Formula solvers starting with M — 105 solvers

M/M/1 Average Number in System

L = \frac{\lambda}{\mu - \lambda}

Computer ScienceProbabilityAverage number of jobs in a single-server queue, waiting plus in service. It rises gently until the server is about 70% busy and then goes vertical.

M/M/1 Average Time in System

W = \frac{1}{\mu - \lambda}

Computer ScienceProbabilityAverage time a job spends in a single-server queue, from arrival to departure. It depends on the difference between the two rates, not on their ratio.

M/M/1 Queue Utilisation

\rho = \frac{\lambda}{\mu}

Computer ScienceFraction of time a single server is busy: the arrival rate divided by the service rate. Below 1 the queue is stable; at 1 or above it grows without limit.

Mach Number

Ma = \frac{v}{\sqrt{\gamma R T / M}}

Fluid MechanicsPhysicsSpeed divided by the local speed of sound in the same gas. Below about Ma = 0.3 a gas can be treated as incompressible, which is the assumption behind most duct and pipe sizing.

Machine Efficiency

\eta = \frac{W_{out}}{W_{in}}

MechanicsPhysicsEfficiency of a machine as useful work out divided by work in, with the shortfall lost to friction, heat, and noise.

Machine Utilisation

U = \frac{T_o}{T_a}

Industrial EngineeringShare of the time a machine was actually running: operating time divided by the time it was available to run. The answer depends entirely on which hours you count as available.

Machinery Cost per Unit Area

c = \frac{C_f}{A} + c_v

Farm OperationsWhat a machine costs per hectare: the annual fixed cost spread over the area it covers, plus the variable cost of covering it. The equation that explains why the same machine is cheap on one farm and ruinous on another.

Machinery Cost per Unit of Output

c_m = \frac{C_m}{Y}

Farm OperationsThe machinery bill expressed per tonne of crop rather than per hectare of land: machinery cost per hectare divided by yield per hectare. The form that makes an over-equipped farm visible.

Machining Time — Turning Pass

t_m = \frac{L}{f \, N}

Machining & TurningHow long one turning pass takes: the length to be travelled divided by how fast the tool advances, which is feed per revolution times revolutions per minute. The building block of every cycle time estimate and every quotation.

Magnetic Field of a Solenoid

B = \frac{\mu_0 N I}{L}

Electricity & MagnetismPhysicsThe uniform field inside a long coil of N turns and length L carrying current I.

Magnetic Flux (Φ = BA cos θ)

\Phi = B A \cos\theta

Electricity & MagnetismPhysicsThe magnetic field threading a surface: field times area times the cosine of the tilt angle.

Magnetic Force on a Current-Carrying Wire

F = B I L \sin\theta

Electricity & MagnetismPhysicsMagnitude of the force on a straight current-carrying wire in a uniform magnetic field.

Magnetic Force on a Moving Charge

F = q v B \sin\theta

Electricity & MagnetismPhysicsMagnitude of the magnetic (Lorentz) force on a charge moving through a magnetic field.

Magnification Factor of a Forced Vibration

M = \frac{1}{\sqrt{(1 - r^{2})^{2} + (2\zeta r)^{2}}}

Machine DesignHow much bigger the steady-state motion is than the deflection the same force would cause if it were applied slowly. At resonance it collapses to 1/(2ζ), which is why lightly damped structures amplify so violently.

Magnification from Heights (m = h_i/h_o)

m = \frac{h_i}{h_o}

OpticsPhysicsMagnification as the ratio of image height to object height.

Magnitude of a 2D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2}

Vectors & MatricesAlgebraGeometryFinds the length of a two-dimensional vector from its x- and y-components, the Pythagorean theorem written for arrows instead of triangles.

Magnitude of a 3D Vector

|\vec{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2}

Vectors & MatricesAlgebraGeometryFinds the length of a three-dimensional vector from its x-, y-, and z-components, extending Pythagoras into space.

Magnitude of a Scaled Vector

|k\vec{v}| = |k|\,|\vec{v}|

Vectors & MatricesAlgebraGives the length of a vector after multiplying it by a scalar, which stretches or shrinks the arrow without ever bending it.

Maintenance Dose Rate from Clearance

R = \mathrm{CL} \cdot C_p

Biomedical & ClinicalThe rate of drug administration that exactly replaces what clearance removes, holding a steady-state plasma concentration.

Major Poisson's Ratio of a Lamina

\nu_{12} = \nu_f V_f + \nu_m (1 - V_f)

Strength of MaterialsThe sideways contraction of a lamina pulled along its fibres, taken as the same volume-weighted average that gives the longitudinal modulus. It is the third of the four constants classical lamination theory needs, and it comes with a trap the isotropic world never prepares anyone for: the ratio measured across the fibres is a different, much smaller number.

Malt Colour Units

MCU = \frac{L \times W}{V}

Brewing & FermentationThe colour a grain bill contributes, weighted by how much of it there is and how far it is diluted — the raw input to every beer-colour correlation.

Manning's Equation for Flow

Q = \frac{1}{n} A R^{2/3} S^{1/2}

Water & WastewaterFluid MechanicsOpen-channel discharge by Manning's equation in SI form, from flow area, roughness, hydraulic radius and slope.

Manning's Equation for Velocity

v = \frac{1}{n} R^{2/3} S^{1/2}

Water & WastewaterFluid MechanicsOpen-channel velocity by Manning's equation in SI form, from the roughness coefficient, hydraulic radius and channel slope.

Manure Rate for a Nitrogen Target

R_m = \frac{R_n}{c \, a}

Crop ProductionHow much manure supplies a target of available nitrogen, from the manure's own analysis and the fraction of its nitrogen that becomes available in the first season.

Manure Storage Volume

V = n \, p \, t

Livestock & FeedThe storage a herd fills over a given number of days, from the head count and the manure produced per head per day. The calculation behind every regulated storage period.

Margin of Error for a Mean

E = z \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraHalf-width of a confidence interval for a mean, built from the critical z-value, the standard deviation and the sample size.

Margin of Error for a Proportion

E = z \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraThe plus-or-minus quoted with a poll result, built from the critical z-value, the sample proportion and the number of respondents.

Marginal Return on an Input

M = \frac{\Delta R}{\Delta C}

Farm OperationsWhat the last increment of an input actually returned: the extra revenue it produced divided by the extra it cost. The test for whether one more pass, one more unit of nitrogen or one more fungicide pays.

Marginal Revenue from Elasticity (Amoroso–Robinson)

\mathit{MR} = P\left(1 + \frac{1}{E_d}\right)

Prices & MarketsRevenue brought in by the next unit sold, when selling it requires cutting the price on every unit. Below the price whenever demand slopes down, and NEGATIVE once demand turns inelastic — which is why no firm with any pricing power sells there.

Markup and Margin Conversion

m = \frac{g}{1 - g}

Money & BusinessConverts directly between markup on cost and gross margin on price. A 50% markup is a 33.3% margin; a 20% margin is a 25% markup.

Markup Percentage (on Cost)

m = \frac{P - C}{C}

Money & BusinessMarkup states profit as a fraction of what the item COST you. A 25% markup on an $80 cost gives a $100 price.

Mash Strike Water Temperature

T_w = \frac{0.2}{R}\,(T_2 - T_1) + T_2

Brewing & FermentationHow hot the water must be so that grain at room temperature and water together settle at the mash temperature you want. Palmer's form, from a heat balance between grain and water.

Mass Law Transmission Loss

TL = 20\log_{10}\!\left(\frac{\pi m f}{\rho_0 c}\right) - 5

Acoustics & NoiseFor a single limp panel, blocking sound is almost entirely about weight: transmission loss climbs about 6 dB every time the surface density doubles, and another 6 dB every time the frequency doubles. It is the reason there is no light way to stop bass.

Mass Percent of a Solution

c = \frac{m_{\text{solute}}}{m_{\text{solution}}} \times 100\%

ChemistryExpresses a solution's concentration as the solute's share of the total solution mass in percent.

Mass-Energy Equivalence (E = mc²)

E = m c^{2}

Modern PhysicsPhysicsThe rest energy locked in mass: multiply by the speed of light squared.

Material Removal Rate — Milling

Q = a_e \, a_p \, v_f

Machining & TurningHow fast metal leaves the part under a milling cutter: the cross-section of the cut — width times depth — multiplied by how fast the table carries the work through it. A rectangle sweeping through a solid, and nothing more complicated than that.

Material Removal Rate — Turning

Q = V \, f \, a_p

Machining & TurningHow fast metal is leaving the part on a lathe: cutting speed times feed per revolution times depth of cut. Three numbers multiplied, and the product is the honest measure of how hard a roughing pass is working.

Max Bending Moment — Centre Point Load

M = \frac{P L}{4}

Strength of MaterialsMechanicsPhysicsMaximum bending moment in a simply supported beam carrying one point load at midspan, M = PL/4, occurring under the load.

Max Bending Moment — Uniform Load

M = \frac{w L^{2}}{8}

Strength of MaterialsMechanicsPhysicsMaximum bending moment at midspan of a simply supported beam under a uniformly distributed load, the classic M = wL²/8.

Max Moment — Simple Beam, Off-Centre Point Load

M = \frac{P a (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsMaximum bending moment in a simply supported beam with a single point load at distance a from one support, M = Pab/L, occurring directly under the load.

Maximum Cable Tension at the Support (Parabolic)

T_{\max} = \sqrt{H^{2} + \left( \dfrac{w L}{2} \right)^{2}}

Strength of MaterialsAt the support the cable is pulling in two directions at once: horizontally with H, which is the same everywhere along it, and vertically with half the total load, which is what the tower has to hold up. The tension in the cable is the resultant of those two, and it is the largest tension anywhere in the span. This is the number the fitting, the insulator, the anchorage and the factor of safety are all sized on.

Maximum Cable Tension from the Sag Ratio (Parabolic)

T_{\max} = H \sqrt{1 + \frac{16 d^{2}}{L^{2}}}

Strength of MaterialsThe same support tension as the load-based form, written the way it is actually used: the horizontal tension multiplied by a factor that depends on nothing but the sag ratio. It shows in one line why a shallow cable can be designed on H alone and a deep one cannot — at 1 in 20 the multiplier is 1.005, at 1 in 5 it is 1.077.

Maximum Clock Frequency from the Critical Path

f_{max} = \frac{1}{t_{pd} + t_{su}}

Semiconductors & ChipsFastest clock a synchronous design can be run at: the reciprocal of the slowest register-to-register path plus the setup time the receiving flip-flop demands. Everything else in timing closure is a correction to these two terms.

Maximum Ground-Level Concentration

C_{max} = \frac{2Q}{e \pi u H^{2}} \cdot \frac{\sigma_z}{\sigma_y}

Air Quality & DispersionThe worst ground-level concentration a plume ever produces, wherever downwind it occurs. Falls with the square of effective stack height, which is the whole argument for building tall.

Maximum Heart Rate from Age (220 − Age)

\mathrm{HR}_{max} = 220 - a

Everyday & HealthThe 220-minus-age estimate of maximum heart rate in beats per minute, the rule of thumb printed on every gym treadmill.

Maximum In-Plane Shear Stress

\tau_{max} = \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsRadius of Mohr's circle: the largest shear stress on any plane through a plane-stress element, equal to half the difference of the two principal stresses.

Maximum Possible Heat Transfer (Qmax)

\dot{Q}_{max} = \dot{m}_{min} c_{min} (T_{h,in} - T_{c,in})

Heat TransferThermodynamicsThe thermodynamic ceiling on exchanger duty: the minimum capacity rate multiplied by the full inlet-to-inlet temperature difference.

Maximum Power Transfer to a Matched Load

P_{max} = \frac{V_{Th}^{2}}{4 R_{Th}}

Electricity & MagnetismElectrical TradeThe most power a source can hand to a load, which happens when the load resistance is made equal to the source's own Thevenin resistance.

Maximum Principal Stress (Mohr's Circle)

\sigma_1 = \frac{\sigma_x + \sigma_y}{2} + \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsLarger of the two principal stresses for a plane-stress element, from the normal stresses σx and σy and the shear τxy — the circle's centre plus its radius.

Maximum Speed on a Flat Curve

v_{\max} = \sqrt{\mu_s g r}

MechanicsPhysicsFastest a vehicle can round a flat, unbanked curve before friction can no longer supply the centripetal force.

Maximum Static Friction (f = μₛN)

f_{s,\max} = \mu_s N

MechanicsPhysicsLargest static friction force available before an object breaks loose and slides, from the static coefficient and normal force.

Maximum Unsupported Span (Barton)

B_{max} = 2\,ESR\,Q^{0.4}

Rock MechanicsBarton's estimate of the largest span that will stand with no support at all, from the rock mass quality Q and the excavation support ratio the opening's purpose earns. The leading 2 is 2 METRES, not a pure number: this is a metric equation, and the answer an imperial reader sees has been converted from one.

McKee Box Compression Formula (Long Form)

BCT = 2.028 \; ECT^{0.746} \left( D_{MD} D_{CD} \right)^{0.127} Z^{0.492}

Packaging & Box StrengthMcKee's original and more general result, before caliper was substituted in as a proxy. It takes the board's measured bending stiffness in the machine and cross directions and uses their geometric mean, which is what the panel buckling analysis actually calls for. Use it whenever you have real stiffness data, and especially on board that has been crushed in converting, where caliper flatters the stiffness that is left.

McKee Box Compression Formula (Short Form)

BCT = 5.874 \; ECT \; h^{0.508} \, Z^{0.492}

Packaging & Box StrengthThe 1963 relation that turns an edge crush value, a board caliper and a box perimeter into the compression strength of a regular slotted container. It is the single most used equation in corrugated packaging, and the exponents are not the halves it is usually written with: 0.508 on caliper and 0.492 on perimeter, chosen so the constant 5.874 comes out a pure number that works in any consistent unit system.

Mean Arterial Pressure (MAP)

\mathrm{MAP} = \mathrm{DBP} + \frac{\mathrm{SBP} - \mathrm{DBP}}{3}

Biomedical & ClinicalThe time-weighted average arterial pressure over a cardiac cycle, estimated as the diastolic pressure plus one third of the pulse pressure.

Mean Cell Residence Time (Sludge Age)

\text{SRT} = \frac{V \, X}{Q_w \, X_w}

Water & WastewaterWater TreatmentSolids retention time of an activated sludge plant: the mass of solids under aeration divided by the mass wasted each day.

Mean Residence Time from a Tracer

\bar{t} = \frac{\int_0^{\infty} t\,C\,dt}{\int_0^{\infty} C\,dt} = \frac{M_1}{M_0}

Chemical EngineeringThe average time fluid actually spends in a vessel, taken as the first moment of a tracer curve divided by its zeroth moment. Compared with the nominal V/v₀, it is the single most useful diagnostic a reactor engineer has: it finds dead volume and short-circuiting that no drawing will show.

Mechanical Advantage of a Lever

MA = \frac{d_e}{d_l}

MechanicsPhysicsMechanical advantage of a lever as the ratio of effort arm to load arm, showing how much the lever multiplies force.

Median Rank Failure Probability

P_{f,i} = \frac{i - 0.3}{N + 0.4}

Ceramics & GlassBefore a set of broken specimens can be plotted, each one needs a failure probability attached to it. Bernard and Bos-Levenbach's 1953 approximation to the median rank is the estimator almost everyone uses: sort the strengths, and the i-th of N gets (i − 0.3)/(N + 0.4).

Merchant Shear Angle from Chip Thickness Ratio

\tan\phi = \frac{r \cos\alpha}{1 - r \sin\alpha}

Machining & TurningThe angle of the plane the chip shears along, recovered from two things you can measure with a micrometer: how thick the chip came out compared with how deep the cut was, and the rake angle of the tool. The foundation of the orthogonal cutting model Merchant published in 1945.

Michaelis–Menten Equation

v = \frac{V_{max} \, [S]}{K_m + [S]}

Chemical EngineeringThe rectangular hyperbola every enzyme assay is fitted to: the rate climbs almost linearly with substrate while substrate is scarce, then flattens onto a ceiling V_max as every active site fills. K_m is the substrate concentration that gets you halfway to that ceiling, and it is a concentration, not a rate.

Mid-Parent Heterosis

MPH = 100 \, \frac{F_1 - MP}{MP}

Plant Breeding & GeneticsHow far an F₁ hybrid exceeds the average of its two parents, as a percentage — the measure of hybrid vigour that justifies the whole hybrid seed industry.

Midpoint Formula

x_m = \frac{x_1 + x_2}{2}

AlgebraGeometryThe coordinate halfway between two endpoints along one axis; apply it to x and again to y for the midpoint of a segment.

Midrange

M = \frac{x_{\max} + x_{\min}}{2}

StatisticsAlgebraThe midpoint between the largest and smallest observations, a quick centre estimate computed from just the two extremes.

Miles per Gallon ↔ L/100 km

C \cdot M = 235.215

Everyday & HealthThe reciprocal relation between litres per hundred kilometres and US miles per gallon: their product is always 235.215.

Milling Table Feed Rate

v_f = N \, z \, f_z

Machining & TurningThe speed the table actually travels under a milling cutter: spindle speed times the number of teeth times the chip each tooth is asked to take. The one number a milling machine is programmed in, built from the one number the cutter cares about.

Mine Air Power

P_{air} = p \, Q

Rock MechanicsThe power actually delivered to the air: pressure times volume flow. It is the useful output of a mine fan and the floor under its electricity bill — the shaft power is this divided by the fan efficiency, and the motor draw is that divided by the drive and motor efficiencies again.

Miner's Cumulative Damage Rule (Three Blocks)

D = \frac{n_1}{N_1} + \frac{n_2}{N_2} + \frac{n_3}{N_3}

Strength of MaterialsMechanicsLinear damage summation over three blocks of a variable-amplitude load history. Failure is predicted when the damage fraction D reaches 1.

Minimum Pipe Bore at the Erosional Limit

d = \sqrt{\frac{4Q}{\pi V_e}}, \qquad V_e = \frac{C}{\sqrt{\rho_m}}

Erosion & SedimentThe line-sizing form of API RP 14E: for a known flow and mixture density, the smallest inside diameter that still keeps the velocity at or under the erosional limit. Round up to the next available schedule and the pipe is compliant; round down and it is not.

Minimum Principal Stress (Mohr's Circle)

\sigma_2 = \frac{\sigma_x + \sigma_y}{2} - \sqrt{\left(\frac{\sigma_x - \sigma_y}{2}\right)^{2} + \tau_{xy}^{2}}

Strength of MaterialsCivil & SurveyingMechanicsSmaller of the two principal stresses for a plane-stress element — the circle's centre minus its radius. Often the compressive one, and the value brittle materials care about least.

Minimum Reflux Ratio (Underwood, Binary)

R_{min} = \frac{1}{\alpha - 1}\left(\frac{x_D}{z_F} - \frac{\alpha\left(1 - x_D\right)}{1 - z_F}\right)

Chemical EngineeringThe least reflux that could ever achieve a given split, at the price of an infinite number of stages. The lower limit of the design, opposite Fenske's upper one, and the number every real reflux ratio is quoted as a multiple of.

Minor Loss from K Factor

h_L = K \, \frac{v^{2}}{2g}

HVAC & HydronicsFluid MechanicsWater TreatmentHead lost through a valve or fitting as a multiple of velocity head, with g = 9.80665 m/s² and K taken from a fitting table.

Minute Ventilation (Tidal Volume × Respiratory Rate)

\dot{V}_E = V_T \cdot \mathrm{RR}

Biomedical & ClinicalThe total volume of gas moved into the lungs each minute: the volume of one breath multiplied by the number of breaths.

Mixed Air Temperature

T_m = f \, T_{oa} + (1-f) \, T_{ra}

HVAC & HydronicsThermodynamicsTemperature of the blend leaving a mixing box, weighted by the outdoor air fraction — the reading that verifies an economizer's damper position.

Modulus of Rupture Predicted from Compressive Strength

f_r = k \sqrt{f'_c}

ConcreteThe flexural tensile stress at which plain concrete cracks, PREDICTED from its compressive strength. It is the stress a deflection calculation uses to decide when a section cracks, and it is the basis of rigid pavement and slab-on-grade thickness design. The coefficient k is a USER INPUT, because the codes do not agree on it and one of them does not even use a square root.

Modulus of Rupture, Four-Point Bending

\sigma = \frac{3PL}{4bd^2}

Ceramics & GlassHow ceramic strength is actually measured: a rectangular bar on two supports, loaded by two inner rollers at the quarter points, broken, and the peak load converted to a stress by beam theory. Nearly every strength figure on a ceramic datasheet came from this test.

Mohr–Coulomb Shear Strength

\tau_f = c' + \sigma'\tan\phi'

Soil MechanicsMechanicsMohr–Coulomb failure criterion giving the shear strength of soil from effective cohesion and the friction mobilised by effective normal stress.

Moist Air Density at Altitude (and the 1.08 Correction)

\rho = \frac{p_z\,(1 + W)}{R_{da}\,T\,(1 + 1.6078\,W)}, \quad p_z = 101\,325\,(1 - 2.25577 \times 10^{-5} z)^{5.25588}

HVAC & HydronicsThermodynamicsAir Quality & DispersionDensity of moist air from altitude, temperature and humidity ratio, and the correction factor it forces on the 1.08, 0.68 and 4.5 rules — which are sea-level, standard-air numbers and nothing else.

Moist Air Enthalpy (per kg DRY air)

h = 1.006\,t + W\,(2501 + 1.86\,t)

HVAC & HydronicsThermodynamicsTotal heat content of moist air per kilogram of DRY air — the sensible term for the air plus the latent and sensible terms for the water it carries. The basis is per kg of dry air, not per kg of mixture.

Moist Air Specific Volume (per kg DRY air)

v = \frac{0.287042\,(t + 273.15)\,(1 + 1.6078\,W)}{p}

HVAC & HydronicsThermodynamicsCubic metres of moist air per kilogram of DRY air, from temperature, humidity ratio and barometric pressure — the number that turns a fan's volume flow into a mass flow.

Molality (b = n/m)

b = \frac{n}{m_{\text{solvent}}}

ChemistryDefines molality as moles of solute per kilogram of solvent.

Molarity (C = n/V)

C = \frac{n}{V}

ChemistryDefines molar concentration as moles of solute per volume of solution.

Mole Fraction

x_1 = \frac{n_1}{n_1 + n_2}

ChemistryGives the fraction of all moles in a two-component mixture contributed by the solute.

Moles from Mass (n = m/M)

n = \frac{m}{M}

ChemistryConverts a measured mass into an amount of substance by dividing by the molar mass.

Moment Magnitude (Mw)

M_w = \dfrac{2}{3}\left(\log_{10} M_0 - 9.05\right)

SeismologyThe magnitude every agency now reports, defined directly from seismic moment by Hanks and Kanamori in 1979. Unlike the older scales it does not saturate, so it stays meaningful for the largest earthquakes there are.

Moment of Inertia — I-Beam or Built-Up Section

I = \frac{B H^{3} - (B - t_w)(H - 2t_f)^{3}}{12}

Strength of MaterialsCivil & SurveyingGeometryStrong-axis second moment of area of a doubly symmetric I-shape, taken as the full bounding rectangle minus the two rectangular voids beside the web.

Moment of Inertia: Point Mass

I = m r^{2}

MechanicsPhysicsRotational inertia of a compact mass circling at radius r from the axis.

Moment of Inertia: Solid Disk

I = \tfrac{1}{2} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid disk or cylinder about its central axis.

Moment of Inertia: Solid Sphere

I = \tfrac{2}{5} m r^{2}

MechanicsPhysicsRotational inertia of a uniform solid sphere about an axis through its center.

Money Multiplier (Textbook Reserve Model)

m = \frac{1}{r_r}

Prices & MarketsThe textbook relation between a reserve requirement and the deposits a banking system can support: the reciprocal of the required reserve ratio. Read the learning zone before using it — the Bank of England's own 2014 bulletin says plainly that this is not how money is created in a modern economy.

Monkman–Grant Relation

t_r \, \dot{\varepsilon}_s^{\,m} = C_{MG}

Metallurgy & Heat TreatmentThe empirical observation that a metal creeping twice as fast lasts about half as long, whatever the stress and temperature that made it creep. Monkman and Grant found it in 1956 across a wide range of alloys, and it is the reason a short test that never reaches rupture can still predict one.

Monod Growth Equation

\mu = \frac{\mu_{max} \, S}{K_s + S}

Chemical EngineeringJacques Monod's 1949 description of how fast a culture grows when one nutrient is limiting: the specific growth rate rises with substrate and levels off at µ_max. It has exactly the algebra of Michaelis–Menten and Monod was careful to say it is nothing like the same claim — this one is a curve that fits, not a mechanism that explains.

Moore's Law Doubling

N = N_0 \, 2^{t/T}

Semiconductors & ChipsTransistor count on a chip after a stretch of time, doubling on a stated period. Moore's 1965 paper claimed a doubling every year; he revised it to every two in 1975, and the two-year figure is the one that held for four decades.

MOSFET Drain Current in Saturation (Square Law)

I_D = \tfrac{1}{2} k_n \frac{W}{L} (V_{GS} - V_{th})^2

Semiconductors & ChipsDrain current of a MOSFET held above threshold with enough drain voltage to pinch off the channel. The current depends on the square of the overdrive and, to first order, not on the drain voltage at all — which is what makes the device a current source.

MOSFET Drain Current in the Linear (Triode) Region

I_D = k_n \frac{W}{L}\left[(V_{GS} - V_{th})V_{DS} - \frac{V_{DS}^2}{2}\right]

Semiconductors & ChipsDrain current when the drain-source voltage is below the overdrive and the channel reaches all the way across. At small drain voltages the square term vanishes and the transistor is simply a resistor the gate can adjust.

Motion Blur Speed Limit

V_{max} = \frac{k \; \mathrm{GSD}}{t_{exp}}

Photogrammetry & Remote SensingThe fastest ground speed that keeps the smear from forward motion inside a stated fraction of a pixel. The image moves across the sensor while the shutter is open, and that smear is a hard limit on real resolution that no amount of fine GSD can rescue.

Motional EMF (ε = BLv)

\varepsilon = B L v

Electricity & MagnetismPhysicsVoltage generated across a conductor of length L moving at speed v through a field B.

Motor Efficiency

\eta = \frac{100 \, P_{out}}{P_{in}}

Electrical TradeElectricity & MagnetismPercentage efficiency of a motor or drive as mechanical output power divided by electrical input power, times one hundred.

Motor Locked-Rotor Starting Current

I_{LR} = \frac{1000 \, k \, P}{\sqrt{3} \, V}

Electrical TradeInrush a three-phase motor draws at the instant of across-the-line starting, from the locked-rotor kVA per horsepower on its nameplate code letter.

Motor Torque from Power and Speed

T = \frac{P}{2\pi N}

Electrical TradeElectricity & MagnetismShaft torque a motor develops at a given output power and speed — the trade's 5252·HP/RPM rule written in SI, with N in revolutions per second.

Mould Shrinkage

S = \frac{L_{mould} - L_{part}}{L_{mould}}

Polymers & Plastics ProcessingThe part comes out smaller than the steel it was made in, and this is by how much — expressed as a fraction of the MOULD dimension, which is the convention the tool is cut to. Getting the reference wrong, or the axis wrong, is how a first-off comes back out of tolerance on a tool that was cut exactly to the drawing.

MQH Hot Gas Layer Temperature

\Delta T_g = 6.85 \left( \dfrac{\dot{Q}^{2}}{A_0 \sqrt{H_0} \; h_k A_T} \right)^{1/3}

Fire Protection & Fire DynamicsMcCaffrey, Quintiere and Harkleroad's 1981 correlation for how hot the upper layer gets in a naturally ventilated compartment fire. It balances what the fire puts in against what the opening carries out and what the boundaries absorb, and it is the workhorse hand calculation of enclosure fire analysis.

Multiplication Rule (Independent Events)

P(A \cap B) = P(A) \, P(B)

ProbabilityStatisticsWhen one event has no influence on the other, the chance that both occur is the product of their separate probabilities.

Munden's Stitch Density for Plain Knit

S = \frac{K_s}{\ell^{2}}, \qquad c = \frac{K_c}{\ell}, \qquad w = \frac{K_w}{\ell}

Textiles & FabricMunden's 1959 finding, and one of the cleanest results in textile science: the dimensions a relaxed plain-knit fabric settles at depend on the STITCH LENGTH and on essentially nothing else — not the yarn count, not the fibre, not the machine gauge. Courses and wales per unit length each go as the reciprocal of the loop length, so stitch density goes as its reciprocal square, with constants that are properties of the knitted structure itself.

Muskingum Channel Storage

S = K\left[X I + (1 - X) O\right]

Water & WastewaterCivil & SurveyingFluid MechanicsStorage held in a river reach by the Muskingum method: a prism term set by the outflow plus a wedge term set by the inflow, weighted by X and scaled by the travel time K.