of the retained physical regime, parameter constraints and readout; it does not justify selecting whichever nominal curve is closer.

Heating or a geometry change can make a coefficient vary between runs. If that effect can span the separation, the test no longer has the same interpretation. Restore the shared condition, include the changed dependence or choose another contrast. Increasing speed without this return can defeat the very accounts being compared.

PHY.10:5.2 - Change a drive that makes storage and conduction coincide

A two-terminal element is driven with V(t)=V_0 exp(t/tau). One proposed account is an ideal resistor, I=G V. Another is an ideal capacitor, I=C dV/dt, prepared with charge C V_0 at the start of the recorded ramp.

Since dV/dt=V/tau, choosing C=G tau makes the complete current traces identical during this ramp. More accurate recording of that same drive cannot distinguish the accounts.

Use V_0=1 V, tau=2 s, G=1 mS and C=2 mF. Stop increasing the voltage when it reaches 2 V and hold it there. After the drive and readout have settled, the resistor predicts 2 mA and the ideal capacitor predicts zero current. The change removed dV/dt while retaining V.

The waiting interval must be interpreted physically. A source with finite output resistance and a detector with finite response can create a transient after the change. Include it or choose a readout time after its consequential effect. The ideal predictions also exclude a significant leakage path.

Now suppose the held-voltage current is 0.6 mA, while the earlier ramp still has I/V=1 mS. The ideal pair is inadequate for these records. A parallel conductance and capacitance give

I=G V+C dV/dt.

The hold gives G=0.3 mS. Substitution into the ramp relation G+C/tau=1 mS gives C=1.4 mF. The test has opened a physically different account in which conduction and storage coexist. These two records determine its two parameters under the stated idealization; they do not establish that this account suffices at every frequency or voltage.

The resulting distinction changes use. A continuously held voltage dissipates power through the conductance, while the capacitance stores charge and supplies a transient response. Subsequent pulse or frequency use can therefore ask a question that the original exponential ramp could not resolve.

PHY.10:5.3 - Refocus a hidden physical difference

A prepared ensemble has a transverse phase signal. Consider two idealized accounts of its free decay. In account S, each member has a fixed frequency offset delta, drawn from the Lorentzian density

p(delta)=Gamma/[pi (delta²+Gamma²)], with Gamma>0.

Each phase advances by delta t. Averaging over the ensemble gives M_S(t)=exp(-Gamma t) for t≥0. The individual offsets remain fixed even though the mean signal decays.

In account D, the phase instead has independent Gaussian increments with variance 2 Gamma dt over an interval dt. This Markov dephasing gives the same free signal, M_D(t)=exp(-Gamma t). The two accounts agree on this free-decay observation.

Apply a refocusing rotation at time tau and read the signal at 2 tau. In the ideal pulse comparison, the sign of phase accumulation is reversed for the second interval. Under S, each accumulated phase becomes delta tau-delta tau=0; the ensemble signal returns to 1. Under D, the two intervals have independent phase increments. Subtracting them leaves variance 4 Gamma tau, so the signal remains exp(-2 Gamma tau).

For Gamma=10 s⁻¹ and tau=0.1 s, the predicted refocused signals are 1 and approximately 0.135. The intervention exposes a difference hidden by the equal free decays.

A physical pulse has finite duration, range and accuracy. Its response over the occupied frequency range, other relaxation during the sequence and readout error must be included where they can change this separation. For illustration, if their combined effect on each predicted normalized signal is bounded by 0.05, the predicted indication intervals around 1 and 0.135 remain disjoint. This bound is a condition of that proposed implementation, not supplied by the ideal calculation.

If the pulse cannot refocus a consequential part of the ensemble, a small return can have that cause as well as irreversible dephasing. Change the pulse or reference comparison, retain its bounded effect, or leave the interpretation conditional. A partial echo can also motivate an account with both static variation and changing noise. The performed comparison then guides which hidden dynamics to retain.