Free FE Other Disciplines Exam Flashcards

For ay" + by' + cy = 0 you solve the algebraic equation ar^2 + br + c = 0. The sign of the discriminant b^2 - 4ac sets the solution form: positive gives two real roots (overdamped, pure exponentials), zero gives a repeated root (critically damped, with a te^(rt) term), and negative gives complex roots (underdamped, a decaying sinusoid).

x(k+1) = x(k) - f(x(k))/f'(x(k)), which replaces the curve at each step with its tangent line at the current guess. It converges very quickly near a simple root but stalls or diverges wherever the derivative is zero or near zero, so FE problems always supply a starting value close to the answer.

The matrix is singular: it has no inverse and its rows and columns are linearly dependent, so Ax = b has either no solution or infinitely many rather than one unique solution. It also means zero is an eigenvalue, since eigenvalues are the values of lambda that make det(A - lambda*I) = 0.

Where f'(x) = 0, a positive f"(x) means a local minimum (concave up) and a negative f"(x) means a local maximum (concave down). If f"(x) = 0 the test is inconclusive and the point may be an inflection point, so you must instead check whether f' actually changes sign there.

Use t when the population standard deviation is unknown and must be estimated from the sample, which is the usual case for small samples; use z when the population standard deviation is known or the sample is large. The t interval is wider at the same confidence level, and that extra width is the price of estimating the spread from the data.

A Type I error rejects a null hypothesis that is actually true - a false alarm whose probability is the significance level alpha. A Type II error fails to reject a false null hypothesis, with probability beta. Tightening alpha to avoid false alarms raises beta unless you also increase the sample size.

R-squared is the fraction of variation in the dependent variable explained by the fitted model, running from 0 to 1. A high value means the curve tracks the data points; it does not prove causation and does not confirm that the chosen model form is physically correct.

pH = -log10 of the hydrogen-ion molar concentration, and pH + pOH = 14 at 25 degrees C, so a neutral solution sits at pH 7. Because the scale is logarithmic, a pH 4 solution has 100 times the hydrogen-ion concentration of a pH 6 solution.

A system at equilibrium shifts in the direction that partially offsets any imposed change in concentration, pressure, or temperature. Adding a reactant drives the reaction forward, and raising the pressure on a gas-phase equilibrium shifts it toward the side with fewer moles of gas.

Oxidation (loss of electrons) always happens at the anode and reduction (gain of electrons) at the cathode. In galvanic corrosion the more active metal becomes the anode and corrodes preferentially, which is exactly why a sacrificial zinc or magnesium anode is attached to steel to protect it.

A signal must be sampled at more than twice its highest frequency component to be reconstructed without distortion. Sampling too slowly causes aliasing, where high-frequency content masquerades as a lower frequency - an error no downstream filtering can undo, so the anti-aliasing filter must sit ahead of the A/D converter.

Accuracy is how close readings sit to the true value (systematic error or bias); precision is how tightly repeated readings cluster (random error or scatter). An instrument can be precise but inaccurate, and that repeatable offset can be calibrated out, while random scatter can only be reduced by averaging more readings.

Licensees must be cognizant that safeguarding the health, safety, and welfare of the public comes first when performing services for clients and employers. The same section requires them to notify the employer or client, and any other appropriate authority, when their professional judgment is overruled and public safety is endangered.

Licensees may undertake assignments only when qualified by education or experience in the specific technical field involved, and may not seal any document dealing with subject matter in which they lack competence or that was not prepared under their responsible charge. Holding a license is not permission to practice outside your own specialty.

LCA tallies the energy, material, and emission burdens of a product or system from raw-material extraction through manufacture, use, and final disposal - a cradle-to-grave accounting. Its purpose is to catch decisions that look clean at one stage while shifting a larger burden onto another.

Under 29 CFR 1910.146, an atmosphere below 19.5 percent oxygen by volume is oxygen-deficient and one above 23.5 percent is oxygen-enriched; either condition makes a confined space hazardous. Enriched air is dangerous not because it is toxic but because it sharply increases the flammability of ordinary materials.

The LFL is the lowest fuel-in-air concentration that will support combustion and the UFL is the highest; between them the mixture is flammable. OSHA 29 CFR 1910.146 calls an atmosphere hazardous once a flammable gas, vapor, or mist exceeds 10 percent of its LFL, building in a tenfold margin before ignition is possible.

The half-life is the time for half of a radioactive material to decay, so the remaining amount is N = N0 * (1/2)^(t/half-life) and the decay constant is ln(2)/half-life. Decay is exponential, not linear: after three half-lives 12.5 percent still remains, which is why shielding and waiting periods are sized from this relation.

(P/F, i%, n) = (1 + i)^(-n), converting one future amount into today's dollars. Because the factor shrinks as either the interest rate or the number of periods grows, distant cash flows contribute very little to present worth - the reason long-horizon projects are so sensitive to the assumed interest rate.

IRR is the interest rate at which a project's net present worth equals zero. A project is acceptable when its IRR exceeds the minimum attractive rate of return. When ranking mutually exclusive alternatives you must run an incremental IRR analysis, because the alternative with the highest individual IRR is not necessarily the best economic choice.

Annual depreciation = (initial cost - salvage value) / recovery period, giving the same deduction every year. Accelerated methods such as MACRS shift larger deductions into the early years, which raises the present worth of the tax savings even though the total amount depreciated is identical.

Three: sum of forces in x equals zero, sum of forces in y equals zero, and sum of moments about any point equals zero. If the unknown reactions outnumber those three equations, the structure is statically indeterminate and statics alone cannot solve it. A three-dimensional body supplies six equations instead.

At an unloaded joint connecting only two non-collinear members, both carry zero force; at an unloaded joint of three members where two are collinear, the third carries zero force. They are not useless - they brace the loaded members against buckling and pick up load under other loading cases, so they stay in the structure.

I = Ic + A*d^2, where Ic is the moment of inertia about the centroidal axis, A is the area, and d is the distance between the two parallel axes. Because the transfer term is always positive, the centroidal axis always yields the smallest possible moment of inertia for that shape.

Friction supplies whatever force is needed to prevent sliding, up to a maximum of the static coefficient times the normal force; it equals that maximum only when motion is impending. Once sliding begins the force drops to the kinetic coefficient times the normal force, and because kinetic is lower than static, a block lurches forward the instant it breaks loose.

The net work done on a particle equals its change in kinetic energy: W = (1/2)m(v2^2 - v1^2). Because time never appears, it is the fastest route whenever a problem gives distances and speeds but no time, unlike the impulse-momentum approach.

e is the ratio of relative separation velocity to relative approach velocity along the line of impact, spanning e = 0 for a perfectly plastic impact (the bodies move off together) to e = 1 for a perfectly elastic one. Linear momentum is conserved in every impact, but kinetic energy is conserved only when e = 1.

Mass moment of inertia (kg*m^2 or slug*ft^2) resists angular acceleration and appears in M = I*alpha. Area moment of inertia (m^4 or in^4) describes how cross-sectional area is distributed and governs bending stress and deflection. They share a name but have different units and different uses, so they can never be substituted for one another.

Natural frequency in radians per second is the square root of stiffness divided by mass; divide by 2*pi to get hertz. It depends only on stiffness and mass, never on amplitude, so stiffening the structure or reducing its mass are the only ways to move a resonance away from a forcing frequency.

Elongation = PL/(AE), which follows directly from stress = P/A and Hooke's law stress = E * strain. Doubling the length doubles the stretch while doubling the area halves it, and it is the elastic modulus E - not the yield strength - that controls stiffness.

stress = Mc/I, where M is the bending moment, c is the distance from the neutral axis to the fiber of interest, and I is the area moment of inertia about that axis. Maximum stress occurs at the extreme fibers where c is largest, which is why an I-beam concentrates material in flanges far from the neutral axis.

Critical load = pi^2 * E * I / (K*L)^2, where K is the effective-length factor - theoretically 1.0 pinned-pinned, 0.5 fixed-fixed, and 2.0 fixed-free. Buckling depends on stiffness and end restraint, not on yield strength, so specifying a stronger steel of the same modulus does not raise the buckling load.

It equals half the difference between the two principal stresses, which is the radius of Mohr's circle, and it acts on planes oriented 45 degrees from the principal planes. The principal planes themselves carry zero shear stress - that is the defining property that makes the transformation useful.

Inside a two-phase region, the fraction of one phase equals the length of the tie-line segment on the opposite side of the overall composition divided by the full tie-line length. The tie line tells you the composition of each phase; the lever rule tells you how much of each phase is present.

For materials with no sharp yield point, such as aluminum and many alloys, yield strength is read where a line parallel to the elastic slope but offset by 0.2 percent strain crosses the stress-strain curve. It is a reporting convention rather than a physical transition, so quoted values only compare fairly when the same offset was used.

Quenching cools austenite fast enough to trap it as martensite, which is very hard but brittle. Tempering then reheats the steel below its transformation temperature to trade some hardness back for toughness. Full annealing, by contrast, uses slow furnace cooling to maximize ductility and relieve residual stress.

Gauge pressure = density * g * depth, so it depends only on the fluid's specific weight and the vertical depth, never on the shape or volume of the container. For water the specific weight is about 9.81 kN/m^3 (62.4 lbf/ft^3), roughly 9.81 kPa of pressure per meter of depth.

Conservation of mass gives density1*A1*V1 = density2*A2*V2; for an incompressible liquid the densities cancel and A1*V1 = A2*V2. That is why velocity rises where a pipe narrows, and you normally apply continuity first so Bernoulli can convert that velocity change into a pressure change.

Pressure head plus velocity head plus elevation head is constant only for steady, incompressible, inviscid flow along a streamline with no shaft work and no heat transfer. Real piping violates the frictionless assumption, so you must extend it to the energy equation with head-loss and pump or turbine head terms.

Reynolds number is velocity times diameter divided by kinematic viscosity, the ratio of inertial to viscous forces. Pipe flow is treated as laminar below about Re = 2,100 (some texts cite 2,300) and fully turbulent above roughly Re = 4,000, with an unpredictable transition band between - and the regime decides which friction relationship you are allowed to use.

Head loss = f * (L/D) * (V^2 / 2g). In laminar flow the Darcy friction factor is exactly 64/Re; in turbulent flow it comes from the Moody diagram or the Colebrook equation using Reynolds number and relative roughness. Because loss scales with velocity squared, doubling the flow roughly quadruples the pumping head.

pV = nRT with a universal gas constant of 8.314 J/(mol*K), or pv = RT per unit mass using a gas-specific constant. Near high pressure or the saturation line, real gases need pv = zRT, where z departs from 1.0 to account for molecular volume and intermolecular attraction.

KCL says the algebraic sum of currents entering any node is zero, which expresses conservation of charge. KVL says the algebraic sum of voltages around any closed loop is zero, which expresses conservation of energy. Reach for KCL when node voltages are the unknowns and KVL when loop currents are.

For a pure sinusoid the RMS value is the peak divided by the square root of 2, about 0.707 times peak. RMS is defined as the DC value that would dissipate the same average power in a resistor, so P = Vrms^2 / R; using peak values instead overstates the power by a factor of two.

P = sqrt(3) * line voltage * line current * power factor. A poor power factor forces more line current for the same real power, which raises I^2*R losses and drives up the required conductor, switchgear, and transformer sizing.

Change in internal energy = heat added minus work done by the system, an energy-conservation bookkeeping statement. It is silent about direction: it permits heat to flow from cold to hot just as readily as hot to cold, and only the second law rules that out.

Maximum efficiency = 1 - (cold reservoir temperature / hot reservoir temperature), using absolute temperatures in kelvin or degrees Rankine. It is the ceiling no real heat engine between those two reservoirs can beat, and because the cold reservoir can never reach absolute zero, no cycle can reach 100 percent.

Heat rate = -k*A*(temperature gradient); for a plane wall this reduces to k*A*(temperature difference)/thickness, and the thermal resistance is thickness/(k*A). Writing conduction as a resistance lets you add composite wall layers in series exactly like electrical resistors.

Heat rate = h*A*(surface temperature - fluid temperature), where h is the convection coefficient. Unlike thermal conductivity, h is not a material property - it depends on fluid velocity, geometry, and whether the flow is free or forced, which is why forced convection moves far more heat than natural convection.

Net radiation = emissivity * sigma * A * (surface temperature^4 - surroundings temperature^4), with sigma = 5.67 x 10^-8 W/(m^2*K^4) and emissivity between 0 and 1. Because it scales with absolute temperature to the fourth power, radiation is minor near room temperature but dominates at furnace and flame temperatures.