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All equations

All 548 formulas in the library, grouped by subject area. Open any one for its calculator, worked example and practice questions.

Aerospace (7)

Automotive and vehicle dynamics (35)

Wheel drive force from engine torque VEH-01
Vehicle speed from engine speed VEH-02
Equivalent mass with rotating inertia VEH-03
Vehicle acceleration from road load VEH-04
Available tractive force (torque or grip limited) VEH-05
Engine swept displacement VEH-06
Mean piston speed VEH-07
Brake mean effective pressure VEH-08
Brake thermal efficiency VEH-09
Electric vehicle range estimate VEH-10
Electric motor torque envelope VEH-11
Total stopping distance with reaction time VEH-12
Front axle load during braking VEH-13
Rear axle load during acceleration VEH-14
Ideal front brake force fraction VEH-15
Brake torque from clamp force VEH-16
Brake disc temperature rise VEH-17
Friction ellipse remaining longitudinal force VEH-18
Tyre force load sensitivity VEH-19
Linear tyre lateral force VEH-20
Tyre longitudinal slip ratio VEH-21
Magic Formula tyre force VEH-22
Kinematic turning radius VEH-23
Ackermann outer wheel steer angle VEH-24
Understeer gradient VEH-25
Steady-state steering angle VEH-26
Yaw rate gain VEH-27
Lateral load transfer VEH-28
Sprung body roll angle VEH-29
Static rollover threshold VEH-30
Quarter-car ride frequency VEH-31
Wheel-hop frequency VEH-32
Aerodynamic downforce VEH-33
Cornering speed limit with downforce VEH-34
Power for steady level-road speed VEH-35

Ballistics (1)

Projectile range (no drag) BAL-01

Chemical (8)

Amount and concentration CHE-01
Ideal gas law CHE-02
Sensible heating of a stream CHE-03
First-order batch kinetics CHE-04
Arrhenius rate constant CHE-05
Stirred-tank reactor volume CHE-06
Steady diffusion flux through a plane layer CHE-07
Molar diffusion rate through a plane layer CHE-08

Civil (7)

Bending stress in a rectangular beam CIV-01
Simply supported beam: central point load CIV-02
Simply supported beam: uniform load CIV-03
Cantilever with a tip load CIV-04
Euler buckling load CIV-05
Effective stress in saturated soil CIV-06
Open-channel flow (Manning) CIV-07

Communications (4)

Computer (5)

Control engineering (24)

Proportional feedback on a first-order plant CTL-01
First-order step response CTL-02
Percentage overshoot of an underdamped second-order system CTL-03
Peak time of an underdamped second-order system CTL-04
Approximate 2 % settling time of a second-order system CTL-05
Damping ratio from measured overshoot CTL-06
Steady-state error to a step input CTL-07
Steady-state error to a ramp input CTL-08
PID gains from integral and derivative times CTL-09
Gain of a filtered derivative term CTL-10
Integrator rate with back-calculation antiwindup CTL-11
Discrete integral term after n samples CTL-12
Critical gain from the Routh–Hurwitz cubic test CTL-13
Maximum phase lead of a lead compensator CTL-14
Lead compensator design from required phase CTL-15
Delay margin from phase margin CTL-16
Gain margin in decibels CTL-17
Frequency response of a first-order lag CTL-18
Sensitivity at one frequency CTL-19
State-feedback gains for a double integrator CTL-20
Scalar LQR feedback gain CTL-21
Zero-order-hold discretisation of a first-order plant CTL-22
Tustin frequency warping CTL-23
Aliased frequency after sampling CTL-24

Driveline and transmission (19)

Overall driveline reduction ratio DRV-01
Total tractive force required DRV-02
Engine power required at a given speed DRV-03
Minimum first-gear ratio for climbing DRV-04
Top-gear ratio for a target top speed DRV-05
Minimum friction coefficient to climb a slope DRV-06
Gearbox ratio span DRV-07
Number of gears for a ratio span DRV-08
Geometric step between gears DRV-09
Ratio of gear j in a geometric set DRV-10
Torque converter efficiency DRV-11
Torque converter power loss DRV-12
Carrier speed of a simple planetary gear DRV-13
Planetary ratio, sun to carrier (ring held) DRV-14
Planetary ratio, ring to carrier (sun held) DRV-15
Planetary ratio, sun to ring (carrier held) DRV-16
Order excitation frequency DRV-17
Gear mesh frequency DRV-18
Track-pitch excitation frequency DRV-19

Electrical (43)

Electric charge ELE-01
Electrical work done ELE-02
Electrical power ELE-03
Ohm’s law ELE-04
Resistance and temperature ELE-05
Peak and RMS values (AC) ELE-06
Resistors in parallel ELE-07
Resistivity ELE-08
Battery internal resistance ELE-09
Energy stored in a capacitor ELE-10
Electric field strength ELE-11
Capacitor time constant ELE-12
Capacitor discharge half time ELE-13
Force on a current ELE-14
Transformer equation ELE-15
Potential divider ELE-16
RC circuit response ELE-17
Inductive reactance ELE-18
Capacitive reactance ELE-19
Real power, single-phase AC ELE-20
Real power, three-phase AC ELE-21
Inverting op-amp gain ELE-22
Non-inverting op-amp gain ELE-23
Joule heating ELE-24
Three resistors in parallel ELE-25
Series RLC impedance ELE-26
Resonant frequency ELE-27
Quality factor of a series RLC circuit ELE-28
Skin depth ELE-29
Load current from a Thevenin source ELE-30
Load voltage from a Thevenin source ELE-31
Power delivered to a load by a Thevenin source ELE-32
Maximum power transfer from a Thevenin source ELE-33
Inductor current after a switching event ELE-34
Time constant of an RL circuit ELE-35
Half-power bandwidth of a series RLC circuit ELE-36
Reactive power in a single-phase circuit ELE-37
Apparent power in a single-phase circuit ELE-38
Cutoff frequency of a first-order RC filter ELE-39
Gain of a first-order RC low-pass filter ELE-40
Gain of a first-order RC high-pass filter ELE-41
Shockley diode current ELE-42
Thermal voltage ELE-43

Flight dynamics and control (4)

Euler angle rates from body rates FLT-01
Pitch-rate response to an elevator step FLT-02
Specific total energy and energy height FLT-03
Specific total energy rate FLT-04

Fluid Mechanics (15)

Force from a change in stream velocity FLU-01
Total head (Bernoulli) FLU-02
Pump head from the extended head balance FLU-03
Pipe friction and fitting losses FLU-04
Laminar flow in a pipe (Hagen–Poiseuille) FLU-05
Pump shaft power FLU-06
Viscous shear stress FLU-07
Buoyant force FLU-08
Speed from a pressure difference FLU-09
Dynamic pressure FLU-10
Drag force FLU-11
Pump flow rate after a speed change FLU-12
Pump head after a speed change FLU-13
Pump power after a speed change FLU-14
Darcy friction factor from the Haaland equation FLU-15

Foundations: Chemistry (35)

Relative formula mass FCH-01
Percentage by mass of an element FCH-02
Moles from mass FCH-03
Number of particles (Avogadro) FCH-04
Reacting masses from an equation FCH-05
Concentration of a solution FCH-06
Volume of a gas FCH-07
Percentage yield FCH-08
Atom economy FCH-09
Mean rate of reaction FCH-10
Enthalpy change from bond energies FCH-11
R_f value in chromatography FCH-12
Gas released in an open flask FCH-13
Titration calculation FCH-14
Dilution FCH-15
Enthalpy change from calorimetry FCH-16
Hess’s law from formation enthalpies FCH-17
Gibbs free energy FCH-18
Relative atomic mass from isotopes FCH-19
Percentage uncertainty FCH-20
Rate equation FCH-21
Equilibrium constant Kc FCH-22
pH of a solution FCH-23
pH of a weak acid FCH-24
pH of a buffer FCH-25
Electrochemical cell potential FCH-26
Equilibrium constant Kp from partial pressures FCH-27
Partial pressure from mole fraction FCH-28
Hydrogen-ion concentration of a weak acid without the approximation FCH-29
Concentration of excess acid or base after mixing FCH-30
pH after mixing a strong acid and a strong base FCH-31
Concentration over time for a zero-order reaction FCH-32
Concentration over time for a second-order reaction FCH-33
Rate constant at a second temperature (Arrhenius) FCH-34
Absorbance from the Beer–Lambert law FCH-35

Foundations: Maths (62)

Percentage change FMA-01
Repeated percentage change FMA-02
Gradient of a straight line FMA-03
Arithmetic sequence: nth term FMA-04
Mean from a frequency table FMA-05
Independent events: both and either FMA-06
Area of a trapezium FMA-07
Circle: circumference and area FMA-08
Arc length and sector area (degrees) FMA-09
Volume of a cone and cylinder FMA-10
Sphere: volume and surface area FMA-11
Pythagoras’ theorem FMA-12
Right-angled triangle trigonometry FMA-13
Sine rule FMA-14
Cosine rule FMA-15
Triangle area from two sides and an angle FMA-16
Arithmetic series: sum of n terms FMA-17
Geometric series: sum to infinity FMA-18
Quadratic formula FMA-19
Binomial expansion (first three terms) FMA-20
Logarithms FMA-21
Arc length and sector area (radians) FMA-22
Small-angle approximation FMA-23
Product rule FMA-24
Quotient rule FMA-25
Chain rule (power of a linear function) FMA-26
Definite integral of a power FMA-27
Trapezium rule (4 strips) FMA-28
Newton–Raphson step FMA-29
Conditional probability FMA-30
Binomial probability FMA-31
Normal distribution: z-score FMA-32
Standard deviation from the mean of squares FMA-33
Distance from average velocity FMA-34
Heron’s formula FMA-35
Distance between two points FMA-36
Geometric series sum FMA-37
Exponential growth and decay FMA-38
Change of logarithm base FMA-39
Dot product FMA-40
Modulus of a complex number FMA-41
Argument (phase) of a complex number FMA-42
Complex division in polar form FMA-43
Two simultaneous equations: solution for x FMA-44
Two simultaneous equations: solution for y FMA-45
First-order response to a constant input FMA-46
Laplace transform of a decaying exponential FMA-47
Laplace transform of a sine FMA-48
Laplace transform of a cosine FMA-49
Relative uncertainty of a product or power FMA-50
Midpoint of a line segment FMA-51
Radius of a circle from its centre and a point on it FMA-52
Direct and inverse proportion FMA-53
Sum of the interior angles of a polygon FMA-54
Interior angle of a regular polygon FMA-55
Exterior angle of a regular polygon FMA-56
Total surface area of a closed cylinder FMA-57
Total surface area of a cone FMA-58
Sine of a sum of angles FMA-59
Cosine of a sum of angles FMA-60
Angle between two vectors in three dimensions FMA-61
Scalar projection of one vector onto another FMA-62

Foundations: Physics (56)

Mean velocity FPH-01
Mean acceleration FPH-02
Final speed from distance (v² = u² + 2as) FPH-03
Weight FPH-04
Momentum FPH-05
Average force from momentum change FPH-06
Work done by a force at an angle FPH-07
Limiting friction on a slope FPH-08
Gravitational potential energy FPH-09
Spring force and stored energy FPH-10
Power from energy and time FPH-11
Density FPH-12
Pressure from force and area FPH-13
Wave speed FPH-14
Specific heat capacity FPH-15
Specific latent heat FPH-16
Moment of a force FPH-17
Young’s modulus from a test FPH-18
Simple harmonic motion: peak speed and acceleration FPH-19
Period of a mass on a spring FPH-20
Period of a simple pendulum FPH-21
Centripetal force FPH-22
Speed in circular motion FPH-23
Capacitance FPH-24
Capacitor discharge FPH-25
Gravitational field and orbit FPH-26
Electric field of a point charge FPH-27
Magnetic force on a moving charge FPH-28
Induced emf (Faraday’s law) FPH-29
Phase and path difference FPH-30
Double-slit fringe spacing FPH-31
Diffraction grating FPH-32
Refraction (Snell’s law) FPH-33
Critical angle FPH-34
Refractive index FPH-35
Photon energy FPH-36
Photoelectric effect FPH-37
de Broglie wavelength FPH-38
Mass–energy equivalence FPH-39
Radioactive decay and half-life FPH-40
Newton’s law of gravitation FPH-41
Circular orbital speed FPH-42
Escape speed FPH-43
Doppler shift (moving source) FPH-44
Inverse-square law for intensity FPH-45
Image distance for a thin lens FPH-46
Magnification of a thin lens FPH-47
Gas pressure from molecular motion FPH-48
Root-mean-square speed of gas molecules FPH-49
Mean translational kinetic energy per molecule FPH-50
Orbital period of a circular orbit FPH-51
Speed of a simple harmonic oscillator at a displacement FPH-52
Total energy of a simple harmonic oscillator FPH-53
Potential energy of a simple harmonic oscillator FPH-54
Kinetic energy of a simple harmonic oscillator FPH-55
Sound intensity level FPH-56

Fundamentals (7)

Newton’s second law FUN-01
Work and kinetic energy FUN-02
Mechanical power FUN-03
Efficiency FUN-04
Mass flow and volume flow FUN-05
Hydrostatic pressure FUN-06
Shaft speed and angular frequency FUN-07

Guidance and trajectory following (5)

Wrapped course error GUI-01
Cross-track error from a straight path GUI-02
Course command for straight-path following GUI-03
Lookahead (pure-pursuit) lateral acceleration GUI-04
Bank angle for a commanded lateral acceleration GUI-05

Heat Transfer (19)

Heat flow through a composite wall HTR-01
Heat loss through a pipe wall HTR-02
Net thermal radiation HTR-03
Lumped transient cooling HTR-04
Thermal diffusivity and Fourier number HTR-05
Nusselt number HTR-06
Prandtl number HTR-07
Heat exchanger duty (LMTD method) HTR-08
Blackbody emissive power HTR-09
Biot number HTR-10
Dittus–Boelter correlation HTR-11
Log-mean temperature difference HTR-12
Fin parameter of a straight fin HTR-13
Heat transfer from a straight fin with an insulated tip HTR-14
Efficiency of a straight fin with an insulated tip HTR-15
Effectiveness of a counterflow heat exchanger HTR-16
Heat duty of an exchanger from its effectiveness HTR-17
Number of transfer units of a heat exchanger HTR-18
Critical insulation radius of a cylinder HTR-19

Mechanical (94)

Constant-acceleration displacement MEC-01
Axial stress and extension MEC-02
Twist of a solid circular shaft MEC-03
Shaft power from torque and speed MEC-04
Natural frequency of a mass on a spring MEC-05
Damped natural frequency MEC-06
Heat conduction through a wall MEC-07
Convective heat transfer MEC-08
Acceleration from the velocity gradient MEC-09
Projectile height at a given time MEC-10
Acceleration along a curved path MEC-11
Radial acceleration in polar coordinates MEC-12
Transverse acceleration in polar coordinates MEC-13
Rolling without slipping MEC-14
Rigid-body kinetic energy MEC-15
Parallel-axis theorem MEC-16
Direct impact with restitution MEC-17
Angular impulse and momentum (fixed axis) MEC-18
Angular acceleration from a moment (Euler’s equation) MEC-19
Resultant of a trapezoidal distributed load MEC-20
Friction limit MEC-21
Second moment of a built-up section MEC-22
Transverse shear stress in a beam MEC-23
Principal stresses in plane stress MEC-24
Von Mises equivalent stress MEC-25
Thermal stress in a restrained bar MEC-26
Forced vibration amplitude MEC-27
Base-excitation transmissibility MEC-28
Damping from a ring-down MEC-29
Free underdamped response MEC-30
Tresca equivalent stress MEC-31
Direct stress MEC-32
Hooke’s law MEC-33
Torsional shear stress in a solid shaft MEC-34
Bending stress MEC-35
Second moment of area, rectangle MEC-36
Maximum shear stress (plane stress) MEC-37
Factor of safety MEC-38
Projectile maximum height MEC-39
Projectile time of flight MEC-40
Braking distance MEC-41
Lamé radial stress in a thick cylinder MEC-42
Lamé hoop stress in a thick cylinder MEC-43
Axial stress in a closed-end thick cylinder MEC-44
Maximum hoop stress at the bore MEC-45
Hoop stress at the outside surface MEC-46
Thin-walled cylinder hoop stress MEC-47
Thin-walled cylinder axial stress MEC-48
Mohr’s circle centre (mean in-plane stress) MEC-49
Normal stress on a rotated element, x′ face MEC-50
Normal stress on a rotated element, y′ face MEC-51
Shear stress on a rotated element MEC-52
Principal angle in plane stress MEC-53
Principal stresses in order, including the zero MEC-54
Absolute maximum shear stress in plane stress MEC-55
Tresca to von Mises ratio MEC-56
Yield utilisation against a required factor of safety MEC-57
Normal stress on any plane (three dimensions) MEC-58
Shear stress magnitude on any plane (three dimensions) MEC-59
First stress invariant (trace) MEC-60
Second stress invariant MEC-61
Third stress invariant (determinant) MEC-62
Principal stresses in three dimensions MEC-63
Tresca equivalent stress from principal stresses MEC-64
Von Mises equivalent stress from principal stresses MEC-65
Von Mises equivalent stress from stress components (three dimensions) MEC-66
Poisson’s ratio from measured strains MEC-67
Elastic–plastic interface radius from a target plastic fraction MEC-68
First-yield pressure of a thick cylinder MEC-69
Fully plastic limit pressure of a thick cylinder MEC-70
Autofrettage pressure for a target plastic fraction MEC-71
Contact pressure at the elastic–plastic interface MEC-72
Peak hoop stress at radius r during autofrettage MEC-73
Residual hoop stress at radius r after autofrettage MEC-74
Residual radial stress at radius r after autofrettage MEC-75
Residual hoop stress at the bore after autofrettage MEC-76
Reverse-yielding screening ratio at the bore MEC-77
Autofrettage pressure for a target bore compression MEC-78
Plastic zone radius from a prescribed autofrettage pressure MEC-79
Bore hoop stress in service after autofrettage MEC-80
Normal strain from three-dimensional stress and temperature change MEC-81
Shear strain from shear stress MEC-82
Shear modulus from Young’s modulus and Poisson’s ratio MEC-83
Bulk modulus from Young’s modulus and Poisson’s ratio MEC-84
Strain energy stored in an axially loaded member MEC-85
Strain energy stored in a twisted shaft MEC-86
Stress amplitude of a fatigue cycle MEC-87
Mean stress of a fatigue cycle MEC-88
Stress ratio of a fatigue cycle MEC-89
Modified Goodman factor of safety MEC-90
Miner’s rule cumulative damage (three load levels) MEC-91
Capstan equation for rope friction MEC-92
Hertz contact radius of a sphere on a flat surface MEC-93
Hertz maximum contact pressure for a sphere on a flat surface MEC-94

Navigation (6)

Ground speed and course from NED velocity NAV-01
Ground speed from airspeed and wind NAV-02
Range and bearing to a point NAV-03
Position error from accelerometer bias NAV-04
Pitch from a stationary accelerometer NAV-05
GNSS pseudorange with receiver clock bias NAV-06

Operational analysis (1)

Lanchester square law OPS-01

Propulsion (2)

Tsiolkovsky rocket equation PRP-01
Rocket thrust PRP-02

Reliability (3)

Sensors & detection (8)

Statistics (12)

Sample mean STA-01
Standard error of the mean STA-02
Confidence interval half-width STA-03
Sample size for a target precision STA-04
One-sample t statistic STA-05
Pearson correlation coefficient STA-06
Regression slope STA-07
Coefficient of determination STA-08
Poisson probability STA-09
Normal probability density STA-10
Probability of at least one success STA-11
Margin of error for a proportion STA-12

Terramechanics and off-road mobility (29)

Drawbar pull TER-01
Thrust to climb a grade TER-02
Maximum climbable grade TER-03
Traction limit on a slope TER-04
Approach or departure angle TER-05
Breakover angle TER-06
Force to push a rigid wheel over a step TER-07
Mean ground pressure of a tracked vehicle TER-08
Bekker pressure–sinkage relation TER-09
Sinkage under uniform pressure TER-10
Soil compaction resistance TER-11
Mohr–Coulomb soil shear strength TER-12
Janosi–Hanamoto shear stress TER-13
Terramechanics driving slip TER-14
Maximum soil thrust under a flat contact TER-15
Tractive efficiency TER-16
Ideal differential steering radius TER-17
Track thrust with finite slip TER-18
Idealised track steering resistance TER-19
Undrained bearing capacity of clay TER-20
Rigid wheel resistance on clay TER-21
Remoulded cone index TER-22
Undrained strength from cone index TER-23
Mean maximum pressure (MMP) TER-24
Effect of weight and track width on MMP TER-25
Limiting cone index, tracked vehicle TER-26
Limiting cone index, wheeled vehicle TER-27
Vehicle cone index for one pass (VCI₁) TER-28
Vehicle cone index for 50 passes (VCI₅₀) TER-29

Thermodynamics (20)

Ideal-gas internal energy and enthalpy change THE-01
Steady-flow energy balance THE-02
Boundary work at constant pressure THE-03
Ideal-gas entropy change THE-04
Isentropic turbine efficiency THE-05
Isentropic compressor efficiency THE-06
Carnot efficiency THE-07
Refrigeration coefficient of performance THE-08
Wet-steam quality and mixture enthalpy THE-09
First law of thermodynamics THE-10
Isothermal expansion work THE-11
Adiabatic compression temperature THE-12
Otto cycle efficiency THE-13
Brayton cycle efficiency THE-14
Ideal refrigeration coefficient of performance THE-15
Pressure after a polytropic process THE-16
Temperature after a polytropic process THE-17
Boundary work in a polytropic process THE-18
Ideal Diesel cycle efficiency THE-19
Ideal heat pump coefficient of performance THE-20

Tracking and estimation (6)

Scalar Kalman filter measurement update TRK-01
Steady-state Kalman gain for a random-walk state TRK-02
Alpha–beta tracker update TRK-03
Process noise for a constant-velocity model TRK-04
Innovation gate statistic TRK-05
RMSE from bias and spread TRK-06

Vibration exposure (11)

Apparent mass at a driving point VIB-01
Measured transmissibility VIB-02
Weighted RMS acceleration over three segments VIB-03
Combined vibration dose value of separate segments VIB-04
Motion sickness dose value at a steady level VIB-05
Seat effective amplitude transmissibility (SEAT) VIB-06
Partial daily vibration exposure on one axis VIB-07
Daily vibration exposure from three tasks on one axis VIB-08
Daily vibration exposure A(8), governing axis VIB-09
Time to reach a chosen exposure value VIB-10
Remaining time to an exposure value with prior dose VIB-11