Formula solvers starting with P — 138 solvers

P_cm Cracking Parameter (Ito–Bessyo)

P_{cm} = C + \frac{Si}{30} + \frac{Mn + Cu + Cr}{20} + \frac{Ni}{60} + \frac{Mo}{15} + \frac{V}{10} + 5B

Welding & JoiningIto and Bessyo's 1968 composition parameter, fitted to cracking tests on low-carbon steels rather than to hardenability on the higher-carbon steels the IIW expression came from. Carbon is weighted far more heavily relative to everything else, which is what makes it the fairer screen for a modern microalloyed grade.

P-Wave Velocity from Elastic Moduli

V_p = \sqrt{\dfrac{K + \frac{4}{3}G}{\rho}}

SeismologyHow fast a compressional wave travels through an elastic solid: bulk modulus plus four-thirds of the shear modulus, over density, square-rooted. The first arrival on every seismogram, and the wave that gives the P its name.

P₂O₅ to Elemental Phosphorus

P = 0.4364 \times P_2O_5

Crop ProductionConverts the phosphate figure on a fertiliser label to the elemental phosphorus a soil test or tissue analysis reports, using the mass share of P in P₂O₅.

Packed Column Height from HTU and NTU

Z = H_{OG} \, N_{OG}

Chemical EngineeringA packed column's height split into the two things that set it: how hard the separation is, and how good the packing is at it. Chilton and Colburn's transfer-unit idea, and the reason packing is specified in metres of HTU rather than in stages.

Paint Quantity from Coverage Rate

N = \frac{A \, c}{a}

Trades & ConstructionContainers of paint needed: the surface area times the number of coats, divided by the area one container covers at the manufacturer's spread rate.

Parabolic Grain Growth Law

D^{\,n} - D_0^{\,n} = K \, t

Metallurgy & Heat TreatmentOnce recrystallisation is finished, the new grains start consuming each other to reduce total boundary area, and the mean size grows as a power of time. Beck and co-workers fitted it in 1948. It is why an over-long soak undoes exactly the strength that Hall–Petch said the fine grain had bought.

Parallel Axis Theorem (I = I_c + Ad²)

I = I_c + A d^{2}

Strength of MaterialsCivil & SurveyingGeometrySecond moment of area of a shape about an axis parallel to its own centroidal axis: add A times the offset squared. The transfer term that builds every plate girder and flitch beam.

Parallel RLC Admittance

Y = \sqrt{G^{2} + (B_{C} - B_{L})^{2}}

Electricity & MagnetismElectrical TradeTotal admittance of a parallel RLC branch in siemens, where the capacitive and inductive susceptances cancel before joining the conductance.

Parallel System Reliability (Redundancy)

R_s = 1 - (1 - R)^{n}

Computer ScienceProbabilityReliability of n identical components when any single one is enough: the chance that they do not all fail together. Redundancy buys nines fast, provided the failures are genuinely independent.

Parallelogram Area

A = b \cdot h

GeometryArea of a parallelogram from its base and perpendicular height.

Parallelogram Area from Two Vectors

A = \left|a_x b_y - a_y b_x\right|

Vectors & MatricesGeometryAlgebraFinds the area of the parallelogram spanned by two plane vectors directly from their four components, no angle or height required.

Parallelogram Perimeter

P = 2(a + b)

GeometryPerimeter of a parallelogram from its two side lengths.

Paris Law Crack Growth Rate

\dfrac{da}{dN} = C \left( \Delta K \right)^{m}

Strength of MaterialsParis and Erdogan's 1963 observation that fatigue crack growth plots as a straight line on log-log axes against the range of stress intensity. It is the equation damage-tolerant design is built on — the reason a cracked aircraft structure can be flown, inspected and flown again rather than scrapped.

Paris Law Cycles to Failure

N = \dfrac{a_f^{\,1 - m/2} - a_i^{\,1 - m/2}}{\left( 1 - \tfrac{m}{2} \right) C \left( Y \, \Delta\sigma \sqrt{\pi} \right)^{m}}

Strength of MaterialsThe Paris rate integrated from the flaw you found to the flaw that fails: how many more cycles the part has. This is the arithmetic behind every damage-tolerance inspection interval in aviation and every fitness-for-service remaining-life assessment on a pressure vessel. It assumes constant amplitude loading and a geometry factor Y that does not change as the crack grows.

Parshall Flume Free Flow

Q = 4 \, W \, H^{1.522 \, W^{0.026}}

Water & WastewaterFluid MechanicsFree-flow discharge through a Parshall flume of 1 to 8 ft throat width, with W and the head H in feet and Q in cubic feet per second.

Partial Pressure from Mole Fraction

P_i = x_i \, P_{\text{total}}

ChemistryThermodynamicsGives a gas component's partial pressure as its mole fraction times the total pressure, the practical form of Dalton's law of partial pressures.

Partially Filled Horizontal Cylindrical Tank

V = L \left[ r^{2} \cos^{-1}\!\left(\frac{r-h}{r}\right) - (r-h)\sqrt{2rh - h^{2}} \right]

HVAC & HydronicsFluid MechanicsGeometryLiquid volume in a horizontal cylinder from the wetted depth, using the circular segment area times the tank length.

Particles from Moles (Avogadro's Number)

N = n\,N_A

ChemistryConverts an amount in moles into an actual particle count using the Avogadro constant NA = 6.02214076 × 10²³ per mole.

Particulate Collection Efficiency

\eta = \frac{C_{in} - C_{out}}{C_{in}}

Air Quality & DispersionThe fraction of the dust entering a control device that does not leave it, from inlet and outlet loadings. The number on every baghouse, cyclone and precipitator datasheet, and the one most often quoted with too many nines.

Pasquill–Gifford Dispersion Coefficient

\sigma = a \, x^{b}

Air Quality & DispersionThe crosswind or vertical spread of a plume at a downwind distance, in the power-law form fitted to the Pasquill stability classes. The width term the Gaussian model needs.

Passphrase Entropy (Diceware and Word Lists)

H = w\log_2 D

Cryptography & Key SizingEntropy in bits of a passphrase built by drawing w words uniformly at random from a list of D. Diceware's standard list holds 7776 words — 6^5, one word per roll of five dice — so each word contributes exactly log2(7776) = 12.925 bits. The dice matter: rolling them is what makes the draw uniform, and choosing words that 'feel random' is not the same operation.

Password Entropy (Uniform Random Generation)

H = L\log_2 b

Cryptography & Key SizingEntropy in bits of a password GENERATED by picking L symbols uniformly at random from an alphabet of b. This page takes a length and an alphabet size and has nowhere to type a password, on purpose: entropy describes how a secret was produced, not what it looks like, so no calculation on the characters of a human-chosen password can recover it.

Peak Sun Hours from Insolation

PSH = \frac{E}{G_{ref} A}

Solar & Wind PowerA day's solar energy on a surface, re-expressed as the number of hours of full 1000 W/m² sun that would have delivered the same total. The tidy fiction that turns a curve across the sky into a single number a designer can multiply by.

Péclet Number for Mass Transfer

\mathrm{Pe} = \frac{u L}{D}

Chemical EngineeringWhether the stream carries the solute along faster than diffusion can spread it out: bulk advection divided by molecular diffusion. The number that decides if a reactor behaves like a plug or like a stirred tank.

Penetration Rate from Corrosion Current Density

P = \frac{i \, M}{n \, F \, \rho}

Corrosion & ProtectionFaraday's law applied to a corroding metal: a corrosion current density converted into the thickness of metal it removes per year, through the equivalent weight and the density.

Per-Capita Wastewater Flow

q = \frac{Q}{P}

Water & WastewaterWater TreatmentFluid MechanicsAverage wastewater contributed per person per day, from the plant flow and the population served, reported in gallons per capita per day.

Per-Unit Base Impedance

Z_{base} = \frac{V_{base}^{2}}{S_{base}}

Electricity & MagnetismElectrical TradeThe ohms that count as 1.0 per unit on a chosen voltage and apparent-power base — the yardstick every per-unit fault study is measured against.

Percent Backwash Water

\%BW = \frac{V_{bw}}{V_f} \times 100

Water & WastewaterWater TreatmentShare of a filter's production consumed by its own backwash, the housekeeping ratio that flags short runs and poor coagulation.

Percent Blowdown

\%B = \frac{B}{M} \times 100

Water TreatmentFluid MechanicsBlowdown expressed as a percentage of makeup water — the share of purchased water that goes straight to the sewer.

Percent Change

c = \frac{x_{\text{new}} - x_{\text{old}}}{x_{\text{old}}}

AlgebraRelative change from an old value to a new one, as a fraction of the old value.

Percent Composition of an Element

\%X = \frac{a\,M_X}{M_{\text{compound}}} \times 100\%

ChemistryGives the mass percent an element contributes to a compound from the formula subscript and molar masses.

Percent Difference

PD = \frac{|x_1 - x_2|}{(x_1 + x_2)/2}

StatisticsAlgebraCompares two measurements of equal standing by dividing their gap by their average, when neither counts as the accepted value.

Percent Error

PE = \frac{|x_{\text{meas}} - x_{\text{acc}}|}{x_{\text{acc}}}

StatisticsAlgebraHow far a measurement strays from the accepted value, expressed as a fraction of that accepted value for lab reports and calibration checks.

Percent Grade from Rise and Run

G = \frac{100\,\Delta h}{L}

Civil & SurveyingGeometryPercent grade is one hundred times the vertical rise divided by the horizontal run, measured level, never along the slope.

Percent Ionization of a Weak Acid

\%\,\text{ion} = \frac{[\mathrm{H^+}]}{C} \times 100\%

ChemistryExpresses what fraction of a weak acid has actually donated its proton, comparing the equilibrium hydrogen ion concentration to the formal concentration.

Percent Solution in the Tank

P = 100 \, \frac{V_p}{T}

Crop ProductionThe strength of a spray mix stated as a percentage of the tank: product volume over tank volume. How handgun, spot-treatment and greenhouse labels are written, in place of a rate per unit area.

Percent Voltage Drop

\%V_{d} = \frac{100 \, V_{d}}{V_{s}}

Electrical TradeElectricity & MagnetismVoltage drop expressed as a percentage of the supply voltage, the form codes use for the 3% branch and 5% total limits.

Percent Yield

\%\,\text{yield} = \frac{m_{\text{actual}}}{m_{\text{theoretical}}} \times 100\%

ChemistryCompares the mass actually isolated from a reaction to the maximum mass stoichiometry predicts.

Perfectly Inelastic Collision

v = \frac{m_1 u_1 + m_2 u_2}{m_1 + m_2}

MechanicsPhysicsCommon velocity of two bodies that stick together after a perfectly inelastic collision, from conservation of momentum.

Performance Rating (Linear Approximation)

R_p = R_{avg} + \frac{400 \, (W - L)}{N}

Games & RatingsStatisticsWhat a player's results in one event were worth, expressed on the rating scale: the average rating of the opponents faced, adjusted by 400 points for every whole point of score above or below an even split. This is the linear form used for quick estimates, and it is an approximation of a definition that is otherwise iterative.

Period of a Spring-Mass Oscillator

T = 2\pi \sqrt{\tfrac{m}{k}}

Waves & OscillationsMechanicsPhysicsPeriod of a mass bouncing on a spring, set only by the mass and the spring stiffness.

Period-Frequency Relation

T = \frac{1}{f}

Waves & OscillationsPhysicsPeriod and frequency are reciprocals: seconds per cycle versus cycles per second.

Periodic Temperature Profile

T(x,t) = T_m + A \, e^{-x/d} \sin\!\left(\frac{2\pi t}{P} - \frac{x}{d}\right)

Heat TransferThe full harmonic solution: the temperature at depth x and time t when the surface swings sinusoidally about a mean. The exponential is the decrement factor and the term subtracted inside the sine is the phase lag, so this one line contains both of the previous pages — and it is what you use when you want a temperature rather than a ratio.

Permutations (nPr)

{}_{n}P_{r} = \frac{n!}{(n-r)!}

ProbabilityAlgebraNumber of ways to choose r items from n distinct items when the order of the selection matters, as with podium finishes.

Perpendicular Slope Relation

m_2 = -\frac{1}{m_1}

AlgebraGeometrySlope of a line perpendicular to a given line — the negative reciprocal, so that the two slopes always multiply to −1.

Petroff's Equation (Journal Bearing Friction Torque)

T_f = \frac{4 \pi^{2} \mu N r^{3} L}{c}

Machine DesignThe friction torque of a journal running concentric in its bearing, from Petroff's 1883 analysis: shear the oil film uniformly around the journal and add it up. It assumes the journal sits centred, so it is exact only for a lightly loaded bearing — but it is remarkably close for real ones, and it is where the Sommerfeld number came from.

Peukert's Law (Battery Runtime)

t = H \left( \frac{C}{I H} \right)^{k}

Electricity & MagnetismElectrical TradeHow long a battery really lasts at a chosen current, given its rated capacity, the discharge time that rating was measured over, and its Peukert exponent.

PFR Conversion (First Order)

X = 1 - e^{-\mathrm{Da}}

Chemical EngineeringHow far a first-order reaction gets in a plug-flow tube, as a function of the Damköhler number alone. Reactant decays exponentially along the length, so conversion closes on 100% far faster than a stirred tank can: Da = 2.3 gives 90%, Da = 4.6 gives 99%.

PFR Design Equation (First Order)

V = \frac{v_0}{k}\ln\!\left(\frac{1}{1 - X}\right)

ChemistryVolume of a plug-flow (tubular) reactor for a first-order liquid-phase reaction. Concentration falls smoothly along the tube instead of dropping to the outlet value at the inlet, so a PFR always needs less volume than a CSTR for the same conversion.

pH and pOH Relation

\mathrm{pH} + \mathrm{pOH} = 14

ChemistryLinks the acidity and basicity scales of any aqueous solution at 25 °C, where pKw = 14.

pH from Hydrogen Ion Concentration

\mathrm{pH} = -\log_{10}\,[\mathrm{H^+}]

ChemistryWater TreatmentExpresses acidity as the negative base-10 logarithm of the hydrogen ion concentration in mol/L.

pH of a Weak Acid from Ka

\mathrm{pH} = -\log_{10}\sqrt{K_a\,C}

ChemistryEstimates the pH of a weak monoprotic acid solution from its dissociation constant and formal concentration using the standard x-is-small approximation.

Phase Angle from Power Factor

\varphi = \arccos(\text{PF})

Electrical TradeElectricity & MagnetismThe angle by which current lags or leads voltage, found from the power factor — the link between the ratio and the triangle.

Photo Scale

S = \frac{f}{H - h}

Photogrammetry & Remote SensingThe scale of a vertical photograph: focal length divided by the height of the camera above the ground being imaged. Written as a fraction — 1:20 000 means one on the photo is twenty thousand on the ground. The classical form of the same similar triangles that give the ground sample distance, and the one every photogrammetry course starts with.

Photoelectric Effect

KE_{max} = E_{photon} - \phi

Modern PhysicsPhysicsMaximum kinetic energy of an ejected electron: photon energy (hf) minus the work function.

Photon Energy (E = hf)

E = h f

Modern PhysicsPhysicsThe energy of a single photon: Planck's constant times the light's frequency.

Photon Energy from Wavelength (E = hc/λ)

E = \frac{h c}{\lambda}

Modern PhysicsPhysicsPhoton energy written in terms of wavelength: shorter waves, more energetic photons.

Photon Momentum (p = h/λ)

p = \frac{h}{\lambda}

Modern PhysicsPhysicsMassless but not momentum-less: a photon carries h divided by its wavelength.

Pieces from a Stock Length (with Saw Kerf)

N = \frac{L_s + k}{L_p + k}

Trades & ConstructionHow many finished pieces come out of one stock length once every cut eats a kerf of saw blade. The last piece needs no trailing kerf, which is where the +k on top comes from.

Pieces per Load

n = \min\!\left(\frac{V_a}{v_p},\; \frac{W_{\max}}{w_p}\right)

Freight & ShippingHow many identical pieces fit before one of the two limits stops you. The smaller of the cube answer and the weight answer, because a load has to satisfy both and satisfying one is no help.

Pielou's Evenness (J′)

J' = \frac{H'}{\ln S}

Ecology & BiodiversityStatisticsShannon diversity as a fraction of the largest value it could take for that many species, so a community with every species equally abundant scores 1 and a community dominated by one species scores near 0. A proportion, never a count.

Pillar Factor of Safety

FS = \frac{S_p}{\sigma_p}

Rock MechanicsPillar strength divided by pillar stress: the number the whole room-and-pillar exercise exists to produce. Both terms come from correlations with real scatter in them, so the factor of safety is a way of carrying that uncertainty explicitly rather than a promise about any one pillar.

Pipe Internal Volume

V = \frac{\pi D^{2}}{4} L

HVAC & HydronicsFluid MechanicsWater TreatmentThe liquid a run of pipe holds, from inside diameter and developed length — the starting point for every flush, fill or chemical dose.

Pipe Velocity from Flow and Diameter

v = \frac{4Q}{\pi D^{2}}

HVAC & HydronicsFluid MechanicsWater TreatmentAverage velocity in a full round pipe from volumetric flow and inside diameter — the first check on any piping design.

Pitting Resistance Equivalent Number (PREN)

PREN = \%Cr + 3.3\,\%Mo + 16\,\%N

Corrosion & ProtectionThe single index that ranks stainless steels for resistance to pitting and crevice attack in chloride: chromium, plus molybdenum weighted 3.3, plus nitrogen weighted 16. A composition arithmetic, not a measurement.

pKa from Acid Dissociation Constant

\mathrm{p}K_a = -\log_{10} K_a

ChemistryConverts an acid dissociation constant into its logarithmic pKa form and back, compressing a huge range of acid strengths onto one readable scale.

pKb from Base Dissociation Constant

\mathrm{p}K_b = -\log_{10} K_b

ChemistryConverts a base dissociation constant into its logarithmic pKb form and back, the basic-side counterpart of the pKa scale.

Plant Population from Row and Seed Spacing

P = \frac{1}{w \, s}

Crop ProductionThe plant population a geometry produces: one plant occupies a rectangle of row spacing by in-row spacing, so the population is simply the reciprocal of that rectangle's area.

Plastic Moment Capacity (Mp = Z fy)

M_p = Z f_y

Strength of MaterialsCivil & SurveyingMoment at which a compact steel section is fully plastic, with every fibre at yield. The nominal flexural strength Mn used in LRFD steel design before the φ factor.

Plastic Section Modulus — Rectangle

Z = \frac{b h^{2}}{4}

Strength of MaterialsCivil & SurveyingGeometryPlastic section modulus of a solid rectangle, the first moment of the two half-areas about the equal-area axis. Exactly 1.5 times the elastic section modulus bh²/6.

Plastic Zone Radius (Mohr–Coulomb)

R_p = R \left[ \frac{2\left(p_0(k-1) + \sigma_{cm}\right)}{(1+k)\left((k-1)p_i + \sigma_{cm}\right)} \right]^{\frac{1}{k-1}}

Rock MechanicsHow far out from a circular tunnel the ground has failed, for a Mohr–Coulomb rock mass in a hydrostatic stress field. This is the classical closed-form convergence–confinement result, and it is the page that shows what support actually buys: in the plastic range, a modest confinement shrinks the failed ring sharply, which is the opposite of what happens in the elastic range.

Plasticity Index (PI = LL − PL)

PI = LL - PL

Soil MechanicsStrength of MaterialsPlasticity index of a fine-grained soil as the liquid limit minus the plastic limit, the width of the moisture range where clay behaves plastically.

Plume Centreline Temperature Rise

\Delta T_0 = 25.0 \, \dfrac{\dot{Q}_c^{2/3}}{(z - z_0)^{5/3}}

Fire Protection & Fire DynamicsHow hot the gases are on the axis of a buoyant plume, a stated height above the fire. Heskestad's far-field solution: the convective part of the fire drives the column, and the temperature rise falls off as the 5/3 power of the height above the plume's virtual origin.

Point Load Strength Index and the UCS it Implies

\sigma_c \approx k \, I_{s(50)}, \qquad I_{s(50)} = \left( \frac{D_e}{50\,\mathrm{mm}} \right)^{0.45} \frac{P}{D_e^{2}}

Rock MechanicsThe ISRM Suggested Method's field strength test: break a lump of rock between two conical platens, divide the breaking load by the square of the equivalent core diameter, correct it to a 50 mm standard size, and multiply by a conversion factor to estimate the uniaxial compressive strength. The test is cheap and portable. The conversion factor is where all the uncertainty lives, so it is an input here and not a constant.

Point Source Radiant Heat Flux

\dot{q}'' = \dfrac{\chi_r \, \dot{Q}}{4 \pi R^{2}}

Fire Protection & Fire DynamicsThe simplest radiation model in fire protection: treat the fire as a point radiating a fixed fraction of its heat release rate equally in all directions, and the flux at a target falls off as the inverse square of the distance. Crude, quick, and the first thing anyone reaches for when asked how far away is far enough.

Point-Slope Form of a Line

y = y_1 + m(x - x_1)

AlgebraGeometryEquation of a straight line through one known point with a known slope, giving the y-value at any x you care to choose.

Poiseuille's Law

Q = \frac{\pi \, \Delta P \, r^{4}}{8 \mu L}

Fluid MechanicsPhysicsLaminar flow rate through a round pipe — proportional to the fourth power of the radius.

Poisson Probability

P(X = k) = \frac{\lambda^{k} e^{-\lambda}}{k!}

ProbabilityStatisticsChance of exactly k events in a fixed interval when events occur independently at a constant average rate lambda.

Poisson's Ratio

\nu = \frac{\varepsilon_{lat}}{\varepsilon_{ax}}

Strength of MaterialsMechanicsPhysicsPoisson's ratio, the lateral contraction per unit of axial extension — close to 0.30 for steel and 0.33 for aluminium.

Polar Moment of Inertia — Solid Shaft

J = \frac{\pi d^{4}}{32}

Strength of MaterialsMechanicsGeometryPolar moment of inertia of a solid round shaft, J = πd⁴/32, in m⁴ — exactly twice the diametral moment of inertia.

Polarization Resistance

R_p = \frac{\Delta E}{\Delta I}

Corrosion & ProtectionThe slope of potential against current at the corrosion potential: apply a few millivolts, measure the current that flows, divide. The measurement that feeds the Stern-Geary equation.

Pooled Standard Deviation

s_p = \sqrt{\frac{(n_1 - 1) s_1^{2} + (n_2 - 1) s_2^{2}}{n_1 + n_2 - 2}}

StatisticsAlgebraCombines two sample standard deviations into a single estimate of common spread, weighting each by its degrees of freedom.

Population Equivalent

PE = \frac{W}{w}

Water & WastewaterWater TreatmentPopulation equivalent of a waste stream: its pollutant load divided by the load one person contributes, typically 60 g BOD per day.

Pore Water Pressure (u = γw zw)

u = \gamma_w z_w

Soil MechanicsFluid MechanicsHydrostatic pore water pressure at a point below a static water table, from the depth of water standing above it.

Positive Likelihood Ratio (LR+)

\mathrm{LR^+} = \frac{\mathrm{Se}}{1 - \mathrm{Sp}}

Epidemiology & Diagnostic AccuracyHow many times more often a positive result turns up in someone with the condition than in someone without it. A multiplier on the odds, not a probability.

Positive Predictive Value (PPV)

\mathrm{PPV} = \frac{\mathrm{TP}}{\mathrm{TP} + \mathrm{FP}}

Epidemiology & Diagnostic AccuracyOf everyone the test called positive, the share who genuinely have the condition — read along the test-positive ROW of a 2×2 table rather than down a column, which is what makes it move with prevalence.

Post-Test Odds (Odds × Likelihood Ratio)

O_{post} = O_{pre} \cdot \mathrm{LR}

Epidemiology & Diagnostic AccuracyBayes' theorem with the arithmetic taken out of it: in odds, an update is one multiplication. Pre-test odds times the likelihood ratio gives post-test odds, and O/(1 + O) turns that back into a probability.

Pounds of Active Chemical in a Tank

m = V \times \mathrm{SG} \times \rho_w \times p

Water TreatmentChemistryGives the mass of active chemical held in a storage tank from its volume, the solution's specific gravity and the percent active ingredient.

Powder Consumed by a Build

m = \rho_b \, V

Additive ManufacturingApparent (bulk) density multiplied by the volume of the powder bed. The number that matters is that V is the whole ENVELOPE the build sweeps — plate area times build height — not the volume of the parts. A powder-bed machine fills the entire box whether the parts occupy five per cent of it or fifty.

Powder Factor

PF = \frac{m_e}{V_r}

Rock MechanicsThe mass of explosive spent per unit volume of rock broken — the single number a blast is judged and costed by. Everywhere except North America it is kilograms per cubic metre; in North America it is pounds per cubic yard, or per short ton, and the three are quoted interchangeably by people who then wonder why the figures do not agree.

Power (P = W/t)

P = \frac{W}{t}

MechanicsPhysicsAverage power as work or energy delivered per unit time.

Power Factor from Real and Apparent Power

\text{PF} = \frac{P}{S}

Electrical TradeElectricity & MagnetismPower factor as the ratio of real power in watts to apparent power in volt-amperes, the share of supplied capacity doing work.

Power from Force and Velocity (P = Fv)

P = F v

MechanicsPhysicsInstantaneous power delivered by a force parallel to the velocity.

Power of a Power

\left( b^{m} \right)^{n} = b^{mn}

AlgebraRaising a power to another power multiplies the two exponents, the exponent law behind every nested exponential expression.

Power-Factor Correction Capacitance

C = \frac{Q_{c}}{2\pi f V^{2}}

Electrical TradeElectricity & MagnetismCapacitance needed across a single-phase load to supply a given number of vars at the supply voltage and frequency.

Power-Factor Correction kvar

Q_{c} = P \left( \tan\varphi_{1} - \tan\varphi_{2} \right)

Electrical TradeElectricity & MagnetismSize of the capacitor bank in vars needed to lift a load from its existing power factor up to a chosen target power factor.

Power-Law Reaction Rate

r = k\,C_A^{\,n}

Chemical EngineeringThe rate of a reaction as a power of the reactant concentration. The reaction order n is fitted from data, not read off the balanced equation — and because the units of k depend on n, this is the equation where careless unit handling does the most damage.

Power-Law Viscosity Ratio

\frac{\eta}{\eta_{ref}} = \left( \frac{\dot{\gamma}}{\dot{\gamma}_{ref}} \right)^{\,n-1}

Polymers & Plastics ProcessingThe Ostwald–de Waele power law, written as a ratio between two points on the same flow curve instead of against a consistency index. Give it one measured viscosity at one measured shear rate and the index n, and it gives the viscosity anywhere else on the straight part of the curve. Written this way every term has an honest unit — which the textbook form cannot manage.

ppm to mg/m³ Conversion

C = \frac{ppm \cdot M}{24.45}

Air Quality & DispersionConverts a gas concentration by volume into one by mass, using the molar volume of an ideal gas at 25 °C and 101.325 kPa. The bridge between an analyser that reads in ppm and a limit written in mg/m³.

Prandtl Number

\mathrm{Pr} = \frac{\mu c_p}{k}

Heat TransferFluid MechanicsThermodynamicsFluid property group comparing how fast momentum diffuses with how fast heat does, setting the relative thickness of the two boundary layers.

Predicted Value from a Regression Line

\hat{y} = a + b x

StatisticsAlgebraReads a prediction off a fitted least-squares line for any chosen value of the predictor, given the intercept and slope.

Present Value

\mathit{PV} = \frac{\mathit{FV}}{(1 + r)^{t}}

AlgebraToday's value of a future amount, discounted at rate r per period.

Press Brake Bending Force — Air Bending

F = \frac{K_b \, UTS \, L \, t^{2}}{V}

Machining & TurningThe force an air bend takes: the bending factor times the tensile strength, times the length of the bend, times the square of the thickness, all divided by the die opening. The thickness is squared, which is why a small increase in gauge doubles the tonnage, and the die opening is in the denominator, which is why opening the die is the cheapest way to get a bend a small press can make.

Pressure (P = F/A)

P = \frac{F}{A}

MechanicsPhysicsPressure as perpendicular force spread over an area.

Pressure Head (h = P/ρg)

h = \frac{P}{\rho g}

Fluid MechanicsWater TreatmentPhysicsConverts a pressure into the equivalent height of a fluid column, with g = 9.80665 m/s².

Prevalence

P = \frac{C}{N}

Epidemiology & Diagnostic AccuracyThe share of a population that has the condition at a given moment. Existing cases, not new ones — a stock, not a flow.

Price Elasticity (Midpoint Form)

E = \frac{\dfrac{Q_2 - Q_1}{(Q_1 + Q_2)/2}}{\dfrac{P_2 - P_1}{(P_1 + P_2)/2}}

Prices & MarketsProportional change in quantity divided by the proportional change in price, both measured against the MIDPOINT of the two observations. One formula serves both curves and the SIGN says which: NEGATIVE is demand — quantity fell as price rose — where below −1 is elastic and between −1 and 0 inelastic; POSITIVE is supply — quantity rose with price — where above 1 is elastic and below 1 inelastic.

Priming Sugar for a Carbonation Level

m = \frac{1.9635 \, V \,(C_t - C_r)}{y}

Brewing & FermentationHow much sugar to add at packaging so the yeast produces exactly the carbonation you want, allowing for the CO₂ the beer already holds and for what the sugar actually yields.

Prism Volume (General Cross-Section)

V = B L

GeometryVolume of any prism or extrusion whose cross-section does not change along its length: cross-sectional area times length.

Probability of At Least One Success

P = 1 - (1 - p)^{n}

ProbabilityStatisticsChance that at least one of n independent attempts succeeds, found as one minus the chance that every single attempt fails.

Process Capability (Cp)

C_p = \frac{\mathit{USL} - \mathit{LSL}}{6\sigma}

Industrial EngineeringHow the tolerance band compares with the spread of the process: the width of the specification divided by six standard deviations. Says nothing about whether the process is centred.

Process Capability Index (Cpk)

C_{pk} = \min\!\left(\frac{\mathit{USL} - \mu}{3\sigma},\ \frac{\mu - \mathit{LSL}}{3\sigma}\right)

Industrial EngineeringCapability measured to the nearer specification limit, so an off-centre process is penalised: the smaller of the two one-sided indices. Cpk can never exceed Cp, and equals it only when the process is perfectly centred.

Producer Surplus (Linear Supply)

\mathit{PS} = \tfrac{1}{2}\,(P - P_{\min})\,Q

Prices & MarketsArea of the triangle between the price received and a straight-line supply curve: what sellers were paid, less the least they would have accepted. P_min is the shutdown price, where the supply line meets the vertical axis and the first unit is offered.

Product Dose from Active Strength

D_p = \frac{100 \, D_a}{A}

Water TreatmentChemistryConverts a required active-ingredient dose into the as-supplied product dose when the drum is only a given percent active.

Product of the Roots of a Quadratic

P = \frac{c}{a}

AlgebraVieta's relation giving the product of a quadratic's two roots directly from its coefficients, without solving the equation.

Product Rate for a Nutrient Rate

R_p = \frac{R_n}{f}

Crop ProductionHow much fertiliser product delivers a target nutrient rate, from the product's analysis. The step between an agronomist's recommendation and a spreader setting.

Projectile Maximum Height

H = \frac{v_0^{2} \sin^{2}\theta}{2g}

MechanicsPhysicsPeak height reached by a projectile launched at a given speed and angle above level ground, ignoring air resistance.

Projectile Range on Level Ground

R = \frac{v_0^{2} \sin 2\theta}{g}

MechanicsPhysicsHorizontal distance a projectile covers over level ground, from its launch speed and angle, ignoring air resistance.

Projectile Time of Flight

T = \frac{2 v_0 \sin\theta}{g}

MechanicsPhysicsTotal time a projectile stays airborne before returning to its launch height, set by launch speed and angle with g = 9.80665 m/s².

Propellant Mass Flow Rate from Thrust and Isp

\dot m = \frac{F}{I_{sp} \, g_0}

Aerospace & FlightHow much propellant an engine drinks per second to hold a given thrust at a given specific impulse. The bridge between the thrust a mission needs and the tankage it has to carry to get it.

Puckorius (Practical) Scaling Index

\mathrm{PSI} = 2\,\mathrm{pH_s} - \mathrm{pH_{eq}}, \quad \mathrm{pH_{eq}} = 1.465\,\log_{10}\mathrm{Alk} + 4.54

Water TreatmentChemistryA Ryznar variant that replaces measured pH with an equilibrium pH derived from alkalinity, giving a truer scaling call in poorly buffered cooling water.

Pulley System Effort Force

F = \frac{W}{n}

MechanicsPhysicsEffort force needed to lift a load with a pulley system, divided down by the number of rope sections supporting the load.

Pulse Peak Power, Average Power and Duty Cycle

P_{avg} = E_p f_{rep} = P_{pk} \, \tau \, f_{rep}

OpticsThe bookkeeping that connects the two power numbers on every pulsed-laser datasheet. Average power is what the wall socket and the cooling loop see; peak power is what the workpiece and the optics see, and on a Q-switched laser it is thousands of times larger. The duty cycle τ·f_rep is the whole of the ratio between them.

Pump Affinity Law — Flow vs Impeller Diameter

\frac{Q_2}{Q_1} = \frac{D_2}{D_1}

HVAC & HydronicsFluid MechanicsWater TreatmentCapacity scales directly with trimmed impeller diameter at constant speed, the classic way to de-rate an oversized centrifugal pump permanently.

Pump Affinity Law — Flow vs Speed

\frac{Q_2}{Q_1} = \frac{N_2}{N_1}

HVAC & HydronicsFluid MechanicsWater TreatmentFirst affinity law: a centrifugal pump's capacity changes in direct proportion to shaft speed when the impeller diameter is unchanged.

Pump Affinity Law — Head vs Impeller Diameter

\frac{H_2}{H_1} = \left(\frac{D_2}{D_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentDeveloped head falls with the square of the trimmed impeller diameter, so a 10% trim sheds about 19% of the head at constant speed.

Pump Affinity Law — Head vs Speed

\frac{H_2}{H_1} = \left(\frac{N_2}{N_1}\right)^{2}

HVAC & HydronicsFluid MechanicsWater TreatmentSecond affinity law: pump head varies with the square of shaft speed, so a 20% speed cut costs 36% of the developed head.

Pump Affinity Law — Power vs Speed

\frac{P_2}{P_1} = \left(\frac{N_2}{N_1}\right)^{3}

HVAC & HydronicsFluid MechanicsWater TreatmentThird affinity law: absorbed power varies with the cube of shaft speed — the single relation that pays for every variable-frequency drive.

Pump Brake Horsepower

BHP = \frac{Q \, H \, SG}{3960 \, \eta}

HVAC & HydronicsFluid MechanicsWater TreatmentShaft power the motor must actually supply; the 3960 constant assumes gpm, feet of head and horsepower, with efficiency as a fraction.

Pump Efficiency from Hydraulic and Shaft Power

\eta = \frac{P_{hyd}}{P_{shaft}}

HVAC & HydronicsFluid MechanicsWater TreatmentPump efficiency is the ratio of hydraulic power delivered to the liquid over the mechanical power absorbed at the shaft.

Pump Specific Speed (Ns)

N_s = \frac{N \sqrt{Q}}{H^{0.75}}

HVAC & HydronicsFluid MechanicsWater TreatmentThe dimensional index that classifies impeller type, evaluated in US units with N in rpm, Q in gpm and H in feet at the best efficiency point.

Pump Water Horsepower

WHP = \frac{Q \, H \, SG}{3960}

HVAC & HydronicsFluid MechanicsWater TreatmentUseful power delivered to the liquid; the 3960 divisor assumes US gallons per minute, feet of head and horsepower output.

Pure Live Seed

PLS = p \times g

Crop ProductionThe fraction of a seed lot that is both the species you paid for and capable of growing. Purity times germination, and the only honest basis on which to compare two lots of forage or turf seed.

PV Array Energy Yield

E = P_{dc} \, H \, PR

Solar & Wind PowerThe energy a photovoltaic array delivers over a period: its DC rating, times the peak sun hours the site receives, times a performance ratio covering every loss between the glass and the meter. NREL's PVWatts model in one line.

PV Temperature Derate

P = P_{stc} \left[ 1 + \gamma \left( T_c - 25^\circ\text{C} \right) \right]

Solar & Wind PowerWhat a photovoltaic module actually delivers once its cells are hotter than the 25 °C the nameplate was measured at. The correction that explains why a panel makes less on the hottest afternoon of the year than on a bright cold morning.

Pyramid Volume

V = \frac{1}{3} B h

GeometryVolume of any pyramid as one-third of its base area times its perpendicular height.

Pythagorean Expectation

W\% = \frac{RS^{\,k}}{RS^{\,k} + RA^{\,k}}

Games & RatingsStatisticsEstimates the win percentage a team deserved from the points it scored and the points it allowed, rather than from the games it happened to win. Bill James named it for its resemblance to the Pythagorean theorem when the exponent is two; the resemblance is the only thing the two have in common.

Pythagorean Theorem

a^{2} + b^{2} = c^{2}

GeometryRelates the three sides of a right triangle.