FE Other Disciplines formula sheet

Single-Variable Calculus

Limits, derivatives, optimization, definite and indefinite integrals, the Fundamental Theorem, area, and L'Hopital — the calculus the FE asks across nearly every topic.

Derivative (definition)

$$f'(x)=\lim_{\Delta x\to 0}\dfrac{f(x+\Delta x)-f(x)}{\Delta x}$$

Instantaneous rate / tangent slope; units of $f'$ are units of $f$ per unit of $x$.

Power rule

$$\dfrac{d}{dx}x^{n}=n\,x^{n-1}$$

Differentiates any monomial; the backbone of polynomial differentiation.

Product and quotient rules

$$(uv)'=u'v+uv',\qquad\left(\dfrac{u}{v}\right)'=\dfrac{u'v-uv'}{v^2}$$

For products and quotients of functions $u(x),v(x)$; mind the numerator order in the quotient.

Chain rule

$$\dfrac{d}{dx}f(g(x))=f'(g(x))\,g'(x)$$

Differentiate compositions: outer derivative times inner derivative.

Critical point + 2nd-derivative test

$$f'(a)=0,\quad f''(a)>0\Rightarrow\text{min},\ f''(a)<0\Rightarrow\text{max}$$

Locate then classify extrema; inflection where $f''$ changes sign.

Power rule for integrals

$$\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C\quad(n\ne -1)$$

Antiderivative of a power; for $n=-1$, $\int x^{-1}dx=\ln|x|+C$.

Integration by parts

$$\int u\,dv = uv-\int v\,du$$

For products; pick $u$ to simplify on differentiation (LIATE guideline).

Fundamental Theorem of Calculus

$$\int_a^b f(x)\,dx=F(b)-F(a),\quad F'=f$$

Evaluate a definite integral (signed area) via any antiderivative $F$.

Area between curves

$$A=\int_a^b\big[f_{\text{upper}}(x)-f_{\text{lower}}(x)\big]dx$$

Limits are the intersection points; split where the curves cross.

L'Hopital's rule

$$\lim_{x\to a}\dfrac{f(x)}{g(x)}=\lim_{x\to a}\dfrac{f'(x)}{g'(x)}$$

Valid only for $0/0$ or $\infty/\infty$; repeat until determinate.

Common derivatives

$$\dfrac{d}{dx}\sin x=\cos x,\ \dfrac{d}{dx}e^{x}=e^{x},\ \dfrac{d}{dx}\ln x=\dfrac{1}{x}$$

Table staples; trig arguments are in radians.

Where it lives: NCEES FE Reference Handbook — Mathematics (Differential Calculus, Integral Calculus, Derivatives and Indefinite Integrals tables) · Stewart, Calculus: Early Transcendentals

Probability Distributions and Estimation

Probability rules plus the binomial, Poisson, normal, exponential, and uniform models — choose the right one, then point-estimate and build a confidence interval for the mean.

Addition rule (inclusion-exclusion)

$$P(A\cup B)=P(A)+P(B)-P(A\cap B)$$

Probability that A or B occurs; the overlap term is zero for mutually exclusive events.

Conditional / joint probability

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

Reduces to $P(A)P(B)$ when A and B are independent; basis of the complement trick for "at least one."

Binomial PMF

$$P(X=x)=C(n,x)\,p^{x}(1-p)^{\,n-x}$$

Exactly $x$ successes in $n$ independent fixed trials, each with success probability $p$. Mean $np$, variance $np(1-p)$.

Poisson PMF

$$P(X=x)=\dfrac{\lambda^{x}e^{-\lambda}}{x!}$$

Count of events in a fixed interval at constant rate; $\lambda$ = expected count = mean = variance. Scale $\lambda$ to the asked interval.

Standard-normal transform

$$z=\dfrac{x-\mu}{\sigma}$$

Converts any normal value to standard-normal units so the unit normal table applies; $\mu$, $\sigma$, and $x$ share units.

Normal table relations

$$R(z)=1-F(z),\qquad F(-z)=1-F(z)$$

$F(z)$ = area to the left, $R(z)$ = area to the right; symmetry lets you handle negative z with the positive-z table.

Exponential distribution

$$f(t)=\lambda e^{-\lambda t},\qquad P(T>t)=e^{-\lambda t}$$

Waiting time to the next Poisson event; mean $1/\lambda$, variance $1/\lambda^2$, memoryless. $t\ge 0$.

Continuous uniform

$$f(x)=\dfrac{1}{b-a},\quad \mu=\dfrac{a+b}{2},\quad \sigma^2=\dfrac{(b-a)^2}{12}$$

Constant density on $[a,b]$; everything outside the interval has zero density.

Sample mean and variance

$$\bar X=\dfrac{1}{n}\sum X_i,\qquad s^2=\dfrac{1}{n-1}\sum\left(X_i-\bar X\right)^2$$

Unbiased point estimators of $\mu$ and $\sigma^2$; the $n-1$ (Bessel) divisor is what the FE expects.

Confidence interval, σ known

$$\bar X \pm z_{\alpha/2}\,\dfrac{\sigma}{\sqrt n}$$

Interval for $\mu$ when $\sigma$ is known or $n$ large; $z_{\alpha/2}=1.96$ (95%), $1.645$ (90%), $2.576$ (99%).

Confidence interval, σ unknown

$$\bar X \pm t_{\alpha/2,\,n-1}\,\dfrac{s}{\sqrt n}$$

Interval for $\mu$ with small samples and estimated $s$; t-multiplier from the handbook table with $n-1$ degrees of freedom.

Where it lives: NCEES FE Reference Handbook — Engineering Probability and Statistics · NCEES FE Reference Handbook — Unit Normal Distribution table and t-Distribution table · Montgomery & Runger, Applied Statistics and Probability for Engineers

Stoichiometry and Chemical Equilibrium

The mole, balanced equations, limiting reagent and yield, plus the equilibrium constant and Le Chatelier's principle that govern how far a reaction goes.

Moles from mass

$$n = \dfrac{m}{M}$$

$n$ = moles, $m$ = mass (g), $M$ = molar mass (g/mol). The first conversion in nearly every stoichiometry problem.

Avogadro's number

$$1\;\text{mol} = 6.022\times10^{23}\;\text{particles}$$

Bridges molecule counts and moles. Rarely needed numerically but defines the mole.

Molar volume at STP

$$V = 22.4\;\text{L/mol}\quad(0^\circ\text{C},\,1\,\text{atm})$$

Converts an ideal-gas volume to moles at standard conditions; $359\,\text{ft}^3$/lb-mol in USCS.

Molarity

$$M = \dfrac{n_{\text{solute}}}{V_{\text{solution, L}}}$$

Concentration in mol/L. Used to convert solution volumes to moles in reaction and equilibrium work.

Limiting-reagent test

$$\text{limiter} = \min_i\!\left(\dfrac{n_i}{\nu_i}\right)$$

Divide each reactant's moles $n_i$ by its coefficient $\nu_i$; the smallest quotient is the limiting reagent that sets the yield.

Percent yield

$$\%\,\text{yield} = \dfrac{m_{\text{actual}}}{m_{\text{theoretical}}}\times100\%$$

Actual product relative to the limiting-reagent maximum. A value above 100% flags an error or impurity.

Equilibrium constant

$$K_{eq} = \dfrac{[\mathrm{C}]^{c}[\mathrm{D}]^{d}}{[\mathrm{A}]^{a}[\mathrm{B}]^{b}}$$

Products over reactants, each to its coefficient; pure solids/liquids are taken as activity 1. Large $K_{eq}$ favors products.

Reaction quotient

$$Q = \dfrac{[\mathrm{C}]^{c}[\mathrm{D}]^{d}}{[\mathrm{A}]^{a}[\mathrm{B}]^{b}}\;\text{(any instant)}$$

Same form as $K_{eq}$ but at non-equilibrium concentrations. $Q<K$ shifts forward, $Q>K$ shifts reverse.

Kp–Kc relation

$$K_p = K_c\,(RT)^{\Delta n}$$

$\Delta n$ = moles of gaseous product minus reactant. Use $R=0.08206\,\text{L·atm/(mol·K)}$ with pressures in atm.

Solubility product

$$K_{sp} = [\mathrm{A^{n+}}]^{m}[\mathrm{B^{m-}}]^{n}$$

Equilibrium constant for a slightly soluble salt $\text{A}_m\text{B}_n$ dissolving; predicts precipitation when $Q>K_{sp}$.

Where it lives: NCEES FE Reference Handbook — Chemistry and Biology (Definitions, Equilibrium Constant of a Chemical Reaction, Solubility Product, Bioconversion) · NCEES FE Reference Handbook — Chemistry and Biology (Periodic Table of Elements) · NCEES FE Reference Handbook — Thermodynamics

Data Acquisition, Sampling, and A/D Conversion

The signal chain from sensor to digital code: how the Nyquist rate prevents aliasing, how n-bit quantization sets resolution, and how conditioning, filtering, and gain make a small signal measurable.

Sampling rate

$$f_s = \dfrac{1}{\Delta t}$$

Sample frequency from the sampling interval $\Delta t$ (s). Units: Hz (samples/s).

Nyquist (Shannon) criterion

$$f_s > 2 f_N$$

Sample faster than twice the highest signal frequency to reconstruct it. The FE handbook labels that highest frequency $f_N$ the Nyquist frequency; $f_s/2$ is the folding frequency, which most DSP texts instead call the Nyquist frequency (with $2 f_N$ the Nyquist rate).

Alias frequency

$$f_a = \left|\,f - k\,f_s\,\right|$$

Apparent frequency of an undersampled component; $k$ is the integer placing $f_a$ in $[0, f_s/2]$. Irreversible once it occurs.

A/D voltage resolution (LSB)

$$Q = \dfrac{V_H - V_L}{2^n}$$

Smallest distinguishable voltage step. $V_H, V_L$ = input range limits, $n$ = number of bits (typ. 4, 8, 10, 12, 16).

Number of quantization levels

$$L = 2^n,\qquad N \in [\,0,\ 2^n - 1\,]$$

An $n$-bit converter has $2^n$ levels and codes 0 through $2^n - 1$.

Code to voltage

$$V = N\,Q + V_L$$

Reconstruct the analog voltage from integer code $N$ using resolution $Q$ and range offset $V_L$.

Maximum representable voltage

$$V_{\max} = V_H - Q$$

Top code is $2^n - 1$, so the highest reading is one LSB below $V_H$ — a common off-by-one trap.

Quantization error

$$e_q = \pm\dfrac{Q}{2}$$

Worst-case rounding error of a single conversion, half a least-significant bit.

Quantization SNR

$$\text{SNR}_{\text{dB}} \approx 6.02\,n + 1.76$$

Ideal SNR for a full-scale sinusoid; roughly 6 dB per bit of dynamic range.

Bits required for a resolution

$$2^n \ge \dfrac{V_H - V_L}{\Delta V}$$

Smallest $n$ that resolves a target step $\Delta V$ over the span; round $n$ up to the next integer.

Amplifier gain

$$V_{\text{out}} = G\,V_{\text{in}},\qquad G = \dfrac{V_H - V_L}{V_{\text{sensor,max}}}$$

Scale the sensor signal to fill the converter range without exceeding it.

Input-referred resolution

$$Q_{\text{in}} = \dfrac{Q}{G}$$

Resolution referred back to the sensor input after a gain stage $G$; amplifying before the A/D improves effective resolution.

Where it lives: NCEES FE Reference Handbook — Instrumentation, Measurement, and Control: Sampling and Analog-to-Digital Conversion · NCEES FE Reference Handbook — Instrumentation, Measurement, and Control: Signal Conditioning · Oppenheim & Willsky, Signals and Systems

NCEES Model Rules and Codes of Ethics

How the NCEES Model Rules order an engineer's duties, why public safety is paramount, and the structured way to resolve the conflict-of-interest and competence dilemmas the FE tests.

Priority of obligations

$$\text{Public} \;>\; \text{Employer/Client} \;>\; \text{Other Licensees}$$

The Model Rules' three sections. Public health, safety, and welfare is the binding paramount duty (240.15.A.1); the ordering of the lower two groups is a teaching heuristic, not an explicit NCEES ranking. The single most useful idea for FE ethics questions: nothing outweighs the public.

Paramount duty

$$\textbf{Safeguard public health, safety, and welfare} = \text{first and foremost responsibility}$$

Model Rules 240.15.A.1. Overrides client schedule, employer profit, and collegial loyalty whenever the public is endangered.

Seal-only-if rule

$$\text{Seal} \iff (\text{within competence}) \;\wedge\; (\text{prepared under responsible charge}) \;\wedge\; (\text{meets standards})$$

An engineer may apply a seal only when all three conditions hold (240.15.A.2 and B.2). Violating any one is plan-stamping.

Competence test

$$\text{Accept assignment} \iff \text{qualified by education or experience in that specific field}$$

Accept assignments only when qualified by education or experience in that specific field (240.15.B.1). Coordinating a multi-discipline project is allowed if each technical segment is signed and sealed by its own responsible licensee (240.15.B.3).

Conflict-of-interest response

$$\text{Potential influence on judgment} \Rightarrow \text{disclose to employer/client (test = appearance, not just reality)}$$

240.15.B.6. The standard is anything that could influence OR appear to influence judgment.

Dual compensation rule

$$\text{Pay from} \;{>}1\; \text{party, same project} \iff \text{full disclosure} \;\wedge\; \text{written consent of all parties}$$

240.15.B.7. Otherwise prohibited. Gratuities from contractors are barred outright (B.5).

Overruled-on-safety duty

$$\text{Judgment overruled} \;\wedge\; \text{public endangered} \Rightarrow \text{notify employer/client + appropriate authority}$$

240.15.A.3. Silence violates the code; you must escalate, not merely comply.

Public-statement competence

$$\text{Public professional opinion} \iff \text{adequate facts} \;\wedge\; \text{competent evaluation}$$

240.15.A.5. Paid or interested-party statements must identify the party and disclose your interest (A.6).

Responsible charge definition

$$\text{Responsible charge} = \text{authority to review/change/reject/approve} + \text{aware of scope} + \text{can answer} + \text{full responsibility}$$

Model Law 110.20.E. The four-part test for whether you may legitimately seal the work.

Confidentiality rule

$$\text{Reveal client info} \iff \text{prior consent (client/employer/public body)} \;\vee\; \text{authorized or required by law or rules}$$

240.15.B.4 permits disclosure only with prior consent or as authorized/required by law or rules. The public-safety override — reporting a genuine hazard even over confidentiality — flows from the paramount duty (240.15.A.1) and the overruled-on-safety duty (A.3), not from B.4 itself.

Where it lives: NCEES FE Reference Handbook — Ethics and Professional Practice (Model Rules §240.15 Rules of Professional Conduct) · NCEES Model Rules §240.15 — Obligations to the Public, Employers/Clients, and Other Licensees · NCEES Model Law §110.20.E — Definition of Responsible Charge

Industrial Hygiene and Exposure Limits

PEL, TLV, and IDLH; dose-response and LD50/LC50 toxicology; the additive rule for chemical mixtures; and radiation half-life — the quantitative core of FE-OTH safety.

Risk definition

$$\text{Risk} = \text{Hazard}\times\text{Exposure}$$

Handbook Safety chapter. Hazard is the inherent capacity to harm; exposure (or probability) is the chance of contact. Distinguishing the two is the most-tested conceptual point.

8-hour time-weighted average

$$\text{TWA} = \dfrac{C_1 t_1 + C_2 t_2 + \cdots}{8\ \text{h}}$$

$C_i$ = concentration during interval $i$ (ppm or mg/m³), $t_i$ = duration (h). Compare to the 8-h PEL.

Additive mixture rule

$$E_{\text{mix}} = \sum_i \dfrac{C_i}{L_i}$$

$C_i$ = measured concentration of component $i$, $L_i$ = its PEL or TLV. $E_{\text{mix}}>1$ means the combined exposure exceeds the limit even if each component is individually compliant (additive-toxicity assumption).

Equivalent mixture limit

$$L_{\text{mix}} = \dfrac{\sum_i C_i}{\sum_i (C_i/L_i)}$$

The single limit a mixture of total concentration $\sum C_i$ would have under the additive rule; useful for reporting a combined PEL.

Lethal dose from LD50

$$D_{\text{lethal}} = LD_{50}\,(\text{mg/kg})\times m\,(\text{kg})$$

Converts a per-kilogram median lethal dose into an absolute dose (mg) for a body of mass $m$. Smaller $LD_{50}$ = more toxic.

Carcinogen incremental risk

$$\text{Risk} = CDI\times CSF$$

$CDI$ = chronic daily intake [mg/(kg·day)], $CSF$ = cancer slope factor [(mg/(kg·day))⁻¹]. EPA acceptable range $10^{-4}$ to $10^{-6}$. No-threshold, linear-at-low-dose model.

Noncarcinogen hazard index

$$HI = \dfrac{CDI}{RfD}$$

$RfD$ = reference dose, a chronic daily dose with no adverse effect. $HI>1.0$ indicates possible harm. Threshold model (NOAEL-based).

Reference dose from NOAEL

$$RfD = \dfrac{NOAEL}{UF}$$

$NOAEL$ = no-observable-adverse-effect level [mg/(kg·day)], $UF$ = total uncertainty factor for animal-to-human extrapolation.

Radioactive decay (halving form)

$$A(t) = A_0\left(\tfrac{1}{2}\right)^{t/T_{1/2}}$$

$A_0$ = initial activity or dose rate, $T_{1/2}$ = half-life. $t/T_{1/2}=n$ leaves a fraction $2^{-n}$.

Decay constant

$$\lambda = \dfrac{\ln 2}{T_{1/2}} = \dfrac{0.693}{T_{1/2}}$$

Continuous-decay rate; $A(t)=A_0 e^{-\lambda t}$. Units are inverse time, matching $t$.

Inverse-square dose falloff

$$\dfrac{\dot D_2}{\dot D_1} = \left(\dfrac{r_1}{r_2}\right)^2$$

Point-source dose rate $\dot D$ falls with the square of distance $r$; doubling distance quarters the dose rate.

Where it lives: NCEES FE Reference Handbook — Safety · OSHA 29 CFR 1910.1000 — Air Contaminants (PELs and the mixture formula) · ACGIH — Threshold Limit Values (TLVs) and Biological Exposure Indices

Time Value of Money

The six discrete compound-interest factors, gradients, and nominal-vs-effective conversions that turn any cash-flow diagram into a single present, annual, or future value.

Single payment, future from present (F/P)

$$F = P(1+i)^n$$

Compounds a single present amount $P$ forward $n$ periods at rate $i$ to its future value $F$ (dollars).

Single payment, present from future (P/F)

$$P = F(1+i)^{-n}$$

Discounts a single future amount $F$ back $n$ periods to present worth $P$ (dollars).

Uniform series present worth (P/A)

$$P = A\,\dfrac{(1+i)^n-1}{i(1+i)^n}$$

Present worth $P$ of $n$ equal end-of-period payments $A$; $P$ falls one period before the first $A$.

Capital recovery (A/P)

$$A = P\,\dfrac{i(1+i)^n}{(1+i)^n-1}$$

Equal end-of-period payment $A$ that recovers a present amount $P$ over $n$ periods at $i$; reciprocal of (P/A).

Uniform series compound amount (F/A)

$$F = A\,\dfrac{(1+i)^n-1}{i}$$

Future value $F$ of $n$ equal payments $A$; $F$ occurs at the same time as the last payment.

Sinking fund (A/F)

$$A = F\,\dfrac{i}{(1+i)^n-1}$$

Equal deposit $A$ needed each period to accumulate a future amount $F$; reciprocal of (F/A).

Arithmetic gradient present worth (P/G)

$$P = G\left[\dfrac{(1+i)^n-1}{i^2(1+i)^n}-\dfrac{n}{i(1+i)^n}\right]$$

Present worth of a flow $0,G,2G,\dots,(n-1)G$; gradient $G$ in dollars per period, starting in year 2.

Arithmetic gradient uniform series (A/G)

$$A = G\left[\dfrac{1}{i}-\dfrac{n}{(1+i)^n-1}\right]$$

Converts an arithmetic gradient $G$ into an equivalent uniform annual amount $A$.

Geometric gradient present worth

$$P = A_1\dfrac{1-\left(\frac{1+g}{1+i}\right)^n}{i-g}\ (i\neq g)$$

Present worth of a flow growing at rate $g$ from year-1 base $A_1$; for $i=g$, $P=A_1 n/(1+i)$.

Effective annual rate

$$i_e=\left(1+\dfrac{r}{m}\right)^m-1$$

Effective annual rate from nominal annual $r$ with $m$ compoundings per year; continuous limit (not printed in the handbook) $i_e=e^r-1$.

Simple interest

$$F = P(1+in)$$

Interest on principal only; used only when a problem explicitly states simple interest.

Inflation-adjusted (combined) rate

$$d = i + f + if$$

Combines real rate $i$ and inflation $f$ into the market rate $d$ used to discount actual (then-current) dollars.

Where it lives: NCEES FE Reference Handbook — Engineering Economics · Newnan, Eschenbach & Lavelle, Engineering Economic Analysis

Equilibrium, Free-Body Diagrams, and Reactions

Draw correct free-body diagrams, identify support reactions, and apply the equilibrium equations to solve for unknown forces and reactions.

2-D force equilibrium

$$\sum F_x = 0, \qquad \sum F_y = 0$$

Net force is zero along two perpendicular axes; signs follow your chosen positive directions.

2-D moment equilibrium

$$\sum M_O = 0$$

Net moment about any point $O$ is zero. Pick $O$ to eliminate unknowns from the equation.

Three independent 2-D equations

$$\sum F_x = 0,\ \sum F_y = 0,\ \sum M_O = 0$$

Solves up to three unknowns for a single rigid body in a plane.

3-D equilibrium

$$\sum \mathbf{F} = 0, \qquad \sum \mathbf{M} = 0$$

Six scalar equations (three force, three moment); solves up to six unknown components.

Two-force member condition

$$\mathbf{F}_A = -\mathbf{F}_B \ \text{(equal, opposite, collinear)}$$

Loaded at only two points; force direction is along the line connecting them.

Three-force member condition

$$\text{three forces concurrent or all parallel}$$

Equilibrium of a body under exactly three forces requires their lines of action to meet at a point (or be parallel).

Roller / smooth-surface reaction

$$R \perp \text{surface},\ \ 1\ \text{unknown}$$

Single reaction normal to the contact surface or the roller's travel direction.

Pin (hinge) reaction

$$A_x,\ A_y\ \ (2\ \text{unknowns, no moment})$$

Two force components; allows free rotation, so it transmits no reaction moment.

Determinacy criterion (single body)

$$r = 3 \ (\text{2-D}), \quad r = 6 \ (\text{3-D})$$

$r$ = number of unknown reaction components for a statically determinate rigid body.

Where it lives: NCEES FE Reference Handbook — Statics · Hibbeler, Engineering Mechanics: Statics · Beer & Johnston, Vector Mechanics for Engineers: Statics

Kinematics of Particles and Rigid Bodies

Position, velocity, and acceleration in rectilinear, projectile, normal-tangential, and polar form, plus relative motion and rigid-body rotation with instant centers.

Kinematic derivatives

$$v=\dfrac{ds}{dt},\qquad a=\dfrac{dv}{dt}=v\,\dfrac{dv}{ds}$$

Definitions of velocity and acceleration. The $v\,dv/ds$ form integrates motion that depends on position with no time given. $s$ in m (or ft), $v$ in m/s, $a$ in m/s².

Constant-acceleration set

$$v=v_0+a t,\quad s=s_0+v_0 t+\tfrac{1}{2}a t^2,\quad v^2=v_0^2+2a(s-s_0)$$

Use only when $a$ is constant (free fall, braking). The last form eliminates time.

Projectile equations

$$x=v_0\cos\theta\,t,\qquad y=v_0\sin\theta\,t-\tfrac{1}{2}g t^2$$

Constant acceleration on independent axes: $a_x=0$, $a_y=-g$. $\theta$ measured from horizontal.

Max height and range (level ground)

$$H=\dfrac{(v_0\sin\theta)^2}{2g},\qquad R=\dfrac{v_0^2\sin 2\theta}{g}$$

Apex height and horizontal range for launch and landing at the same elevation. Range is maximized at $\theta=45^\circ$.

Normal-tangential acceleration

$$a_t=\dfrac{dv}{dt},\qquad a_n=\dfrac{v^2}{\rho},\qquad a=\sqrt{a_t^2+a_n^2}$$

$a_t$ changes speed, $a_n$ changes direction (points to center of curvature). $\rho$ = radius of curvature in m.

Polar velocity and acceleration

$$v_r=\dot r,\quad v_\theta=r\dot\theta,\qquad a_r=\ddot r-r\dot\theta^2,\quad a_\theta=r\ddot\theta+2\dot r\dot\theta$$

Radial and transverse velocity and acceleration. $-r\dot\theta^2$ is centripetal; $2\dot r\dot\theta$ is Coriolis. $\dot\theta=\omega$ in rad/s.

Relative motion

$$\mathbf{v}_A=\mathbf{v}_B+\mathbf{v}_{A/B},\qquad \mathbf{a}_A=\mathbf{a}_B+\mathbf{a}_{A/B}$$

Vector equations; resolve into components before adding. $\mathbf{v}_{A/B}$ = velocity of $A$ relative to $B$.

Rotation kinematics (constant α)

$$\omega=\omega_0+\alpha t,\quad \theta=\theta_0+\omega_0 t+\tfrac{1}{2}\alpha t^2,\quad \omega^2=\omega_0^2+2\alpha(\theta-\theta_0)$$

Angular analogs of constant-acceleration motion. $\omega$ in rad/s, $\alpha$ in rad/s², $\theta$ in rad.

Point velocity on a rotating body

$$v=r\omega,\qquad a_t=r\alpha,\qquad a_n=r\omega^2$$

Speed and acceleration components of a point a distance $r$ from the rotation axis. $r$ in m.

Instantaneous center relation

$$v_P=\omega\,d_{P/IC}$$

Any point's speed equals $\omega$ times its distance to the instant center. For a wheel rolling without slip, IC is the contact point.

Where it lives: NCEES FE Reference Handbook — Dynamics (Particle Kinematics, Kinematics of a Rigid Body) · Hibbeler, Engineering Mechanics: Dynamics

Axial, Bending, Torsion, and Shear Stress

The four elementary stress formulas — P/A, Mc/I, Tr/J, VQ/Ib — plus Hooke's law and Poisson's ratio that every Strength-of-Materials question is built on.

Axial (normal) stress

$$\sigma=\frac{P}{A}$$

Uniform normal stress from a centroidal axial load. $P$ in N (or lbf), $A$ in $\text{mm}^2$ (or $\text{in}^2$); tension positive.

Engineering strain

$$\varepsilon=\frac{\delta}{L}$$

Fractional change in length (dimensionless). $\delta$ = elongation, $L$ = original gauge length.

Hooke's law (1D)

$$\sigma=E\varepsilon,\qquad \tau=G\gamma$$

Linear-elastic relation. $E$ = modulus of elasticity, $G$ = shear modulus (both in $\text{GPa}$ or $\text{Mpsi}$); $\gamma$ in radians.

Poisson's ratio and modulus link

$$\nu=-\frac{\varepsilon_{lat}}{\varepsilon_{long}},\qquad G=\frac{E}{2(1+\nu)}$$

$\nu\approx0.30$ for steel; ties the three elastic constants together so only two are independent.

Bending stress (flexure formula)

$$\sigma=\frac{Mc}{I}=\frac{M}{S}$$

Outer-fiber bending stress. $M$ = moment, $c$ = distance to extreme fiber, $I$ = centroidal moment of inertia, $S=I/c$ = section modulus.

Fiber bending stress with sign

$$\sigma_x=-\frac{My}{I}$$

Linear variation across depth; $y$ measured from the neutral axis. The minus sign puts top fibers in compression for positive (sagging) $M$.

Rectangular moment of inertia

$$I=\frac{bh^3}{12}$$

Centroidal $I$ of a rectangle, base $b$, depth $h$ (bending about the horizontal centroidal axis).

Torsional shear stress

$$\tau=\frac{Tr}{J}$$

Circular shafts only. $T$ = torque, $r$ = radius to point ($r$ at surface for max), $J$ = polar moment of inertia.

Polar moment of inertia (circular)

$$J=\frac{\pi d^4}{32}\;(\text{solid}),\qquad J=\frac{\pi\left(d_o^4-d_i^4\right)}{32}\;(\text{hollow})$$

Note the 32 (torsion) versus 64 for the bending $I=\pi d^4/64$ of the same circle.

Transverse shear stress

$$\tau=\frac{VQ}{Ib}$$

$V$ = shear force, $Q=A'\bar{y}'$ = first moment of area beyond the level, $I$ = full-section inertia, $b$ = width at that level.

Max transverse shear (common sections)

$$\tau_{max}=\frac{3V}{2A}\;(\text{rect}),\qquad \tau_{max}=\frac{4V}{3A}\;(\text{circle})$$

Peak shear at the neutral axis; handy shortcuts that avoid computing $Q$ for solid sections.

Shear flow

$$q=\frac{VQ}{I}$$

Force per unit length along a joint (e.g., bolted/glued built-up beams); $\tau=q/b$.

Where it lives: NCEES FE Reference Handbook — Mechanics of Materials · NCEES FE Reference Handbook — Materials Science/Structure of Matter · Hibbeler, Mechanics of Materials

Phase Diagrams and the Lever Rule

Read a binary phase diagram to find which phases exist, their compositions, and their weight fractions via the lever rule — the core materials skill on FE-Other.

Lever rule — fraction of α

$$W_\alpha = \dfrac{x_\beta - x}{x_\beta - x_\alpha}$$

Weight fraction of the α phase. $x$ = overall composition, $x_\alpha,\,x_\beta$ = phase compositions at the tie-line ends (all in same units, e.g. wt% B). Uses the FAR (β-side) arm.

Lever rule — fraction of β

$$W_\beta = \dfrac{x - x_\alpha}{x_\beta - x_\alpha}$$

Weight fraction of the β phase, using the α-side arm. Always $W_\alpha + W_\beta = 1$.

Phase-fraction closure

$$W_\alpha + W_\beta = 1$$

Mass-balance check for any two-phase tie line; if your two fractions do not sum to 1 you swapped an arm or misread a boundary.

Liquid/solid fractions

$$W_L = \dfrac{x_s - x}{x_s - x_L}, \quad W_S = \dfrac{x - x_L}{x_s - x_L}$$

Lever rule in the $L+\text{solid}$ region; $x_L$ = liquidus composition, $x_s$ = solidus composition at temperature $T$.

Eutectic reaction

$$L \rightarrow \alpha + \beta$$

Liquid transforms to two solid phases at the fixed eutectic temperature and composition.

Eutectoid reaction

$$\gamma \rightarrow \alpha + \text{Fe}_3\text{C}$$

One solid phase transforms to two solid phases; in steel: austenite → pearlite (ferrite + cementite) at 0.76% C, 727°C.

Peritectic reaction

$$L + \alpha \rightarrow \beta$$

Liquid plus one solid transform to a second solid on cooling.

Pearlite fraction in hypoeutectoid steel

$$W_{pearlite} = \dfrac{C_0 - 0.022}{0.76 - 0.022}$$

At just below 727°C, fraction of pearlite for a steel of carbon content $C_0$ (wt%); ferrite solubility limit ≈ 0.022% C, eutectoid = 0.76% C.

Gibbs phase rule (condensed)

$$F = C - P + 1$$

Degrees of freedom $F$ for a binary ($C=2$) system at fixed pressure; $P$ = number of phases. $F=0$ at an invariant (e.g. eutectic) point.

True/engineering composition basis

$$x = \dfrac{\text{mass of B}}{\text{total mass}}\times 100\%$$

Weight-percent basis used throughout; do not mix wt% with at% on the same tie line.

Where it lives: NCEES FE Reference Handbook — Materials Science/Structure of Matter (Binary Phase Diagrams, Lever Rule, Iron-Iron Carbide) · Callister & Rethwisch, Materials Science and Engineering: An Introduction · Van Vlack, Elements of Materials Science and Engineering

Continuity, Bernoulli, and Momentum

The three control-volume workhorses — conservation of mass, the Bernoulli energy equation in head form, and the impulse-momentum principle for forces on bends, nozzles, and jets.

Continuity (mass)

$$\dot m = \rho A v = \text{const}$$

Steady-flow mass conservation; $\dot m$ in kg/s. Same at every cross section.

Continuity (volume)

$$Q = A_1 v_1 = A_2 v_2$$

Incompressible flow; velocity scales as $1/A$, i.e. as $(D_1/D_2)^2$ for round pipes.

Bernoulli equation (head form)

$$\frac{P}{\gamma} + \frac{v^2}{2g} + z = \text{const}$$

Steady, incompressible, frictionless, no machine. Each term is a head (length).

Energy equation with machines and losses

$$\frac{P_1}{\gamma}+\frac{v_1^2}{2g}+z_1+h_p = \frac{P_2}{\gamma}+\frac{v_2^2}{2g}+z_2+h_t+h_f$$

$h_p$ pump head added, $h_t$ turbine head removed, $h_f$ friction loss; all in head units.

EGL minus HGL

$$EGL - HGL = \frac{v^2}{2g}$$

Hydraulic grade line lies below the energy grade line by the velocity head.

Torricelli efflux velocity

$$v = \sqrt{2gh}$$

Speed of fluid leaving a small orifice a depth $h$ below a large free surface.

Stagnation (dynamic) pressure

$$P_0 = P_s + \tfrac{1}{2}\rho v^2$$

Pitot-tube relation; $\tfrac12\rho v^2$ is the dynamic pressure, valid for $Ma<0.3$.

Impulse-momentum principle

$$\sum \vec F = \rho Q\,(\vec v_2 - \vec v_1)$$

Vector equation; resolve by component. Include $PA$ pressure forces on inlet/outlet faces.

Force on a pipe bend (x-component)

$$P_1 A_1 - P_2 A_2\cos\alpha - F_x = \rho Q\,(v_2\cos\alpha - v_1)$$

$F_x$ = force of bend on fluid; the fluid's force on the bend is equal and opposite.

Jet force on a fixed flat plate (normal)

$$F = \rho Q v = \rho A v^2$$

Force on a plate held perpendicular to a free jet of velocity $v$, area $A$.

Where it lives: NCEES FE Reference Handbook — Fluid Mechanics (One-Dimensional Fluid Flow) · NCEES FE Reference Handbook — Fluid Mechanics (Impulse-Momentum Principle)

AC and DC Circuit Analysis

Phasors, reactance, and impedance turn AC steady-state into the same algebra as DC, then the power triangle and series resonance close out the AC and DC circuits topic.

RMS value of a sinusoid

$$V_{\text{rms}}=\frac{V_{\max}}{\sqrt{2}}, \qquad \omega=2\pi f$$

Effective value used in all power and meter calculations; $f$ in Hz, $\omega$ in rad/s.

Inductive reactance

$$X_L=\omega L$$

Ohms; rises with frequency. Inductor current lags voltage by $90^\circ$. $L$ in henrys.

Capacitive reactance

$$X_C=\frac{1}{\omega C}$$

Ohms; falls with frequency. Capacitor current leads voltage by $90^\circ$. $C$ in farads.

Impedance

$$\mathbf{Z}=R+jX=|\mathbf{Z}|\angle\theta, \quad |\mathbf{Z}|=\sqrt{R^2+X^2}, \quad \theta=\tan^{-1}\frac{X}{R}$$

Complex ratio of phasor voltage to phasor current; $X=X_L-X_C$ for a series branch.

AC Ohm's law

$$\mathbf{I}=\frac{\mathbf{V}}{\mathbf{Z}}$$

Phasor current from phasor voltage and impedance; magnitudes divide, angles subtract.

Complex power

$$\mathbf{S}=\mathbf{V}\mathbf{I}^{*}=P+jQ$$

Uses conjugate of current phasor; $P$ in W, $Q$ in var, $S=|\mathbf{S}|$ in VA.

Real, reactive, apparent power

$$P=V_{\text{rms}}I_{\text{rms}}\cos\theta, \quad Q=V_{\text{rms}}I_{\text{rms}}\sin\theta, \quad S=V_{\text{rms}}I_{\text{rms}}$$

$\theta$ = angle by which voltage leads current = impedance angle.

Power in terms of R and X

$$P=I_{\text{rms}}^2 R, \qquad Q=I_{\text{rms}}^2 X$$

Real power dissipates only in resistance; reactive power only in reactance.

Power factor

$$pf=\cos\theta=\frac{P}{S}$$

Lagging (inductive, current lags) or leading (capacitive, current leads).

Power factor correction

$$Q_C=P(\tan\theta_1-\tan\theta_2), \qquad C=\frac{Q_C}{\omega V^2}$$

Parallel capacitor sizing to raise pf from $\cos\theta_1$ to $\cos\theta_2$ at fixed $P$, $V$. The FE Reference Handbook 10.6 prints the power triangle ($P$, $Q$, pf) but not this correction formula.

Resonant frequency

$$\omega_0=\frac{1}{\sqrt{LC}}, \qquad f_0=\frac{1}{2\pi\sqrt{LC}}$$

Where $X_L=X_C$; series impedance equals $R$ and current peaks.

Quality factor and bandwidth (series)

$$Q=\frac{\omega_0 L}{R}=\frac{1}{R}\sqrt{\frac{L}{C}}, \qquad \text{BW}=\frac{\omega_0}{Q}$$

Sharpness of resonance; bandwidth in rad/s between half-power points.

Where it lives: NCEES FE Reference Handbook — Electrical and Computer Engineering (AC Circuits, Impedance) · NCEES FE Reference Handbook — Electrical and Computer Engineering (AC Power, Complex Power) · NCEES FE Reference Handbook — Electrical and Computer Engineering (Resonance)

The First Law and Energy Balances

Conservation of energy for closed and open systems: internal energy, enthalpy, the steady-flow energy equation, and the heat/work sign conventions every thermo problem rides on.

Closed-system first law

$$Q - W = \Delta U + \Delta KE + \Delta PE$$

No mass crosses the boundary. $Q$ positive into system, $W$ positive out (done by system); usually $\Delta KE=\Delta PE=0$. Energy in $\text{kJ}$.

Definition of enthalpy

$$H = U + PV, \qquad h = u + Pv$$

Enthalpy combines internal energy with flow work $Pv$; $h$ in $\text{kJ/kg}$, $P$ in $\text{kPa}$, $v$ in $\text{m}^3/\text{kg}$.

Ideal-gas energy changes

$$\Delta u = c_v\,\Delta T, \qquad \Delta h = c_p\,\Delta T$$

Constant specific heats (cold-air-standard). $c_v,c_p$ in $\text{kJ/(kg·K)}$; valid for ideal gases regardless of process path.

Boundary (moving-boundary) work

$$W_b = \int_{1}^{2} P\,dV$$

Reversible work of a moving boundary in a closed system; equals area under the process curve on a $P$-$V$ diagram.

Constant-pressure boundary work

$$w_b = P\,\Delta v = R\,\Delta T$$

Charles' law process for an ideal gas; the second form uses $Pv = RT$. $R$ is the specific gas constant in $\text{kJ/(kg·K)}$.

Steady-flow mass balance

$$\sum \dot m_i = \sum \dot m_e, \qquad \dot m = \rho A V = \frac{AV}{v}$$

Conservation of mass for a control volume at steady state; $\dot m$ in $\text{kg/s}$, $A$ area, $V$ velocity, $v$ specific volume.

General SFEE (per unit mass)

$$q - w = (h_e - h_i) + \frac{V_e^2 - V_i^2}{2} + g(Z_e - Z_i)$$

Single-inlet/single-exit steady flow. Divide velocity and gravity terms by $1000$ to convert $\text{J/kg}$ to $\text{kJ/kg}$ before adding to enthalpy.

Adiabatic turbine / compressor

$$w = h_i - h_e$$

Negligible heat, $KE$, $PE$. $w>0$ for a turbine (work out); $w<0$ for a compressor (work in).

Throttling valve

$$h_i = h_e$$

No work, no heat, negligible $KE$; enthalpy is conserved (isenthalpic). Used for expansion valves and the Joule-Thomson effect.

Adiabatic nozzle

$$h_i + \frac{V_i^2}{2} = h_e + \frac{V_e^2}{2}$$

No work or heat; enthalpy converts to kinetic energy. Solve for exit velocity $V_e=\sqrt{2(h_i-h_e)}$ with $h$ in $\text{J/kg}$.

Heat exchanger / boiler (one stream)

$$\dot Q = \dot m\,(h_e - h_i)$$

No work; heat added equals enthalpy rise of the stream. Positive $\dot Q$ heats the fluid.