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 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 Product rule Quotient rule Chain rule (power of a linear function) Definite integral of a power Trapezium rule (4 strips) Newton–Raphson step 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 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 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 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 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 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) 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 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 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 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 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 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 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 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) 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
Probability of at least one success STA-11 · tolerance ±2%
p Probability of success each time0.94 n Number of attempts91 Find P (Probability).
Hint Show solution New problem
Session score: no attempts yet