Practice
Work it out, then check it.
Random values drawn from realistic ranges. Work the problem on paper or with your own calculator, then enter the answer. Results within the stated tolerance count as correct.
Formula
Any formula Any Maths formula Any in this topic Percentage change Repeated percentage change Gradient of a straight line Arithmetic sequence: nth term Quadratic formula Distance from average velocity Direct and inverse proportion Any in this topic Area of a trapezium Circle: circumference and area Arc length and sector area (degrees) Volume of a cone and cylinder Sphere: volume and surface area Pythagoras’ theorem Midpoint of a line segment Radius of a circle from its centre and a point on it Sum of the interior angles of a polygon Interior angle of a regular polygon Exterior angle of a regular polygon Total surface area of a closed cylinder Total surface area of a cone Any in this topic Parallelogram area Kite area from its diagonals Apothem of a regular polygon Area of a regular polygon from its perimeter Number of diagonals in a polygon Chord length from the central angle Rise of a minor circular arc Area of a circular segment Area of a circular ring Tangent length from an external point Any in this topic Volume of a prism Total surface area of a right prism Volume of a pyramid Surface area of a sphere Volume of a conical frustum Area after a similarity transformation Volume after a similarity transformation Enlargement about a chosen centre Rotation about a chosen centre Reflection in a vertical line Any in this topic Reverse bearing Endpoint of a bearing traverse Smaller angle between two bearings Height from two angles of elevation Triangulation from two surveyed bearings Any in this topic Right-angled triangle trigonometry Sine rule Cosine rule Triangle area from two sides and an angle Any in this topic Mean from a frequency table Independent events: both and either Conditional probability Any in this topic Arithmetic series: sum of n terms Geometric series: sum to infinity Binomial expansion (first three terms) Logarithms Geometric series sum Exponential growth and decay Change of logarithm base Any in this topic Arc length and sector area (radians) Small-angle approximation Heron’s formula Distance between two points Dot product Sine of a sum of angles Cosine of a sum of angles Angle between two vectors in three dimensions Scalar projection of one vector onto another Any in this topic Inradius of a triangle from area and perimeter Circumradius of a triangle from sides and area Median length from three sides Side split by an internal angle bisector Centroid of a triangle from its vertices Any in this topic Point dividing a segment internally Perpendicular distance from a point to a line Triangle area from Cartesian coordinates Perpendicular bisector of two points Angle between two straight lines Any in this topic Centre and radius of a circle by completing the square Tangent line at a point on a circle Intersection of a line with a circle Common chord of two intersecting circles Parametric point on a circle Any in this topic Tangent addition identity Double-angle cosine from a sine value Double-angle sine from a tangent value Positive half-angle sine Single harmonic equivalent of a sine and cosine sum Any in this topic Period of a transformed sine or cosine Sine equation solutions in one revolution Cosine equation solutions in one revolution General solution of a tangent equation Second angle in the ambiguous sine rule case Any in this topic Product rule Quotient rule Chain rule (power of a linear function) Definite integral of a power Trapezium rule (4 strips) Newton–Raphson step Chain rule: derivative of sin(ax + b) Chain rule: derivative of e^(ax + b) Chain rule: derivative of ln(ax + b) Any in this topic Binomial probability Normal distribution: z-score Standard deviation from the mean of squares Relative uncertainty of a product or power Sample mean Standard error of the mean Confidence interval half-width Sample size for a target precision One-sample t statistic Pearson correlation coefficient Regression slope Coefficient of determination Poisson probability Normal probability density Probability of at least one success Margin of error for a proportion Any in this topic Modulus of a complex number Argument (phase) of a complex number Complex division in polar form Any in this topic Two simultaneous equations: solution for x Two simultaneous equations: solution for y First-order response to a constant input Laplace transform of a decaying exponential Laplace transform of a sine Laplace transform of a cosine Determinant of a 2×2 matrix Determinant of a 3×3 matrix Determinant of a scalar multiple of a matrix Any in this topic Relative uncertainty of a product or power Any in this topic Uniform tolerance: probability density Uniform tolerance: probability below a limit Uniform tolerance: expected value Uniform tolerance: variance Triangular estimate: probability density Triangular estimate: cumulative probability Triangular estimate: expected value Triangular estimate: variance Triangular estimate: planning percentile Any in this topic Weibull life: failure time density Weibull life: probability of failure by a deadline Weibull life: instantaneous failure rate Weibull life: mean time to failure Weibull life: lifetime variance Lognormal measurement: probability density Lognormal measurement: fraction below a threshold Lognormal measurement: arithmetic mean Lognormal measurement: variance Lognormal measurement: most probable value Gamma waiting time: probability density Gamma waiting time: probability of finishing by a deadline Gamma waiting time: expected value Gamma waiting time: variance Gamma waiting time: skewness Any in this topic Beta proportion: probability density Beta proportion: probability below a target Beta proportion: mean Beta proportion: variance Beta proportion: most probable value Laplace error: probability density Laplace error: lower tail probability Laplace error: variance Laplace error: two-sided tolerance exceedance Logistic threshold: probability Logistic threshold: density Logistic threshold: variance Logistic threshold: interquartile range Gumbel maxima: probability of staying below a threshold Gumbel maxima: density Gumbel maxima: expected maximum Gumbel maxima: variance Any in this topic Sampling without replacement: exact count Sampling without replacement: expected count Sampling without replacement: count variance Failures before a target: exact probability Failures before a target: expected number Failures before a target: variance Batch variation: exact successes Batch variation: expected successes Batch variation: variance of the successes Any Physics formula Any in this topic Newton’s second law Constant-acceleration displacement Mean velocity Mean acceleration Final speed from distance (v² = u² + 2as) Weight Momentum Average force from momentum change Moment of a force Any in this topic Work and kinetic energy Mechanical power Efficiency Work done by a force at an angle Gravitational potential energy Spring force and stored energy Power from energy and time Any in this topic Hydrostatic pressure Density Pressure from force and area Specific heat capacity Specific latent heat Any in this topic Wave speed Any in this topic Electric charge Electrical work done Electrical power Ohm’s law Resistors in parallel Force on a current Transformer equation Joule heating Three resistors in parallel Any in this topic Projectile range (no drag) Limiting friction on a slope Young’s modulus from a test Projectile height at a given time Direct impact with restitution Friction limit Direct stress Hooke’s law Projectile maximum height Projectile time of flight Capstan equation for rope friction Any in this topic Natural frequency of a mass on a spring Simple harmonic motion: peak speed and acceleration Period of a mass on a spring Period of a simple pendulum Centripetal force Speed in circular motion Orbital period of a circular orbit Speed of a simple harmonic oscillator at a displacement Total energy of a simple harmonic oscillator Potential energy of a simple harmonic oscillator Kinetic energy of a simple harmonic oscillator Displacement of a simple harmonic oscillator Amplitude from displacement and speed Kinetic fraction of the energy in simple harmonic motion Equivalent stiffness of two springs in series Equivalent stiffness of springs in parallel Any in this topic Gravitational field and orbit Electric field of a point charge Magnetic force on a moving charge Induced emf (Faraday’s law) Newton’s law of gravitation Circular orbital speed Escape speed Coulomb force between two point charges Electric potential of a point charge Magnetic flux through a flat surface Speed of light from the permeability and permittivity of free space Any in this topic Resistance and temperature Peak and RMS values (AC) Resistivity Battery internal resistance Energy stored in a capacitor Electric field strength Capacitor time constant Capacitor discharge half time Potential divider Capacitance Capacitor discharge Any in this topic Phase and path difference Double-slit fringe spacing Diffraction grating Refraction (Snell’s law) Critical angle Refractive index Doppler shift (moving source) Inverse-square law for intensity Image distance for a thin lens Magnification of a thin lens Sound intensity level Any in this topic Ideal gas law Gas pressure from molecular motion Root-mean-square speed of gas molecules Mean translational kinetic energy per molecule First law of thermodynamics Any in this topic Photon energy Photoelectric effect de Broglie wavelength Mass–energy equivalence Radioactive decay and half-life Photon energy from frequency Any Chemistry formula Any in this topic Relative formula mass Percentage by mass of an element Moles from mass Number of particles (Avogadro) Reacting masses from an equation Volume of a gas Gas released in an open flask Any in this topic Concentration of a solution Rf value in chromatography Titration calculation Dilution Enthalpy change from calorimetry Any in this topic Percentage yield Atom economy Mean rate of reaction Enthalpy change from bond energies Any in this topic Hess’s law from formation enthalpies Gibbs free energy Any in this topic Rate equation Equilibrium constant Kc Equilibrium constant Kp from partial pressures Partial pressure from mole fraction Concentration over time for a zero-order reaction Concentration over time for a second-order reaction Rate constant at a second temperature (Arrhenius) Any in this topic pH of a solution pH of a weak acid pH of a buffer Electrochemical cell potential Hydrogen-ion concentration of a weak acid without the approximation Concentration of excess acid or base after mixing pH after mixing a strong acid and a strong base Any in this topic Relative atomic mass from isotopes Percentage uncertainty Absorbance from the Beer–Lambert law Any in this topic Molality from solute and solvent masses Stock molarity from mass fraction and density Concentration after serial dilutions Ionic strength on a concentration basis Missing ion concentration from charge balance Any in this topic Equilibrium constant for a combined reaction Equilibrium constant at a second temperature Reaction Gibbs energy away from equilibrium Equilibrium conversion in a reversible isomerisation Gas dissociation fraction at fixed total pressure Any in this topic Very dilute strong acid including water autoionisation Neutral pH at a specified temperature Hydroxide from a salt of a weak monoprotic acid pH of an amphiprotic intermediate salt Species fractions of a diprotic acid Any in this topic Amounts required for a target buffer Buffer pH after a strong-base addition Strong acid dose to reach a lower buffer pH Differential capacity of a monoprotic buffer Fraction of indicator in its basic colour form Any in this topic Analyte amount from a back titration Mixed hydroxide and carbonate from two endpoints Signed relative titration error Gran extrapolation for a strong acid titre Free silver in a chloride precipitation titration Any in this topic Molar solubility of a binary salt in pure water Onset of precipitation during selective addition Common-ion solubility of a 1-to-2 salt by approximation Solubility of a metal hydroxide at imposed pH Metal precipitation recovery at a fixed free ligand concentration Any in this topic Fraction of metal bound at a fixed free ligand level Complex concentration when both reactants are depleted Conditional EDTA formation constant Distribution ratio of an extractable weak acid Recovery after repeated solvent extractions Any in this topic Reaction rate from one species’ concentration change Partial reaction order from paired initial rates Pseudo-first-order rate constant Relaxation to equilibrium for reversible first-order conversion Product fraction for parallel first-order reactions Any in this topic Intermediate in consecutive first-order reactions Time of maximum intermediate concentration Time for unequal-concentration bimolecular conversion Michaelis–Menten initial rate Initial rate with a competitive inhibitor Any in this topic Solvent vapour pressure above an ideal solution Bubble pressure of an ideal binary liquid Osmotic pressure of a dilute solution Freezing-point depression Boiling-point elevation Any Engineering formula Any in this topic Newton’s second law Work and kinetic energy Mechanical power Efficiency Mass flow and volume flow Hydrostatic pressure Shaft speed and angular frequency Any in this topic Projectile range (no drag) Constant-acceleration displacement Shaft power from torque and speed Natural frequency of a mass on a spring Damped natural frequency Acceleration from the velocity gradient Projectile height at a given time Acceleration along a curved path Radial acceleration in polar coordinates Transverse acceleration in polar coordinates Rolling without slipping Rigid-body kinetic energy Parallel-axis theorem Direct impact with restitution Angular impulse and momentum (fixed axis) Angular acceleration from a moment (Euler’s equation) Friction limit Forced vibration amplitude Base-excitation transmissibility Damping from a ring-down Free underdamped response Projectile maximum height Projectile time of flight Braking distance Capstan equation for rope friction Any in this topic Logarithmic decrement across several cycles Damping ratio from the ring-down decrement Ring-down time to a chosen amplitude fraction Energy lost in one sinusoidal damper cycle Damping from half-power frequencies Critically damped response to an impulse Peak displacement after an underdamped impulse Undamped response to a sudden constant force Residual amplitude after a rectangular force pulse Critically damped response to a force step Any in this topic Uniform beam bending frequencies Simply supported beam modal ordinate Flexural travelling-wave phase speed Flexural wave packet group speed Beam frequency from a trial deflection Transverse string harmonics Rectangular membrane frequencies Circular membrane fundamental frequency Longitudinal frequencies of a fixed–free rod Longitudinal rod characteristic impedance Any in this topic Generalised modal mass Generalised modal stiffness Generalised force from nodal loads Participation factor for base translation Directional effective modal mass Retained directional mass fraction Modal coordinate from an initial displacement Physical motion reconstructed from modal coordinates Modal sum for transfer receptance Residual static flexibility after mode truncation Any in this topic H1 transfer estimate H2 transfer estimate Magnitude-squared coherence Mobility from a measured receptance Accelerance from a measured receptance Modal assurance criterion Least-squares modal scale factor Modal damping from Rayleigh coefficients Rayleigh mass coefficient from two target frequencies Rayleigh stiffness coefficient from two target frequencies Frequency of minimum Rayleigh damping Any in this topic FFT frequency-bin spacing Window coherent gain Equivalent noise bandwidth of a window One-sided windowed acceleration periodogram Number of complete Welch segments Response PSD from a single input PSD Band-limited RMS from PSD samples Mean square over a log–log PSD segment Displacement PSD from acceleration PSD Miles estimate of resonant RMS acceleration Any in this topic Axial stress and extension Twist of a solid circular shaft Resultant of a trapezoidal distributed load Second moment of a built-up section Transverse shear stress in a beam Principal stresses in plane stress Von Mises equivalent stress Thermal stress in a restrained bar Tresca equivalent stress Direct stress Hooke’s law Torsional shear stress in a solid shaft Bending stress Second moment of area, rectangle Maximum shear stress (plane stress) Factor of safety Lamé radial stress in a thick cylinder Lamé hoop stress in a thick cylinder Axial stress in a closed-end thick cylinder Maximum hoop stress at the bore Hoop stress at the outside surface Thin-walled cylinder hoop stress Thin-walled cylinder axial stress Poisson’s ratio from measured strains Normal strain from three-dimensional stress and temperature change Shear strain from shear stress Shear modulus from Young’s modulus and Poisson’s ratio Bulk modulus from Young’s modulus and Poisson’s ratio Strain energy stored in an axially loaded member Strain energy stored in a twisted shaft Hertz contact radius of a sphere on a flat surface Hertz maximum contact pressure for a sphere on a flat surface Any in this topic Stress amplitude of a fatigue cycle Mean stress of a fatigue cycle Stress ratio of a fatigue cycle Modified Goodman factor of safety Miner’s rule cumulative damage (three load levels) Any in this topic Principal stresses in plane stress Von Mises equivalent stress Tresca equivalent stress Maximum shear stress (plane stress) Factor of safety Mohr’s circle centre (mean in-plane stress) Normal stress on a rotated element, x′ face Normal stress on a rotated element, y′ face Shear stress on a rotated element Principal angle in plane stress Principal stresses in order, including the zero Absolute maximum shear stress in plane stress Tresca to von Mises ratio Yield utilisation against a required factor of safety Normal stress on any plane (three dimensions) Shear stress magnitude on any plane (three dimensions) First stress invariant (trace) Second stress invariant Third stress invariant (determinant) Principal stresses in three dimensions Tresca equivalent stress from principal stresses Von Mises equivalent stress from principal stresses Von Mises equivalent stress from stress components (three dimensions) Any in this topic Tresca equivalent stress Lamé radial stress in a thick cylinder Lamé hoop stress in a thick cylinder Axial stress in a closed-end thick cylinder Maximum hoop stress at the bore Hoop stress at the outside surface Thin-walled cylinder hoop stress Thin-walled cylinder axial stress Elastic–plastic interface radius from a target plastic fraction First-yield pressure of a thick cylinder Fully plastic limit pressure of a thick cylinder Autofrettage pressure for a target plastic fraction Contact pressure at the elastic–plastic interface Peak hoop stress at radius r during autofrettage Residual hoop stress at radius r after autofrettage Residual radial stress at radius r after autofrettage Residual hoop stress at the bore after autofrettage Reverse-yielding screening ratio at the bore Autofrettage pressure for a target bore compression Plastic zone radius from a prescribed autofrettage pressure Bore hoop stress in service after autofrettage Any in this topic Bending stress in a rectangular beam Simply supported beam: central point load Simply supported beam: uniform load Cantilever with a tip load Euler buckling load Effective stress in saturated soil Open-channel flow (Manning) Hollow rectangular section second moment of area Plastic section modulus of a symmetric I-section Elastic neutral axis of a multi-material beam Flexural rigidity of a bonded composite beam Force per connector in a built-up beam Reactions to a partial uniformly distributed load Shear under a linearly increasing span load Peak sagging moment from a triangular span load Support reactions with a point load on an overhang Hogging moment at the root of a loaded overhang Deflection beneath an eccentric point load Cantilever tip displacement from an interior point load Cantilever tip deflection under a tapered load Simply supported end rotation under full-span uniform load Cantilever tip deflection from a pure end couple Prop reaction of a cantilever under full-span uniform load Prop reaction from an intermediate point load Fixed-ended beam midspan deflection under uniform load Interior support moment of a two-span continuous beam Moment distribution factor at a rigid joint Moment influence ordinate at a chosen beam section Shear influence ordinate at a chosen section Section moment under a moving axle train Maximum positive section shear from a movable uniform load Peak moment under two equal moving point loads Symmetric two-bar truss member force Joint displacement by virtual work in a truss Truss displacement due to member temperature changes Bending strain energy under a triangular moment diagram Axial stiffness of a diagonal brace against storey sway Horizontal reaction of a three-hinged arch Bending moment at a three-hinged arch section Horizontal tension in a parabolic suspension cable Support tension of a uniformly loaded parabolic cable Exact parabolic cable length from span and sag Rankine active earth pressure coefficient Rankine passive earth pressure coefficient Active force from soil weight and uniform surcharge Height of the active resultant above the wall base Vertical effective stress below a water table Biaxial corner contact pressures of a rectangular footing Peak footing pressure after one edge lifts Boussinesq vertical stress from a surface point load Stress beneath centre of a uniformly loaded circular area Average stress increase by the 2 to 1 method Normally consolidated clay settlement Overconsolidated clay settlement through preconsolidation stress Elastic layer settlement from constrained modulus Time to 90 percent average primary consolidation Secondary compression after primary consolidation Any in this topic Hollow rectangular section second moment of area Plastic section modulus of a symmetric I-section Elastic neutral axis of a multi-material beam Flexural rigidity of a bonded composite beam Force per connector in a built-up beam Any in this topic Reactions to a partial uniformly distributed load Shear under a linearly increasing span load Peak sagging moment from a triangular span load Support reactions with a point load on an overhang Hogging moment at the root of a loaded overhang Any in this topic Deflection beneath an eccentric point load Cantilever tip displacement from an interior point load Cantilever tip deflection under a tapered load Simply supported end rotation under full-span uniform load Cantilever tip deflection from a pure end couple Any in this topic Prop reaction of a cantilever under full-span uniform load Prop reaction from an intermediate point load Fixed-ended beam midspan deflection under uniform load Interior support moment of a two-span continuous beam Moment distribution factor at a rigid joint Any in this topic Moment influence ordinate at a chosen beam section Shear influence ordinate at a chosen section Section moment under a moving axle train Maximum positive section shear from a movable uniform load Peak moment under two equal moving point loads Any in this topic Symmetric two-bar truss member force Joint displacement by virtual work in a truss Truss displacement due to member temperature changes Bending strain energy under a triangular moment diagram Axial stiffness of a diagonal brace against storey sway Any in this topic Horizontal reaction of a three-hinged arch Bending moment at a three-hinged arch section Horizontal tension in a parabolic suspension cable Support tension of a uniformly loaded parabolic cable Exact parabolic cable length from span and sag Any in this topic Rankine active earth pressure coefficient Rankine passive earth pressure coefficient Active force from soil weight and uniform surcharge Height of the active resultant above the wall base Vertical effective stress below a water table Any in this topic Biaxial corner contact pressures of a rectangular footing Peak footing pressure after one edge lifts Boussinesq vertical stress from a surface point load Stress beneath centre of a uniformly loaded circular area Average stress increase by the 2 to 1 method Any in this topic Normally consolidated clay settlement Overconsolidated clay settlement through preconsolidation stress Elastic layer settlement from constrained modulus Time to 90 percent average primary consolidation Secondary compression after primary consolidation Any in this topic Dynamic pressure and lift Induced drag coefficient Mach number and speed of sound Reynolds number Stall speed in level flight Level coordinated turn Net thrust of a jet engine Geopotential altitude from geometric altitude Pressure in a constant temperature-gradient layer Pressure in an isothermal layer Density altitude in the standard troposphere Local pressure scale height Equivalent airspeed Subsonic pitot impact pressure Calibrated airspeed from subsonic impact pressure Temperature indicated by a recovery probe Mach angle of a weak disturbance Trapezoidal wing planform area Wing aspect ratio Mean aerodynamic chord of a trapezoidal wing Span station of the trapezoidal mean aerodynamic chord Finite-wing lift-curve slope Lift coefficient at maximum lift-to-drag ratio Speed for minimum power required Thrust required versus true airspeed Level-flight speed limits for constant available thrust Stall speed under a positive manoeuvre load factor Steady climb angle from excess thrust Still-air horizontal glide distance Steady glide sink rate Breguet jet cruise range Breguet propeller cruise range Stagnation-to-static density ratio Isentropic nozzle area–Mach ratio Sonic speed at a choked throat Corrected engine mass flow Velocity gradient in a varying-area duct Downstream Mach number of a normal shock Static pressure jump through a normal shock Density compression across a normal shock Stagnation-pressure retention through a normal shock Specific entropy increase across an adiabatic shock Ideal choked mass flow through a throat Critical back pressure of a convergent nozzle Exit Mach of a convergent nozzle versus back pressure Exit speed with a nozzle efficiency Gross axial momentum flux from a surveyed exhaust plane Compressor specific work from total pressure ratio Combustor fuel-to-air mass ratio Turbine exit temperature from shaft-power balance Turbine total pressure ratio from actual temperature drop Bypass ratio and fuel contribution to exhaust mass flow Axial induced speed at an actuator disk Ideal actuator-disk power in forward flight Rotor hover figure of merit Helical propeller tip Mach number Propeller efficiency from test coefficients Any in this topic Geopotential altitude from geometric altitude Pressure in a constant temperature-gradient layer Pressure in an isothermal layer Density altitude in the standard troposphere Local pressure scale height Any in this topic Equivalent airspeed Subsonic pitot impact pressure Calibrated airspeed from subsonic impact pressure Temperature indicated by a recovery probe Mach angle of a weak disturbance Any in this topic Trapezoidal wing planform area Wing aspect ratio Mean aerodynamic chord of a trapezoidal wing Span station of the trapezoidal mean aerodynamic chord Finite-wing lift-curve slope Any in this topic Lift coefficient at maximum lift-to-drag ratio Speed for minimum power required Thrust required versus true airspeed Level-flight speed limits for constant available thrust Stall speed under a positive manoeuvre load factor Any in this topic Steady climb angle from excess thrust Still-air horizontal glide distance Steady glide sink rate Breguet jet cruise range Breguet propeller cruise range Any in this topic Stagnation-to-static density ratio Isentropic nozzle area–Mach ratio Sonic speed at a choked throat Corrected engine mass flow Velocity gradient in a varying-area duct Any in this topic Downstream Mach number of a normal shock Static pressure jump through a normal shock Density compression across a normal shock Stagnation-pressure retention through a normal shock Specific entropy increase across an adiabatic shock Any in this topic Ideal choked mass flow through a throat Critical back pressure of a convergent nozzle Exit Mach of a convergent nozzle versus back pressure Exit speed with a nozzle efficiency Gross axial momentum flux from a surveyed exhaust plane Any in this topic Compressor specific work from total pressure ratio Combustor fuel-to-air mass ratio Turbine exit temperature from shaft-power balance Turbine total pressure ratio from actual temperature drop Bypass ratio and fuel contribution to exhaust mass flow Any in this topic Axial induced speed at an actuator disk Ideal actuator-disk power in forward flight Rotor hover figure of merit Helical propeller tip Mach number Propeller efficiency from test coefficients Any in this topic Wheel drive force from engine torque Vehicle speed from engine speed Equivalent mass with rotating inertia Vehicle acceleration from road load Available tractive force (torque or grip limited) Engine swept displacement Mean piston speed Brake mean effective pressure Brake thermal efficiency Electric vehicle range estimate Electric motor torque envelope Total stopping distance with reaction time Front axle load during braking Rear axle load during acceleration Ideal front brake force fraction Brake torque from clamp force Brake disc temperature rise Friction ellipse remaining longitudinal force Tyre force load sensitivity Linear tyre lateral force Tyre longitudinal slip ratio Magic Formula tyre force Kinematic turning radius Ackermann outer wheel steer angle Understeer gradient Steady-state steering angle Yaw rate gain Lateral load transfer Sprung body roll angle Static rollover threshold Quarter-car ride frequency Wheel-hop frequency Aerodynamic downforce Cornering speed limit with downforce Power for steady level-road speed Longitudinal centre of gravity from scale readings Axle reactions on a longitudinal grade Front axle reaction under a horizontal drawbar pull Axle load changes from a vertical tongue load Load transfer ratio from left and right wheel loads Wheel-centre velocity in tyre coordinates Tyre slip angle from local velocity Pneumatic aligning moment Transient tyre force after a slip-angle step Wheel angular acceleration under drive and brake torque Steady cornering axle force distribution Yaw acceleration from axle forces Steady CG sideslip in a linear bicycle model No-side-friction speed on a banked curve Direct yaw moment from unequal longitudinal tyre forces Vertical wheel rate from spring installation Equivalent vertical damping at the wheel Vertical ride rate including tyre compliance Axle roll stiffness from two wheel springs Front share of elastic roll moment Sprung-mass vertical acceleration Unsprung-mass vertical acceleration Coupled body and wheel natural frequencies Road-to-body displacement transfer function Dynamic load coefficient of a tyre Temporal frequency of a spatial road wave Front-to-rear axle road-input delay Amplitude of mean front and rear road input Temporal elevation PSD from a spatial road PSD Road elevation RMS over a spatial band Any in this topic Longitudinal centre of gravity from scale readings Axle reactions on a longitudinal grade Front axle reaction under a horizontal drawbar pull Axle load changes from a vertical tongue load Load transfer ratio from left and right wheel loads Any in this topic Wheel-centre velocity in tyre coordinates Tyre slip angle from local velocity Pneumatic aligning moment Transient tyre force after a slip-angle step Wheel angular acceleration under drive and brake torque Any in this topic Steady cornering axle force distribution Yaw acceleration from axle forces Steady CG sideslip in a linear bicycle model No-side-friction speed on a banked curve Direct yaw moment from unequal longitudinal tyre forces Any in this topic Vertical wheel rate from spring installation Equivalent vertical damping at the wheel Vertical ride rate including tyre compliance Axle roll stiffness from two wheel springs Front share of elastic roll moment Any in this topic Sprung-mass vertical acceleration Unsprung-mass vertical acceleration Coupled body and wheel natural frequencies Road-to-body displacement transfer function Dynamic load coefficient of a tyre Any in this topic Temporal frequency of a spatial road wave Front-to-rear axle road-input delay Amplitude of mean front and rear road input Temporal elevation PSD from a spatial road PSD Road elevation RMS over a spatial band Any in this topic Wheel drive force from engine torque Vehicle speed from engine speed Equivalent mass with rotating inertia Available tractive force (torque or grip limited) Overall driveline reduction ratio Total tractive force required Engine power required at a given speed Minimum first-gear ratio for climbing Top-gear ratio for a target top speed Minimum friction coefficient to climb a slope Gearbox ratio span Number of gears for a ratio span Geometric step between gears Ratio of gear j in a geometric set Torque converter efficiency Torque converter power loss Carrier speed of a simple planetary gear Planetary ratio, sun to carrier (ring held) Planetary ratio, ring to carrier (sun held) Planetary ratio, sun to ring (carrier held) Order excitation frequency Gear mesh frequency Track-pitch excitation frequency Any in this topic Apparent mass at a driving point Measured transmissibility Weighted RMS acceleration over three segments Combined vibration dose value of separate segments Motion sickness dose value at a steady level Seat effective amplitude transmissibility (SEAT) Partial daily vibration exposure on one axis Daily vibration exposure from three tasks on one axis Daily vibration exposure A(8), governing axis Time to reach a chosen exposure value Remaining time to an exposure value with prior dose Any in this topic Proportional feedback on a first-order plant First-order step response Percentage overshoot of an underdamped second-order system Peak time of an underdamped second-order system Approximate 2 % settling time of a second-order system Damping ratio from measured overshoot Steady-state error to a step input Steady-state error to a ramp input PID gains from integral and derivative times Gain of a filtered derivative term Integrator rate with back-calculation antiwindup Discrete integral term after n samples Critical gain from the Routh–Hurwitz cubic test Maximum phase lead of a lead compensator Lead compensator design from required phase Delay margin from phase margin Gain margin in decibels Frequency response of a first-order lag Sensitivity at one frequency State-feedback gains for a double integrator Scalar LQR feedback gain Zero-order-hold discretisation of a first-order plant Tustin frequency warping Aliased frequency after sampling Any in this topic Sample of a phase-offset sinusoid Sample of a linear frequency sweep Zero-order hold amplitude droop Aperture-jitter-limited sine SNR Ideal ADC SNR for a full-scale sine Complex DFT coefficient at one bin One sample from a full complex spectrum Mean square from two-sided DFT coefficients Phase change from a circular sample delay DFT bin relocation by complex modulation Hann window sample Hamming window sample Blackman window sample Kaiser window sample Bartlett window sample Peak sine amplitude from an interior DFT line Tone phase from sine and cosine projections Total harmonic distortion from measured harmonics Power-weighted spectral centroid Sub-bin peak location by parabolic interpolation Linear convolution at a specified sample Circular convolution at a specified sample Cross-correlation at a signed lag Finite-record biased autocovariance Usable input block size for FFT convolution Windowed-sinc low-pass tap High-pass FIR by spectral inversion Windowed-sinc band-pass tap Time delay of a symmetric linear-phase FIR Kaiser window tap-count estimate Moving-average magnitude response Output variance of an FIR driven by white noise Comb difference-filter magnitude Five-point quadratic Savitzky–Golay smoothing Five-point least-squares first derivative One output step of a normalised biquad Decay time constant from a pole radius Oscillation frequency of a complex pole pair Digital notch biquad coefficients Stability test for a real biquad denominator Complex response from filter coefficients Response of cascaded second-order sections Forward–backward filtering amplitude response Group delay from neighbouring phase samples Phase delay at a specified frequency Rational resampling output rate Full upsample–FIR–downsample output length Sinc interpolation of a sampled waveform Envelope from in-phase and quadrature samples Instantaneous frequency from an analytic-signal pair Any in this topic Sample of a phase-offset sinusoid Sample of a linear frequency sweep Zero-order hold amplitude droop Aperture-jitter-limited sine SNR Ideal ADC SNR for a full-scale sine Any in this topic Complex DFT coefficient at one bin One sample from a full complex spectrum Mean square from two-sided DFT coefficients Phase change from a circular sample delay DFT bin relocation by complex modulation Any in this topic Hann window sample Hamming window sample Blackman window sample Kaiser window sample Bartlett window sample Any in this topic Peak sine amplitude from an interior DFT line Tone phase from sine and cosine projections Total harmonic distortion from measured harmonics Power-weighted spectral centroid Sub-bin peak location by parabolic interpolation Any in this topic Linear convolution at a specified sample Circular convolution at a specified sample Cross-correlation at a signed lag Finite-record biased autocovariance Usable input block size for FFT convolution Any in this topic Windowed-sinc low-pass tap High-pass FIR by spectral inversion Windowed-sinc band-pass tap Time delay of a symmetric linear-phase FIR Kaiser window tap-count estimate Any in this topic Moving-average magnitude response Output variance of an FIR driven by white noise Comb difference-filter magnitude Five-point quadratic Savitzky–Golay smoothing Five-point least-squares first derivative Any in this topic One output step of a normalised biquad Decay time constant from a pole radius Oscillation frequency of a complex pole pair Digital notch biquad coefficients Stability test for a real biquad denominator Any in this topic Complex response from filter coefficients Response of cascaded second-order sections Forward–backward filtering amplitude response Group delay from neighbouring phase samples Phase delay at a specified frequency Any in this topic Rational resampling output rate Full upsample–FIR–downsample output length Sinc interpolation of a sampled waveform Envelope from in-phase and quadrature samples Instantaneous frequency from an analytic-signal pair Any in this topic Euler angle rates from body rates Pitch-rate response to an elevator step Specific total energy and energy height Specific total energy rate Any in this topic Force from a change in stream velocity Total head (Bernoulli) Pump head from the extended head balance Pipe friction and fitting losses Laminar flow in a pipe (Hagen–Poiseuille) Pump shaft power Viscous shear stress Buoyant force Speed from a pressure difference Dynamic pressure Drag force Pump flow rate after a speed change Pump head after a speed change Pump power after a speed change Darcy friction factor from the Haaland equation Any in this topic Resultant hydrostatic force on a submerged plane Vertical depth of the centre of pressure on an inclined plane Bottom-hinge moment on a vertical rectangular gate Differential U-tube manometer pressure Free-surface elevation change in a rotating liquid Any in this topic Laminar pipe velocity at a given radius Pipe wall shear stress from frictional pressure loss Kinetic-energy correction factor from sampled velocities Flow per unit width between plates with a moving wall Gravity-driven laminar film discharge per unit width Any in this topic Sudden enlargement mixing loss Equivalent straight length of a fitting Series pipe resistance for quadratic losses Equivalent resistance of two parallel quadratic branches Hardy Cross loop flow correction Any in this topic Differential-pressure Venturi meter discharge Velocity-area discharge by the midsection method Rectangular sharp-crested weir discharge Triangular sharp-crested weir discharge Ideal broad-crested rectangular weir discharge Any in this topic Hydraulic radius of a trapezoidal channel Hydraulic depth of a trapezoidal channel Normal depth of a wide rectangular channel Mean boundary shear stress in uniform channel flow Most efficient rectangular channel width Any in this topic Specific energy in a rectangular channel Critical depth of a rectangular channel Conjugate depth of a rectangular hydraulic jump Specific energy lost across a hydraulic jump Gradually varied flow depth gradient Any in this topic Time to drain a constant-area tank Jet velocity coefficient from its trajectory Normal force of a free jet on an inclined plate Fluid force on a right-angle elbow (x-component) Diameter of a falling free liquid jet Any in this topic Available net positive suction head at a pump inlet Pump operating flow from quadratic curves Combined flow of two dissimilar parallel pumps Euler head transferred by a pump impeller Outlet whirl velocity from the impeller velocity triangle Any in this topic Net extension force of a double-acting cylinder Cylinder retraction speed Hydraulic motor output torque Actual delivery of a displacement pump Blocked-port hydraulic cylinder stiffness Any in this topic Rapid-closure pressure surge Wave speed in an elastic thin-walled pipe Valve-to-reservoir wave return time Surge-tank oscillation period Usable oil volume from a gas-charged accumulator Any in this topic Reynolds number Drag force Number of dimensionless groups (Buckingham pi theorem) Dimensionless force coefficient Model test speed for Reynolds-number matching Full-scale force from a model measurement Any in this topic Mach number and speed of sound Reynolds number Thermal diffusivity and Fourier number Nusselt number Prandtl number Biot number Froude number Weber number Strouhal number Euler number Grashof number Rayleigh number Bulk Richardson number for mixed convection Cauchy number Péclet number Schmidt number Sherwood number Stanton number Any in this topic Pump flow rate after a speed change Pump head after a speed change Pump power after a speed change Pump or fan flow rate at a new speed and diameter Pump head at a new speed and diameter Fan pressure rise at a new speed, diameter and density Pump or fan shaft power at a new speed, diameter and density Propeller advance ratio Propeller thrust from the thrust coefficient Propeller shaft torque from the torque coefficient Propeller shaft power from the torque coefficient Any in this topic Ideal-gas internal energy and enthalpy change Steady-flow energy balance Boundary work at constant pressure Ideal-gas entropy change Isentropic turbine efficiency Isentropic compressor efficiency Carnot efficiency Refrigeration coefficient of performance Wet-steam quality and mixture enthalpy First law of thermodynamics Isothermal expansion work Adiabatic compression temperature Otto cycle efficiency Brayton cycle efficiency Ideal refrigeration coefficient of performance Pressure after a polytropic process Temperature after a polytropic process Boundary work in a polytropic process Ideal Diesel cycle efficiency Ideal heat pump coefficient of performance Entropy change from reversible heat transfer (Clausius) Boltzmann entropy from the number of microstates Entropy of a four-outcome distribution (Gibbs and Shannon) Fundamental thermodynamic relation (change in internal energy) Bekenstein–Hawking entropy of a black hole Any in this topic Heat conduction through a wall Convective heat transfer Heat flow through a composite wall Heat loss through a pipe wall Net thermal radiation Lumped transient cooling Thermal diffusivity and Fourier number Nusselt number Prandtl number Heat exchanger duty (LMTD method) Blackbody emissive power Biot number Dittus–Boelter correlation Log-mean temperature difference Fin parameter of a straight fin Heat transfer from a straight fin with an insulated tip Efficiency of a straight fin with an insulated tip Effectiveness of a counterflow heat exchanger Heat duty of an exchanger from its effectiveness Number of transfer units of a heat exchanger Critical insulation radius of a cylinder Any in this topic Resistance of a hollow sphere Temperature jump across a thermal contact Effective conductivity of parallel paths Plane-wall flux with linearly varying conductivity Temperature within a cylindrical wall Any in this topic Symmetric slab with uniform heat generation Radial temperature in a heat-generating cylinder Radial temperature in a heat-generating sphere Hottest location in a slab with unequal face temperatures Maximum temperature of an internally heated annulus Any in this topic Tip temperature of an insulated-tip fin Distance to a specified temperature fraction on a long fin Base heat rate of a fin with a convecting tip Base heat rate with prescribed fin-tip temperature Overall efficiency of a finned surface Any in this topic Temperature after a sudden surface-temperature change Surface heat flux after a temperature step Cumulative energy absorbed after a temperature step Temperature under a constant imposed surface flux Contact temperature of two semi-infinite solids Any in this topic Plane-wall transient temperature by the first eigenmode Mean temperature ratio of a convectively cooled plane wall Lumped body with constant internal power Equilibration time constant of two finite bodies Penetration depth of a periodic thermal wave Any in this topic Average heat-transfer coefficient on a laminar flat plate Cylinder crossflow mean Nusselt number Mean natural convection on a vertical isothermal plate Dittus–Boelter Nusselt number for heating or cooling Tube outlet temperature at constant wall temperature Any in this topic Exchange between large diffuse grey plates Radiative heat exchange between concentric spheres Radiative flux through a single floating shield Reverse view factor from reciprocity Effective radiant surroundings temperature Any in this topic Blackbody spectral emissive power Wavelength of peak blackbody emission Radiosity of an opaque diffuse grey surface Radiative heat-transfer coefficient to large surroundings View factor between two parallel coaxial disks Any in this topic Parallel-flow exchanger effectiveness Crossflow effectiveness with the larger capacity stream mixed Crossflow effectiveness with both streams unmixed One-shell-pass exchanger effectiveness Overall conductance of a fouled cylindrical tube Any in this topic Heat stored through a complete melting interval Adiabatic mixing temperature of two streams Entropy generation in an insulated heat exchanger Quasi-steady planar solidification front Lumped radiative cooling towards a cold sink Any in this topic Electric charge Electrical work done Electrical power Ohm’s law Resistance and temperature Peak and RMS values (AC) Resistors in parallel Resistivity Battery internal resistance Energy stored in a capacitor Electric field strength Capacitor time constant Capacitor discharge half time Force on a current Transformer equation Potential divider RC circuit response Inductive reactance Capacitive reactance Real power, single-phase AC Real power, three-phase AC Inverting op-amp gain Non-inverting op-amp gain Joule heating Three resistors in parallel Series RLC impedance Resonant frequency Quality factor of a series RLC circuit Skin depth Load current from a Thevenin source Load voltage from a Thevenin source Power delivered to a load by a Thevenin source Maximum power transfer from a Thevenin source Inductor current after a switching event Time constant of an RL circuit Half-power bandwidth of a series RLC circuit Reactive power in a single-phase circuit Apparent power in a single-phase circuit Cutoff frequency of a first-order RC filter Gain of a first-order RC low-pass filter Gain of a first-order RC high-pass filter Shockley diode current Thermal voltage Damping rate of a series RLC circuit Damped angular frequency of a series RLC circuit Critical resistance of a series RLC circuit Damping ratio of a series RLC circuit Any in this topic Current division between two parallel resistors Loaded voltage divider output Unloaded Wheatstone bridge output voltage Node voltage fed through three resistors Delta-to-star resistor conversion Any in this topic Series capacitance of two capacitors Final voltage after capacitor charge sharing Energy dissipated during capacitor charge sharing Capacitor voltage ramp under constant current Magnetic energy stored in an inductor Any in this topic Average output of a half-wave rectifier RMS output of a half-wave rectifier Average output of a full-wave rectifier Reservoir capacitor ripple after full-wave rectification Zener shunt regulator current Any in this topic BJT base current with divider bias and emitter resistor BJT collector-emitter bias voltage BJT small-signal transconductance Hybrid-pi base input resistance Common-emitter gain with an unbypassed emitter resistor Any in this topic NMOS drain current in the triode region NMOS drain current in saturation MOSFET transconductance from overdrive voltage MOSFET small-signal output resistance with channel-length modulation Source-follower voltage gain Any in this topic Weighted inverting summing amplifier Matched four-resistor difference amplifier Inverting integrator output under constant input Inverting differentiator output for a ramp input Transimpedance amplifier output voltage Any in this topic Closed-loop gain with finite open-loop gain Closed-loop bandwidth from gain-bandwidth product Full-power bandwidth from slew rate Output error from input offset voltage Bias-current compensation resistor Any in this topic Sallen-Key low-pass natural frequency Sallen-Key quality factor with equal components Butterworth low-pass magnitude response Minimum Butterworth low-pass filter order Phase lag of a first-order RC low-pass filter Any in this topic Shunt capacitor for single-phase power-factor correction Complex power from voltage and current phasors Total real power by the two-wattmeter method Balanced reactive power from two wattmeters Balanced delta line current Any in this topic Johnson noise voltage of a resistor Shot noise current of a junction Input-referred amplifier and source noise density Noise bandwidth of a first-order low-pass filter ADC quantisation noise Any in this topic Amount and concentration Ideal gas law Sensible heating of a stream First-order batch kinetics Arrhenius rate constant Stirred-tank reactor volume Steady diffusion flux through a plane layer Molar diffusion rate through a plane layer Any in this topic Processor execution time Amdahl’s law Average memory access time Transfer and propagation delay ADC step size and dynamic range Any Defence Engineering formula Any in this topic Radar range equation Passive sonar equation Radar range from echo delay Radar range resolution Radar Doppler shift Radar horizon Sonar transmission loss Active sonar equation Any in this topic Free-space link budget Free-space path loss Thermal noise power Shannon channel capacity Any in this topic Tsiolkovsky rocket equation Rocket thrust Any in this topic Static rollover threshold Drawbar pull Thrust to climb a grade Maximum climbable grade Traction limit on a slope Approach or departure angle Breakover angle Force to push a rigid wheel over a step Mean ground pressure of a tracked vehicle Bekker pressure–sinkage relation Sinkage under uniform pressure Soil compaction resistance Mohr–Coulomb soil shear strength Janosi–Hanamoto shear stress Terramechanics driving slip Maximum soil thrust under a flat contact Tractive efficiency Ideal differential steering radius Track thrust with finite slip Idealised track steering resistance Undrained bearing capacity of clay Rigid wheel resistance on clay Remoulded cone index Undrained strength from cone index Mean maximum pressure (MMP) Effect of weight and track width on MMP Limiting cone index, tracked vehicle Limiting cone index, wheeled vehicle Vehicle cone index for one pass (VCI₁) Vehicle cone index for 50 passes (VCI₅₀) Gravimetric water content of a soil specimen Dry bulk density from field density and water content Void ratio from particle and dry bulk densities Degree of saturation from phase relationships Relative density of a granular test bed Sinkage exponent from two plate observations Bekker moduli from two plate widths Mean shear stress inferred from annular shear torque Shear deformation length from one mobilisation point Mohr–Coulomb parameters from two shear tests Wheel entry angle from geometric sinkage Accumulated shear displacement around a driven wheel Vertical wheel support from measured contact stresses Net horizontal wheel force from contact stresses Wheel torque required to overcome soil shear End pressures beneath an eccentrically loaded flat track Measured longitudinal centre of track pressure Work per area to mobilise soil shear Work input per volume swept by a penetration cone Individual track thrusts for prescribed force and yaw moment Any in this topic Gravimetric water content of a soil specimen Dry bulk density from field density and water content Void ratio from particle and dry bulk densities Degree of saturation from phase relationships Relative density of a granular test bed Any in this topic Sinkage exponent from two plate observations Bekker moduli from two plate widths Mean shear stress inferred from annular shear torque Shear deformation length from one mobilisation point Mohr–Coulomb parameters from two shear tests Any in this topic Wheel entry angle from geometric sinkage Accumulated shear displacement around a driven wheel Vertical wheel support from measured contact stresses Net horizontal wheel force from contact stresses Wheel torque required to overcome soil shear Any in this topic End pressures beneath an eccentrically loaded flat track Measured longitudinal centre of track pressure Work per area to mobilise soil shear Work input per volume swept by a penetration cone Individual track thrusts for prescribed force and yaw moment Any in this topic Ground speed and course from NED velocity Ground speed from airspeed and wind Range and bearing to a point Position error from accelerometer bias Pitch from a stationary accelerometer GNSS pseudorange with receiver clock bias Scalar Kalman filter measurement update Steady-state Kalman gain for a random-walk state Alpha–beta tracker update Process noise for a constant-velocity model Innovation gate statistic RMSE from bias and spread Wrapped course error Cross-track error from a straight path Course command for straight-path following Lookahead (pure-pursuit) lateral acceleration Bank angle for a commanded lateral acceleration Any in this topic Projectile range (no drag) Any in this topic Exponential reliability Series system reliability Availability Weibull life: failure time density Weibull life: probability of failure by a deadline Weibull life: instantaneous failure rate Weibull life: mean time to failure Weibull life: lifetime variance Any in this topic Lanchester square law
Question type
Find the result Mixed: sometimes find an input
Lognormal measurement: fraction below a threshold PRD-16 · tolerance ±2%
m Median m36 µm s Log spread s0.31 x Threshold x43 µm Find F (Fraction at or below x).
Hint Show solution New problem
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