Formula solvers — O

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Obert–Duvall Pillar Strength

S_p = S_1 \left( 0.778 + 0.222 \, \frac{W_p}{h} \right)

Rock MechanicsObert and Duvall's 1967 linear fit for the strength of a hard-rock pillar, scaled from the strength of a cubical specimen by the pillar's width-to-height ratio. Squat pillars are stronger than slender ones because the platens confine the middle of them, and this is the simplest honest expression of that fact.

Odds and Probability

O = \frac{P}{1 - P}

ProbabilityStatisticsConverts between a probability and odds in favour, the ratio of the chance it happens to the chance it does not.

Odds Ratio (Cross-Product)

\mathrm{OR} = \frac{a\,d}{b\,c}

Epidemiology & Diagnostic AccuracyThe odds of exposure among cases divided by the odds among controls, which collapses to the cross-product of a 2×2 table. It is the measure a case-control study can produce, because such a study fixes its own denominators and cannot report a risk.

Ohm's Law

V = I R

Electricity & MagnetismPhysicsRelates the voltage across a conductor to the current through it and its resistance.

Ohnesorge Number (Viscosity against Inertia and Surface Tension)

Oh = \frac{\mu}{\sqrt{\rho \sigma L}} = \frac{\sqrt{We}}{Re}

Fluid MechanicsViscous forces measured against the geometric mean of inertia and surface tension, with no velocity in it at all. Ohnesorge is what decides HOW a drop or jet breaks up, where Weber decides WHETHER — and it is exactly √We/Re, which makes it the clearest illustration on this site that which dimensionless groups you choose is a matter of convenience, while how many you need is not.

Olson's Single-Exponential Litter Decay

X = X_0 \, e^{-k t}

Forestry & Forest EcologyJerry Olson's 1963 model: litter loses a constant PROPORTION of whatever is left in every equal interval, so the mass remaining falls exponentially. One fitted constant k describes a whole litter type, and it is the most-used equation in decomposition ecology despite fitting the late stages of decay badly.

One-Sample T-Test Statistic

t = \frac{\bar{x} - \mu}{s / \sqrt{n}}

StatisticsAlgebraTests a sample mean against a claimed value when the standard deviation is estimated from the sample itself rather than known.

One-Sample Z-Test Statistic

z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}

StatisticsAlgebraTests a sample mean against a claimed population mean when the population standard deviation is known, in standard-error units.

Orbital Period

T = 2\pi \sqrt{\frac{r^{3}}{GM}}

Astronomy & GravitationMechanicsPhysicsTime for one circular orbit of radius r around a central mass M — Kepler's third law in Newtonian form.

Orbital Velocity

v = \sqrt{\frac{GM}{r}}

Astronomy & GravitationMechanicsPhysicsSpeed of a body in a circular orbit of radius r around a central mass M.

Orifice Plate Flow

Q = \frac{C_d}{\sqrt{1 - \beta^{4}}}\cdot\frac{\pi d^{2}}{4}\sqrt{\frac{2\,\Delta P}{\rho}}

Fluid MechanicsPhysicsVolumetric flow through a square-edged orifice plate from the differential pressure across it. The 1/√(1−β⁴) term is the velocity-of-approach factor, with β the bore-to-pipe diameter ratio.

Osmotic Pressure (Π = MRT)

\Pi = M R T

ChemistryGives the osmotic pressure of a dilute solution from its molar concentration and absolute temperature using the van 't Hoff equation.

Otto Cycle Efficiency (Compression Ratio)

\eta = 1 - \frac{1}{r^{\gamma - 1}}

ThermodynamicsPhysicsAir-standard efficiency of a spark-ignition engine from its compression ratio alone. Nothing about fuel, speed or displacement enters — only how hard the charge is squeezed and the heat capacity ratio of the working gas.

Outlier Lower Fence

LF = Q_1 - 1.5 \, IQR

StatisticsAlgebraTukey's lower cutoff for outliers: any observation below one and a half interquartile ranges under the first quartile is flagged.

Outlier Upper Fence

UF = Q_3 + 1.5 \, IQR

StatisticsAlgebraTukey's upper cutoff for outliers: any observation above one and a half interquartile ranges beyond the third quartile is flagged.

Overall Column Efficiency

E_o = \frac{N_t}{N_a}

Chemical EngineeringThe bridge from the calculation to the hardware: theoretical stages divided by the actual trays needed to do their work. Real trays never reach equilibrium, so there are always more of them than the theory asked for.

Overall Equipment Effectiveness (OEE)

\mathit{OEE} = A \times P \times Q

Industrial EngineeringThe single number total productive maintenance is built on: availability times performance times quality. Three losses multiplied, never averaged.

Overall Heat Transfer Coefficient (U)

\frac{1}{U} = \frac{1}{h_i} + \frac{L}{k} + \frac{1}{h_o}

Heat TransferThermodynamicsHVAC & HydronicsOverall coefficient U for a plane wall with fluid on both sides, adding the inside film, the wall and the outside film as resistances in series.

Overall U from Total Resistance

U = \frac{1}{R_{tot} A}

Heat TransferThermodynamicsHVAC & HydronicsConverts an assembly's total resistance in kelvin per watt into the overall coefficient U quoted on exchanger and envelope datasheets.

Overlap Loss from Swath Spacing

O = \frac{w - s}{w}

Farm OperationsThe share of a machine's swath that lands on ground the previous pass already covered, from the working width and the actual pass-to-pass spacing. The number a guidance system is bought to reduce.

Oxygen Transfer Rate — OTR = k_L·a (C* − C_L)

OTR = k_L a \, (C^{*} - C_L)

Chemical EngineeringThe rate at which a stirred, sparged vessel can push oxygen into broth: a lumped coefficient multiplied by the shortfall between saturation and what the probe reads. Oxygen is only about 7 mg/L soluble in water, a dense aerobic culture consumes that much in a few seconds, and so this single number decides how much biomass a bioreactor can carry.