Engineering Fundamentals
Statics, dynamics, engineering mathematics, differential equations, statistics and probability, and engineering economics, with worked numbers.
Every other guide on this site rests on a small set of fundamentals: forces and equilibrium, how things move, the mathematics to describe them, the statistics to make sense of measurements and the economics to decide whether to build them. This guide collects those fundamentals in one place with worked numbers, for students refreshing their first-year courses, people preparing for the FE exam, and working engineers who want a quick reliable reminder.
Start here: what do you need?
| I need to… | Go to |
|---|---|
| find reactions, moments and forces in equilibrium | Statics: Forces, Moments and Equilibrium |
| find a centroid or second moment of area, or solve a friction problem | Centroids, Area Moments and Friction |
| work out how fast, how far and how long | Kinematics: Motion of Particles and Bodies |
| use force, energy or momentum to predict motion | Dynamics: Force, Energy and Momentum |
| brush up on algebra, trigonometry, vectors, matrices and calculus | Engineering Mathematics Essentials |
| model a cooling part, a spring-mass system or a circuit | Differential Equations for Engineers |
| analyze measurements, tolerances and reliability | Statistics and Probability for Engineers |
| decide between projects or compute payback, NPV and loan payments | Engineering Economics |
A suggested order
For mechanics read Statics: Forces, Moments and Equilibrium, Centroids, Area Moments and Friction, Kinematics: Motion of Particles and Bodies and Dynamics: Force, Energy and Momentum in that order; each uses the one before. Engineering Mathematics Essentials and Differential Equations for Engineers support everything, and Statistics and Probability for Engineers and Engineering Economics are independent.
- Statics: Forces, Moments and Equilibrium: Free-body diagrams, resultants, moments, equilibrium, reactions and trusses.
- Centroids, Area Moments and Friction: Centroids, second moments of area, parallel-axis theorem, friction, wedges and belts.
- Kinematics: Motion of Particles and Bodies: Constant acceleration, projectiles, rotation, relative motion and velocity of mechanisms.
- Dynamics: Force, Energy and Momentum: Newton’s laws, work and energy, impulse and momentum, collisions and rotation.
- Engineering Mathematics Essentials: Algebra, trigonometry, vectors, matrices, calculus and series with worked numbers.
- Differential Equations for Engineers: First- and second-order ODEs, the systems that obey them and how to solve them.
- Statistics and Probability for Engineers: Descriptive statistics, the normal distribution, confidence intervals, regression and reliability.
- Engineering Economics: Time value of money, NPV, IRR, payback, depreciation and choosing between projects.
Eight topics in numbers
Worked example: one number from each page
- Statics: a 6 m beam carrying 40 kN in total has reactions of 21.67 and 18.33 kN and a peak moment of 33.6 kN·m.
- Sections and friction: a block on a slope with μ = 0.4 slides at 21.8°.
- Kinematics: a ball thrown at 20 m/s and 45° goes 40.8 m.
- Dynamics: a 1 000 kg car at 100 km/h carries 386 kJ; stopping it in 48 m takes 8 kN.
- Mathematics: the quadratic 2x² − 12x + 10 = 0 has roots 5 and 1.
- Differential equations: a cooling part with τ = 300 s takes 584 s to fall from 90 °C to 30 °C in a 20 °C room.
- Statistics: shafts at 25.00 ± 0.02 mm (1σ) are inside ±0.05 mm 98.76 percent of the time.
- Economics: a $50 000 machine saving $12 000 a year for six years has an IRR of 11.5 percent.
Quick reference: the equations
| What you need | Equation |
|---|---|
| Equilibrium | ΣF = 0, ΣM = 0 |
| Moment | M = F d⊥ |
| Parallel-axis theorem | I = Ic + A d² |
| Friction | F ≤ μ N, φ = arctan μ |
| Constant acceleration | v = v₀ + a t, x = v₀ t + ½ a t², v² = v₀² + 2 a x |
| Newton | ΣF = m a, ΣM = I α |
| Energy | KE = ½ m v², PE = m g h, W = ΔKE |
| Momentum | J = Δ(m v) |
| Quadratic | x = (−b ± √(b² − 4ac)) ÷ 2a |
| First-order response | y = y∞ + (y₀ − y∞) e−t/τ |
| Second-order | ωn = √(k ÷ m), ζ = c ÷ 2√(k m) |
| Normal distribution | z = (x − μ) ÷ σ |
| Compound interest | F = P (1 + i)n |
| NPV | NPV = −I + Σ Ct ÷ (1 + r)t |
Related guides and tools
- Structures and Strength of Materials: stress, beams and fatigue, which build on statics.
- Machine Design and Gears and Power Transmission: where the dynamics are applied.
- Control Systems: differential equations and feedback.
- Thermodynamics and Steam: energy, applied to fluids.
- The Unit Conversions converters and the Downloads library (formula sheets, statics and dynamics, statistics and math, and FE practice sets).
FAQ
Is this enough for the FE exam?
It covers the core of statics, dynamics, mathematics, statistics and economics at the level of the exam, with worked examples; the downloads include practice sets. For the exam use the official reference handbook and your own course material too.
Where are strength of materials and thermodynamics?
In Structures and Strength of Materials and Thermodynamics and Steam.
Do I need to memorize these equations?
Know the few that you use daily and understand where the rest come from. This page is a reference for the rest.
References
- Hibbeler, R. C., Engineering Mechanics: Statics and Dynamics, Pearson; Meriam, J. L., Kraige, L. G., Engineering Mechanics, Wiley.
- Kreyszig, E., Advanced Engineering Mathematics, Wiley; Montgomery, D. C., Runger, G. C., Applied Statistics and Probability for Engineers, Wiley.
- Sullivan, W. G., et al., Engineering Economy, Pearson; NCEES FE Reference Handbook.
Last reviewed: November 2026. Worked values are computed from the stated numbers; this is educational information.