PHY.9:5 - Archetypal Grounding

PHY.9:5.1 - Make mass affect the observation

Two bodies fall in a sufficiently uniform gravitational field with negligible drag. The equation m a=m g gives a=g. Timing their fall cannot distinguish their masses in this idealization, even though gravity exerts different forces.

Choose instead a known horizontal applied force F whose generation does not depend on the unknown mass. Track the body’s acceleration under that force. With other horizontal forces negligible, m a=F gives m=F/a for nonzero a. The physical change is from a force proportional to the unknown mass to an independently supplied force.

Now a constant horizontal force offset b matters during the short observation. Use two otherwise identical preparations with applied forces +F and -F. Assume the same b and the same mass in both, and measure their initial accelerations:

m a_+=F+b, m a_-=-F+b.

Subtracting gives

m=2F/(a_+-a_-).

With F=6 N, a_+=4 m/s² and a_-=-2 m/s², the mass is 2 kg and b=2 N. Using only F/a_+ would give 1.5 kg. The controlled reversal made mass distinguishable from the shared force offset.

The two preparations must preserve that offset. Velocity-dependent drag need not do so after the trajectories diverge. Measuring comparable initial responses, including the changed drag law, or selecting another arrangement can repair that use. Taking more samples of two physically different offsets does not justify the shared-b subtraction.

The acceleration readout and the force reference retain their own calibration and uncertainty. A decision whose margin exceeds those effects can use the mass estimate; a tighter use returns to the consequential contribution.

PHY.9:5.2 - Separate pressure from the probe’s surface force

A large liquid reservoir maintains an unknown gauge pressure p during a small probe’s readout. A vertical circular capillary opens to the atmosphere. Let rho be liquid density, g gravitational acceleration, r the tube radius and h the meniscus height above the pressure reference. Assume hydrostatic equilibrium and a capillary regime in which the meniscus curvature is described by the wetting angle theta.

Hydrostatic pressure and the surface-pressure jump give

p=rho g h - s/r, where s=2 gamma cos(theta)

and gamma is surface tension. The height responds to the wanted pressure and to the capillary surface force.

If s is unknown but the same surface condition can be prepared in two narrow tubes of radii r_1 and r_2, use

rho g h_1=p+s/r_1, rho g h_2=p+s/r_2.

For distinct radii,

p=rho g (r_1 h_1-r_2 h_2)/(r_1-r_2).

Take rho g=10000 Pa/m, r_1=0.1 mm, r_2=0.2 mm, h_1=0.20 m and h_2=0.15 m. The inferred pressure is 1000 Pa and s=0.10 N/m. Interpreting the first height as p=rho g h_1 would give 2000 Pa. Both narrow radii must remain within the chosen capillary approximation.

Now suppose the two surfaces have different, unknown wetting conditions. The equations contain separate s_1 and s_2. The cancellation no longer determines p: two readings with three unknown quantities leave a consequential freedom. Restore the common surface condition, use useful bounds on the surface terms, or select another pressure-sensitive interaction.

The reservoir premise also matters. If filling the capillary changes the pressure whose earlier value was wanted, include the reservoir-probe volume and pressure relation through PHY.6 and C.16.MR. The maintained-pressure calculation cannot by itself recover that earlier value.

PHY.9:5.3 - Prepare a probe whose phase difference can be read

An ideal two-level probe couples to a constant classical field B through

H=(hbar gamma B/2) sigma_z,

where gamma is a known coupling coefficient and sigma_z has eigenvalues +1 and -1 for states |0> and |1>. Treat the field as unchanged by this probe during the interrogation time t. The wanted quantity is B within a supplied range.

Prepare |0>. Evolution multiplies it by a global phase; measurements of this state cannot reveal that phase. Increasing t alone does not create a readable field dependence.

Instead prepare |+>=(|0>+|1>)/sqrt(2). Evolution produces a relative phase phi=gamma B t. Reading sigma_z still gives equal probabilities and is uninformative about phi. A controlled rotation before detection permits a sigma_x or sigma_y measurement. Their means are cos(phi) and sin(phi), so the two preparations for readout can recover phase modulo 2 pi. Each mean requires the corresponding repeated preparation and measurement.

For gamma=1 rad/(s·field-unit), t=1 s and B=pi/3 field-units, the ideal means are 1/2 and sqrt(3)/2. They select phi=pi/3 modulo 2 pi. A supplied range -pi < gamma B t <= pi makes that field value unique.

If the field may instead lie between -4 pi and 4 pi field-units, the same records admit several values. Shorten the initial interrogation, for example to t=0.1 s with the same gamma: the whole supplied range then lies inside the unambiguous phase interval. A later longer interrogation can refine a value once the remaining range supports its interpretation.

Coherence loss, imperfect rotations or an uncertain gamma change the response law. They can be included or constrained where the required estimate needs them. The local phase response alone is not a claim that arbitrarily long interrogation or another quantum resource improves every sensing task.