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Premelting is the persistence of a liquid layer at a surface or interface below the bulk melting temperature. Molecular disjoining pressure and phase equilibrium can stabilize the film. Its hydraulic connection to a reservoir permits frost heave when water freezes onto the base of loaded ice.

Pressure calibration assigns pressure values to a sensor's output by comparison with a reference under specified conditions. It is not the same as making the output more finely resolved. A smooth electrical resistance curve can be repeatable yet have an incorrect pressure scale.

Bridgman used pressure-dependent electrical resistance of manganin and established fixed points, including a bismuth transition. He reported poorer accuracy above the better calibrated range.^[ref]

A fixed-point specimen and a property-measurement sample have distinct roles: one helps assign the scale, while the other is studied on that scale. The reference transition must itself have an assigned pressure at the relevant temperature. Agreement at a single point does not determine a nonlinear sensor response elsewhere. Comparing increasing and decreasing pressure checks hysteresis; returning to a reference checks drift. These are checks one can design from the measurement model, not additional historical controls asserted for the lecture's experiments.

The pressure coefficient compares a local static pressure with a specified reference static pressure, using a fixed reference dynamic pressure:The reference mass density, speed, pressure and location must accompany a reported pressure coefficient. Keep the same $q_{\infty }$ throughout a surface distribution; replacing it by local $\rho U^2/2$ gives a different quantity and fails at a stagnation point. The definition itself requires neither Bernoulli's principle nor inviscid flow, so measured pressure coefficients remain useful when those idealizations fail.

Positive $C_p$ means $p>p_{\infty }$; zero means equality; negative $C_p$ means $p<p_{\infty }$. This sign does not specify the sign of absolute pressure. For stipulated $p_{\infty }=101325\mathrm{Pa}$ absolute and $q_{\infty }=240\mathrm{Pa}$, $C_p=-3$ gives $p=100605\mathrm{Pa}$ absolute: a $720\mathrm{Pa}$ deficit, not a negative absolute pressure. If $p_{\infty }$ equals atmospheric pressure, the numerator is gauge pressure; a wind tunnel's reference static pressure need not equal room atmospheric pressure. These are reference choices, not different physical pressures.

Under the same-height, constant-mass density $\rho ={\rho }_{\infty }$, steady flow and inviscid flow hypotheses of the Bernoulli equation, and with the same Bernoulli invariant as the reference flow,The first equation immediately gives the second after subtracting $p_{\infty }$ and dividing by $q_{\infty }$. Hence a lossless stagnation point has $C_p=1$, $U=U_{\infty }$ gives $C_p=0$, and $U=2U_{\infty }$ gives $C_p=-3$. For a real stationary wall satisfying the no-slip boundary condition, the velocity is zero right at the wall; inserting that wall velocity in an inviscid flow formula would wrongly predict $C_p=1$ everywhere. A thin boundary layer approximation instead relates the wall pressure to the outer-flow pressure and uses the outer velocity where the Bernoulli equation applies.

For potential flow around a circular cylinder of radius $a$ with zero circulation, the surface tangential velocity is $u_{\theta }=-2U_{\infty }\sin \theta$. Thus the ideal pressure coefficient isWith $\theta$ measured from downstream, the front is at $\pi$ and the rear at zero. Both ideal stagnation points have $C_p=1$, while the sides at $\pi /2$ and $3\pi /2$ have $C_p=-3$. This exact model is a useful baseline for an experiment; its complete rearward pressure recovery is not a prediction for a separated real fluid wake.

To obtain a force, integrate the local normal contributions with their directions. Let $\mathbf{n}$ be the unit normal vector from the body into the fluid and $\Sigma$ its closed wetted surface:The second equality removes a spatially constant reference pressure over the closed surface; it does not remove a nonuniform hydrostatic pressure or justify dropping the reference term on an isolated open patch. A negative $C_p$ contributes outwards in this reference-subtracted calculation, even though a positive absolute pressure still pushes inwards locally. Project ${\mathbf{F}}_p$ onto the chosen lift and drag directions. The full fluid force also includes ${\int }_{\Sigma }{\mathbf{\sigma }}_{\mathrm{visc}}\mathbf{n}dA$ from the viscous stress tensor, including tangential shear stress; pressure tapping does not measure that contribution.

For the two-dimensional cylinder, force per unit span follows from $dA=ad\theta$ per unit span and $\mathbf{n}=(\cos \theta ,\sin \theta )$:Substituting $C_p=1-4{\sin }^2\theta$ gives $(0,0)$: opposite points have identical pressure coefficients and opposite normal vectors. Zero drag despite a strongly nonuniform pressure is the instructive limitation associated with D'Alembert's paradox. Real viscosity, boundary layers and flow separation can prevent rearward pressure recovery and produce pressure drag. Thus the measured curve can test the model and estimate a pressure contribution without proving that total drag vanishes.

A motivating measurement is Virginia Tech's Experiment 3 - Flow Past a Circular Cylinder. Its open-jet wind tunnel holds a rotatable $140\mathrm{mm}$ cylinder with end plates and 36 one-millimetre pressure tappings at ten-degree intervals. A reference Pitot-static tube supplies $p_{0\infty }$ and $p_{\infty }$ to one digital manometer; a second connects a surface tap to its positive port and reference static pressure to its negative port. Record the simultaneous readings:The last equality uses the low-speed stopping relation, not an additional speed measurement. The manual requires sealed tubing and zero checks with the tunnel off. It also provides a water-column manometer: read each column's initial level, and subtract the independently measured reference-to-atmosphere pressure from the tap-to-atmosphere readings. Record tap positions after rotation. Compare the two instruments and check spanwise uniformity; end plates do not guarantee two-dimensional flow. ^[ref]

The inferential distinction matters: the manometers directly measure pressure differences, the ratio estimates pressure coefficients, and a surface integral estimates the pressure part of the force. Measure room absolute pressure and temperature if converting the reference difference to speed through the ideal gas law $\rho =p_{\mathrm{abs}}/(RT)$; use thermodynamic temperature in that equation. For force reconstruction, cover the complete contour and use numerical integration with the tap geometry. Finite tap spacing, unknown end effects, probe disturbance and fluctuations limit that reconstruction; tubes and instruments also average or delay changing pressures. Retain the signed raw readings, repeated readings and reference readings so that these limits can be assessed. A small reference difference magnifies any zero error in the ratio, rather than producing a reliably large $C_p$.

At fixed temperature, $dG=VdP$ gives the pressure correction for the participating solids. Constant reaction volume reduces the integral to $\Delta V_{\mathrm{solids}}(P-P_0)$. A gas product treated by chemical fugacity instead contributes $\nu RT\ln (f/P_0)$ relative to its reference state. Adding a separate gas-volume correction to that term would double count the same pressure dependence.

Two fluids at a stationary, freely movable interface require equal pressures when surface tension and other normal stresses are negligible. This local pressure equilibrium does not require equal temperatures or mass densities. Curvature with nonzero surface tension or an applied load changes the normal-stress balance. Comparing points at different heights also requires the hydrostatic equilibrium condition.

A pressure sensor converts a pressure into a recorded signal. An absolute pressure sensor uses a vacuum reference; a gauge pressure sensor uses a specified reference pressure. A closed gas syringe and an absolute pressure sensor can test Boyle's law after allowing the temperature to stabilize.

Suppose a small volume change obeys $dV=-V_1{\kappa }_{T}dP+V_1\alpha dT$, with its displayed coefficients held constant. Work on the body is $\delta W=V_1{\kappa }_{T}PdP-V_1\alpha PdT$. Two rectangular pressure-temperature paths between the same endpoints differ by $W_a-W_b=V_1\alpha \Delta P\Delta T$, where path a changes temperature first. The straight interpolation samples the mean pressure during its temperature change, giving $W_c=(W_a+W_b)/2$. Variable material coefficients need not preserve that average relation.

The inferred history of pressure and temperature experienced by a rock. Dating specific growth domains and linking them to mineral reactions are needed to attach time to the path.

The temperature dependence of electric polarization at fixed strain.

The temperature at which a defined primer duplex is half dissociated under specified conditions. Sequence, salt and concentration matter; a simple composition rule is an approximation rather than a complete assay design.

The primordial atomic cooling curve describes hydrogen-helium gas without heavy elements. Near the hydrogen excitation threshold it rises sharply, then shows hydrogen and helium excitation/ionization features. Bound-state losses weaken when both elements become highly ionized. At sufficiently high temperatures thermal bremsstrahlung gives a slowly rising tail. Composition, density convention and ionization assumptions fix the numerical normalization.

With a weak conversion time proportional to $T^{-5}$ and a radiation-era age proportional to $g_{\ast }^{-1/2}{T}^{-2}$, cosmological weak freeze-out occurs at $T_f\propto g_{\ast }^{1/6}$. Extra radiation therefore raises the freeze-out temperature and the neutron-to-proton ratio $r_f=e^{-Q/T_f}$. Before decay, the primordial helium mass fraction is $Y_f=2r_f/(1+r_f)$, giving $\delta Y_f/Y_f=Q\delta \log T_f/[T_f(1+r_f)]$. A shorter delay through the deuterium bottleneck also leaves more neutrons. After freeze-out, conserve baryon number and evolve the neutron fraction to obtain $Y_p\simeq Y_f{e}^{-(t_N-t_f)/{\tau }_n}$. A precise abundance requires the weak rates and nuclear-burning history, not merely the number of relativistic species.

At fixed cosmological weak freeze-out temperature $T_f$ and fixed neutron survival factor, neutron-proton chemical equilibrium gives $r_f=e^{-Q/T_f}$ and neutron-limited helium synthesis gives $Y_p\propto 2r_f/(1+r_f)$. Differentiating yields the displayed sensitivity. A finite change $Q\mapsto (1+\alpha )Q$ gives $Y_p^{\prime }/Y_p=(1+e^{Q/T_f})/(1+e^{(1+\alpha )Q/T_f})$. Increasing the positive mass difference reduces helium production. Changes to the weak conversion rates, neutron lifetime or nuclear history must be supplied separately for a full abundance prediction.

Primordial abundances require extrapolation to material minimally processed by stars. Helium is inferred from hydrogen and helium emission in low-metallicity ionized regions, with ionization and temperature corrections. Deuterium is inferred from isotope-shifted absorption in low-metallicity quasar systems. Lithium is measured in old metal-poor stellar atmospheres and requires a correction, or a bound, for stellar depletion. These observables probe different astrophysical environments and systematic errors.

A prism minimum-deviation index measurement infers a transparent sample's relative refractive index from two angles: its known apex angle $A$ and the smallest deviation ${\delta }_{\min }$ of a selected spectral line. A spectrometer combines a source and slit, a collimator, a rotating optical prism table, and a viewing optical telescope with an angular readout. The collimator supplies a nearly parallel beam; the optical telescope locates its emergent direction. This turns an otherwise difficult measurement of the phase velocity of light into a measurement of angles.

For the symmetric path through a uniform optical prism, with the same external medium at both faces and positive refractive indices,The angles $i,e,r_1,r_2$ are measured from face normals. This is the minimum deviation relation, not a formula for an arbitrary prism orientation. An absolute refractive index additionally requires $n_{\mathrm{ext}}$.

For example, stipulate $A=60^{\circ }$ and $n_{\mathrm{rel}}=3/2$. Then $i=e=\arcsin (3/4)\simeq 48.5904^{\circ }$ and ${\delta }_{\min }\simeq 37.1808^{\circ }$. Substitution recovers $n_{\mathrm{rel}}=1.5$. These synthetic values illustrate the geometry; they are not readings for the prism in the laboratory protocol below.

A Michigan State University teaching experiment uses a mercury lamp, slit, collimator, prism table and optical telescope, with a subsidiary angular scale read to one arcminute. First focus the optical telescope on a distant object; with the prism removed, align its crosshair to a narrow vertical slit image, focus the collimator, and establish the direct-beam zero. Insert and centre the prism. Select the green mercury spectral line, tabulated at $546.1\mathrm{nm}$, then rotate the prism while tracking its image with the optical telescope. Locate the turning point where continued prism rotation reverses the image motion, and read the deviation. Re-establish the minimum separately for each other spectral line. The protocol assumes $A=60^{\circ }$; that assumed apex angle is not an independently measured result. ^[ref]

For an uncertainty-aware implementation, repeat the turning-point search from both prism-rotation directions, recheck the direct-beam reference, and record individual readings before averaging. Independently determine $A$ or include its calibration uncertainty; record the chosen spectral line, sample identity, temperature and surrounding-air conditions. These are proposed controls deduced from the inference, not claimed additional instructions or results of that manual. Finite slit width, unresolved spectral lines, imperfect collimation, table tilt, angular calibration and sample nonuniformity can limit the inferred refractive index. One arcminute readout resolution alone does not establish one arcminute total measurement uncertainty. A single selected wavelength gives one index, not a measured optical dispersion curve.

A short geometric derivation uses $r_1+r_2=A$ and $\delta =i+e-A$. Reversibility exchanges entrance and exit. At the symmetric stationary path, $i=e$ and $r_1=r_2$; applying Snell's law gives the displayed index relation. The plotted calculation checks that this stationary point is a minimum on the chosen transmitting branch. To assess local sensitivity, differentiate the inferred index using angles in radians:This first-order relation exposes both angular contributions; it is not an instruction to add signed systematic errors as independent random errors. In the synthetic example, a deviation error of $+1$ arcminute at fixed $A$ changes the inferred relative refractive index by approximately $+1.92\times 10^{-4}$.

A mixture of the maximally mixed state and a chosen pure state. Unitary conjugation preserves its identity part; every traceless observable has expectation $p⟨\psi |O|\psi ⟩$. The coefficient $p$ is effective polarization rather than literal purity: $\mathrm{tr}{\rho }^2=1/d+(1-1/d)p^2$. Preserving the largest population of $n$ weakly polarized equal spins and averaging the rest gives $p\simeq nϵ/[2(2^n-1)]$, displaying the signal penalty of this high-temperature ensemble preparation.

An instrument comparing dry-bulb and wet-bulb temperature to investigate atmospheric humidity.

Choose the illustrative local modelHere $t$ is the Celsius temperature, measured relative to $0{}^{\circ }\mathrm{C}$, so $\alpha t$ is dimensionless. For this chosen affine function, the derivative and inverse relation areCalculated values are $R(0{}^{\circ }\mathrm{C})=100\Omega$, $R(20{}^{\circ }\mathrm{C})=107.70\Omega$, $R(25{}^{\circ }\mathrm{C})=109.625\Omega$, and $R(50{}^{\circ }\mathrm{C})=119.25\Omega$. These are model outputs, not measurements or an assertion that any real Pt100 follows this line exactly. The number $0.00385$ here is the chosen slope parameter; it is not a universal local derivative for all platinum resistance thermometers. Curvature, manufacturing differences, and the fitted calibration interval matter. A wider-range calibration uses a validated response curve rather than extrapolating this line indefinitely.

For a separate synthetic thermometer calibration, stipulate $R_a=100.20\Omega$ at $t_a=0{}^{\circ }\mathrm{C}$ and $R_b=119.60\Omega$ at $t_b=50{}^{\circ }\mathrm{C}$. A two-point affine function has slope $0.388\Omega /{}^{\circ }\mathrm{C}$ and givesThus $R=109.90\Omega$ would indicate $25{}^{\circ }\mathrm{C}$. If an additional synthetic check at reference $25{}^{\circ }\mathrm{C}$ instead supplied $110.10\Omega$, the line would indicate $25.515\dots {}^{\circ }\mathrm{C}$. The $0.20\Omega$ residual exposes disagreement that two fitted points cannot reveal. This separate example changes both offset and slope; it is not a replacement specification for the nominal Pt100 local linear model.

A $3\times 1.5\times 1.5\mathrm{m}{\mathrm{m}}^3$ single crystal of gadolinium antimonide carried a $2\mathrm{mm}$ fibre Bragg grating cyanoacrylate-bonded along its $a$ axis. The pulsed magnetic field paralleled that axis. Broadband light, a circulator and a spectrometer resolved reflected-wavelength shifts at $5.4\mathrm{K}$. Pre-pulse readings measured noise; low fibre thermal expansion reduced thermal background. A residual post-pulse shift indicated possible heating. Spectral distortion checked bond uniformity, while the fibre's smaller cross-section limited loading; bonding was not assumed force-free. Longitudinal strain showed a kink near $32.6\mathrm{T}$.^[ref]

The distinction between readout and inference is essential. The fibre Bragg grating senses a change of reflected wavelength; interpreting this as the specimen's strain requires good strain transfer and a controlled temperature. A useful independent control would compare a bonded sensor with a nearby unloaded sensor through the same pulse. Another would vary bond stiffness or specimen cross-section to test loading. These are proposed checks, not additional procedures claimed for the experiment above. The pulsed magnet's winding details were not specified in the inspected account.

This experiment measures one longitudinal component of strain. Establishing the first-order volume change would require three orthogonal normal components under the same conditions. GdSb also has its own antiferromagnetism and field-dependent magnetic structure: this measured curve does not establish the simple saturated isotropic law. Its role here is to make the measurement chain and its limitations concrete.

Edward Mills Purcell, Henry Cutler Torrey, and Robert Vivian Pound placed paraffin wax in a capacitively loaded coaxial cavity near 30 MHz between magnet poles. Coupling loops supplied and extracted power; oscillating and steady fields were perpendicular. Output balanced against a generator reference fed amplification, detection and a microammeter. A slow field sweep revealed absorption above fluctuations at room temperature. Weak drive avoided population equalization. Calibration was rough; field inhomogeneity could contribute to linewidth.^[ref]

These are driven experiments with continuous radiofrequency excitation and changing static field, not demonstrations of modern pulsed Fourier transform nuclear magnetic resonance. Bloch's theoretical paper explicitly distinguishes proposed free precession after a pulse from the forced precession actually studied.^[ref]

An experimental failure need not mean absent nuclear magnetic moments. In the model below, excessive drive reduces the population imbalance through nuclear magnetic resonance saturation; waiting too briefly can leave insufficient polarization. Separately, ordinary electrical losses can cause heating without identifying a nuclear resonance. The absorption experiment measured a balanced electrical signal, rather than using heating as its readout. Useful diagnostic checks include moving off resonance, changing the frequency, checking field uniformity and reducing drive. These are general checks; the historical operations actually documented are specified above.