Your FREE FE Exam (Fundamentals of Engineering) Practice Test 2026 – 250+ Q&A

Correct answer: $P = 500e^{t/5}$

The answer is $P = 500e^{t/5}$. A rate proportional to the current amount gives $P = P_0 e^{kt}$, and here $P_0 = 500$ and $k = 0.2 = 1/5$. An exponent of $5t$ inverts the rate constant and grows far too fast, and a negative exponent describes decay, which needs a negative rate constant. The form $500 + 100t$ holds the starting rate of 100 per year fixed forever, which solves $dP/dt = 100$ rather than $dP/dt = 0.2P$.

Correct answer: Second order, linear, and nonhomogeneous

The equation is second order, linear, and nonhomogeneous. The highest derivative is y'', so it is second order; y, y' and y'' each appear to the first power with constant coefficients, so it is linear. The term sin(x) depends only on the independent variable, so it does not make the equation nonlinear, but because it is nonzero and free of y the equation is nonhomogeneous rather than homogeneous.

Correct answer: $y = e^{x}(C_1 \cos 2x + C_2 \sin 2x)$

The answer is $e^{x}(C_1 \cos 2x + C_2 \sin 2x)$. Roots $a \pm bi$ give an envelope $e^{ax}$ times sine and cosine of $bx$, so $a = 1$ sets $e^{x}$ and $b = 2$ sets the frequency. Putting 2 in the exponent and 1 in the cosine swaps the real and imaginary parts, and $e^{-x}$ flips the sign of the real part, which turns growth into decay. Hyperbolic cosh and sinh belong to real roots $\pm 2$, not to a complex pair.

Correct answer: q ≈ 3.68

The charge is about 3.68. Separating variables gives $q = 10e^{-t/5}$, so at $t = 5$ the charge is $10e^{-1} = 10 \times 0.368 \approx 3.68$. The value 6.32 comes from the charging form $10(1 - e^{-1})$, which describes a capacitor filling, not one discharging. The value 0.37 drops the initial charge of 10, and 27.2 uses $e^{+1}$, a sign error that makes the charge grow.

Correct answer: x = 3

The solution has x = 3. Adding the equations cancels y and leaves 3x = 9, so x = 3 and then y = 2. Stopping at 9 forgets to divide by the coefficient 3, while 1 comes from setting y to zero in the second equation and solving it alone. The value 2 is y, not x, so it is the right number for the wrong unknown.

Correct answer: Rows [1/2, 0] and [0, 1/4]

The inverse has rows [1/2, 0] and [0, 1/4]. The determinant is 2 times 4 = 8, and swapping the diagonal entries to [4, 2] then dividing by 8 gives 1/2 and 1/4, the reciprocal of each diagonal entry. Dividing by 8 without the swap puts 1/4 and 1/2 in the wrong places, and 1/8 on both diagonals uses only the determinant. Negating the entries gives the additive inverse, which does not multiply with the original to give the identity.

Correct answer: a · b = -1

The dot product is -1, from (3)(1) + (-1)(4) + (2)(0) = 3 - 4 + 0. Getting 7 drops the negative sign on the middle product, and 9 adds the two vectors and then sums the components instead of multiplying them pairwise. The value -8 is the first component of the cross product, a different operation that returns a vector.

Correct answer: Trace of 9

The trace is 9, the sum of the main-diagonal entries 6 + 3. The value 16 is the determinant, (6)(3) - (2)(1), which equals the product of the eigenvalues rather than their sum. Adding all four entries gives 12, and multiplying the two diagonal entries gives 18; neither is the trace.

Correct answer: Accuracy improves, because shorter chords follow the curve more closely

Accuracy improves as subintervals are added. The trapezoidal error for a smooth function shrinks roughly with the square of the step width, because shorter chords follow the curve more closely. Rounding error only matters at extreme subinterval counts, so it does not make accuracy worse in ordinary use. The rule is exact only for straight lines, not for curves, and the even-count requirement belongs to Simpson's rule, not to the trapezoidal rule.

Correct answer: The estimate is almost a solution of f(x) = 0

The small residual means the estimate almost satisfies f(x) = 0, so it is very close to a root. The residual is a value of f, not of its slope, so it says nothing about the derivative being zero. A residual this small is the opposite of divergence. The next Newton step is f(x) divided by f'(x), not f(x) itself, so the shift in x is not 0.000003 unless the slope happens to equal 1.

Correct answer: It swaps f'(x) for a chord slope through two prior iterates

The secant method swaps the derivative for the slope of a line through the two most recent points, so no explicit derivative is needed. Keeping the root bracketed is the property of bisection and false position; the secant method can leave the bracket. Its convergence order is about 1.6 per step, below Newton's quadratic rate, and it needs two starting values while Newton's method needs only one.

Correct answer: The limit is 6

The limit is 6. Factoring $x^2 - 9$ as $(x - 3)(x + 3)$ cancels the $x - 3$ in the denominator, leaving $x + 3$, which equals 6 at $x = 3$. The value 9 substitutes into $x^2$ and ignores the denominator, 0 substitutes into the numerator alone, and 3 is just the point being approached. The form $0/0$ is indeterminate, not a value of the limit.

Correct answer: $3t^2 + 4$

The velocity is $v(t) = 3t^2 + 4$. Integrating $6t$ gives $3t^2 + C$, and $v(0) = 4$ sets $C = 4$. Writing $6t^2$ raises the power but skips dividing by the new exponent 2. Writing $4t$ integrates the initial velocity as if it were an acceleration term, and $6t + 4$ adds the acceleration expression to the starting velocity without integrating it.

Correct answer: $-2/(x - 1)^2$

The derivative is $-2/(x - 1)^2$. The quotient rule gives $[(x - 1)(1) - (x + 1)(1)]/(x - 1)^2 = -2/(x - 1)^2$. Reversing the order of subtraction in the numerator gives the positive 2, and adding the two products instead of subtracting them gives $2x$. Squaring the numerator $x + 1$ instead of the denominator puts the wrong expression underneath.

Correct answer: 1 + x + x²/2

The three-term approximation is $1 + x + x^2/2$, since each term of the Maclaurin series for $e^x$ is $x^n/n!$ and $2! = 2$. Leaving out the factorial gives $1 + x + x^2$. The alternating sign belongs to $e^{-x}$, and keeping only even powers gives the series for $\cosh x$, not for $e^x$.

Correct answer: 0.500

The value is 0.500, since the side adjacent to the 60 degree angle in a 30-60-90 triangle is half the hypotenuse. The value 0.866 is sin(60 degrees), the same as cos(30 degrees), and 0.707 is cos(45 degrees). The value -0.952 is what a calculator returns for cos(60) when it is set to radians instead of degrees.

Correct answer: y = 2x + 2

The line is y = 2x + 2. The slope is (10 - 2)/(4 - 0) = 2, and the point (0, 2) gives the intercept directly. A slope of 0.5 divides run by rise, 2.5 divides the second point's y by its x instead of using the changes, and 8 is the rise without dividing by the run of 4.

Correct answer: c ≈ 6.24

Side c is about 6.24. The law of cosines gives $c^2 = 25 + 49 - 2(5)(7)(0.5) = 39$, and $\sqrt{39} \approx 6.24$. Dropping the cosine term gives $\sqrt{74} \approx 8.60$, adding it instead of subtracting gives $\sqrt{109} \approx 10.4$, and 39.0 is $c^2$ with the square root never taken.

Correct answer: x = 0 and y = 4

The point is x = 0 and y = 4. The conversion uses x = r cos(theta) and y = r sin(theta), and with theta = 90 degrees the cosine is 0 and the sine is 1. Swapping sine and cosine puts the 4 on the x axis, measuring the angle clockwise gives -4 for y, and assigning r to both coordinates ignores the trigonometric factors.

Correct answer: $\ln x + C$

The antiderivative is $\ln x + C$. The power rule fails at exponent -1 because it would divide by zero, and the natural logarithm is the function whose derivative is $1/x$. The base-10 logarithm has derivative $1/(x \ln 10)$, so it is off by a constant factor. The expression $-1/x^2$ is the derivative of $1/x$, not its antiderivative, and $x \ln x$ is part of the antiderivative of $\ln x$.

Correct answer: Roots 3 and -2, so a sum of two real exponentials

The roots are 3 and -2, so the solution is $C_1 e^{3x} + C_2 e^{-2x}$, a sum of two real exponentials. Factoring gives $(r - 3)(r + 2) = 0$; the pair -3 and 2 would come from $r^2 + r - 6$, a sign slip on the middle term. Sine and cosine terms appear only when the roots are complex, so real distinct roots never give an oscillating form.

Correct answer: About 95% of intervals built this way would cover the true mean

The confidence level describes the method: across repeated samples, about 95 percent of intervals built this way would contain the true mean. The spread of individual parts is far wider than an interval for the mean, so it does not hold 95 percent of parts. Later sample means scatter around the true mean, not around this one interval, and a range for the next single part would be a prediction interval.

Correct answer: A strong negative linear association

A value of -0.85 means a strong negative linear association. The sign gives the direction, so strength falls as temperature rises, and a magnitude near 1 marks a strong fit. Either positive answer gets the sign wrong, and calling it weak would fit a value near zero, not one at 0.85 in magnitude.

Correct answer: E(X) = 24.0

The expected value is 24: (10)(0.2) + (20)(0.5) + (40)(0.3) = 2 + 10 + 12. The value 20 is just the most likely outcome. Getting 21 pairs 10 with 0.3 and 40 with 0.2, which swaps two probabilities, and 23.3 is the plain average of the three outcomes with no weighting.

Correct answer: Speed accounts for 81% of the variation in flow

R-squared is the share of the variation in pump flow that the regression on motor speed accounts for, here 81 percent. The slope carries units and is a separate quantity. R-squared says nothing about points sitting exactly on the line, and the correlation coefficient is the square root of 0.81, which is 0.9 in magnitude, not 0.81.

Correct answer: P(rejecting H0 when H0 is true)

The significance level alpha is the probability of a Type I error, rejecting the null hypothesis when it is true. Rejecting a false null is the power of the test, 1 minus beta. Retaining a false null is a Type II error with probability beta, and retaining a true null has probability 1 minus alpha, the confidence level.

Correct answer: SE = 2.00

The standard error is 2, since sigma divided by the square root of n is 12/6. The value 4 divides the variance, 144, by 36 instead of dividing the standard deviation by 6. The value 6 is the square root of the sample size itself, and 12 is the population standard deviation with no adjustment for sample size.

Correct answer: P = 1/3

The probability is 1/3, because only 5 and 6 exceed 4, which is 2 of 6 equally likely faces. The value 1/6 counts a single face, and 1/2 counts 4, 5 and 6, as if the stem said 4 or more. The value 2/3 counts the four faces that are 4 or less, the complement of the event asked for.

Correct answer: Mean = 2.00

The expected number of successes is np = 8 times 0.25 = 2. The value 0.25 is the success probability for one trial, and 8 is the number of trials. The value 1.5 is np(1 - p), which is the variance of the count rather than its mean.

Correct answer: The 99 percent width exceeds the 90 percent width

The 99 percent interval is wider. Higher confidence needs a larger critical value, about 2.576 against 1.645 for a z interval, and the margin is that value times the same standard error. So the 90 percent interval is the narrower one and the widths cannot match. The ratio of widths is the ratio of critical values, about 1.57 for z, not a fixed factor of two.

Correct answer: P = 0.72

Both operate with probability 0.9 times 0.8 = 0.72, since the two are independent. The value 0.80 takes the weaker component alone as if it set the joint result, and 0.85 averages the two probabilities. The value 0.98 is the chance that at least one of them operates, 1 - (0.1)(0.2), not the chance that both do.

Correct answer: Sum of squared vertical residuals

Ordinary least squares picks the slope and intercept that minimize the sum of squared vertical residuals, the gaps between each observed y and the fitted line. Squared perpendicular distances belong to orthogonal or total least squares, a different fit. Absolute residuals define least-absolute-deviation regression, whether measured vertically or perpendicularly, so neither absolute form is the least-squares criterion.

Correct answer: EMV = $300 thousand

The expected monetary value is $300 thousand: (600)(0.4) + (100)(0.6) = 240 + 60. The value $350 thousand averages the two outcomes as if each were equally likely. Getting $400 thousand swaps the probabilities, putting 0.6 on the large profit, and $700 thousand simply adds the outcomes without any weighting.

Correct answer: The mean is larger than the median

In a right-skewed data set the mean is larger than the median, because the long upper tail pulls the average toward large values while the median stays at the middle rank. A median above the mean marks left skew, and equal mean and median mark a symmetric shape. The mode sits at the peak, below both the median and the mean, so it is not larger than either.

Correct answer: Not significant at 10%, since the interval includes zero

Since the 90 percent interval includes zero, a zero difference is consistent with the data, so the difference is not significant at the 10 percent level. Failing to find a difference does not prove the means equal; it only means the data cannot rule equality out. Leaning positive still leaves zero inside, and the width of an interval says nothing on its own about whether a difference exists.

Correct answer: Variance = 4.00

The variance is 4, because a Poisson distribution has its variance equal to its mean. The value 0.25 is the reciprocal of the rate, and 2 is the standard deviation, the square root of the variance. The value 16 squares the mean, which treats the mean as if it were the standard deviation.

Correct answer: It is cut to half its value

The standard error is halved. It equals sigma divided by the square root of n, and going from 25 to 100 raises the square root from 5 to 10. A quarter would need n to rise sixteen-fold, since the dependence is on the square root and not on n itself. The value cannot stay the same or grow, because a larger sample always shrinks the standard error.

Correct answer: Mole and mass ratios linking reactants and products

Stoichiometry is the quantitative bookkeeping of a reaction: the mole and mass ratios, fixed by the balanced equation, that link reactants and products. Rates and mechanisms are chemical kinetics, and heat absorbed or released is thermochemistry. Where an equilibrium settles is set by the equilibrium constant, which is also a separate topic from the amount ratios.

Correct answer: 5 O2 molecules

Five O2 are needed: C3H8 + 5 O2 -> 3 CO2 + 4 H2O. The 3 CO2 and 4 H2O hold 6 + 4 = 10 oxygen atoms, which is 5 O2 molecules. The value 10 counts oxygen atoms instead of molecules, 7 adds the product molecules 3 + 4, and 3 balances only the carbon dioxide and forgets the oxygen in the water.

Correct answer: 2.0 mol of H2O

The reaction gives 2.0 mol of water, because the equation sets a 2:2 ratio between H2 and H2O and hydrogen is the limiting reactant. The value 1.0 mol is the oxygen consumed, not the water formed. Getting 3.0 mol adds all the reactant moles, and 4.0 mol counts hydrogen atoms instead of water molecules.

Correct answer: 22 g of CO2

The reaction gives 22 g of CO2: 6.0 g of carbon is 0.50 mol, the 1:1 ratio gives 0.50 mol of CO2, and 0.50 times 44 g/mol is 22 g. The value 6 g keeps only the carbon's mass, and 12 g reports the molar mass of carbon. The value 16 g is the oxygen consumed, 0.50 mol times 32 g/mol, not the product mass.

Correct answer: Transfer of electrons between species

Oxidation-reduction is defined by electron transfer: one species loses electrons and is oxidized while another gains them and is reduced. Proton transfer defines acid-base reactions. Gaining oxygen was the historical meaning of oxidation, but many redox reactions involve no oxygen at all, so it is not the definition, and swapping whole ions between partners is a double-displacement reaction with no change in oxidation state.

Correct answer: Zn is oxidized and acts as the reducing agent

Zinc goes from 0 to +2, so it loses electrons and is oxidized, and because it supplies the electrons that reduce Cu2+ it is the reducing agent. The species that is oxidized is never the oxidizing agent. The copper ion goes from +2 to 0, so it gains electrons and is reduced rather than oxidized, which makes both copper answers false.

Correct answer: pH 3, an acidic solution

The pH is 3, since pH = -log(1.0 x 10^-3), and a pH below 7 at 25 degrees C is acidic. The value 11 is 14 minus 3, which is the pOH, not the pH. A pH of 3 cannot be basic, and pH 11 cannot be acidic, so the mixed answers fail on the label too.

Correct answer: A weak acid plus a salt of its conjugate base

A buffer needs a weak acid and a salt of its conjugate base, so one partner absorbs added base and the other absorbs added acid. The conjugate base of a strong acid, such as chloride, is too weak to take up added acid, so that pair does not buffer. An excess of strong base uses up the weak acid entirely, and equal moles of strong acid and base just give a neutral salt solution.

Correct answer: A pH of 9.0

The pH is 9. The ion product gives [H+] = (1.0 x 10^-14)/(1.0 x 10^-5) = 1.0 x 10^-9 mol/L, and -log of that is 9. The value 5 is the pOH, 7 assumes the solution is neutral even though it has excess hydroxide, and 14 is pKw itself before the pOH is subtracted.

Correct answer: Forward and reverse rates stay equal, so concentrations hold

Equilibrium is dynamic: the forward and reverse reactions keep running at equal rates, so concentrations stay constant. The reactions do not stop, even though the amounts look frozen. Concentrations at equilibrium are constant but generally unequal to one another, and a reversible reaction never runs one way until the reactants are used up.

Correct answer: Yield falls, since equilibrium shifts toward N2 and H2

The ammonia yield falls. The forward reaction releases heat, so heat acts like a product, and raising the temperature pushes the equilibrium back toward nitrogen and hydrogen; the equilibrium constant gets smaller. Heating speeds both directions, not just the forward one, but it also changes the equilibrium constant, so the yield does not hold steady, and the shift runs away from ammonia rather than toward it.

Correct answer: Toward A, the side with fewer moles of gas

The equilibrium moves toward A, the side with one mole of gas rather than two. Shrinking the volume raises every partial pressure, and the system relieves that by forming fewer gas molecules, so moving toward B would add molecules and push the pressure higher. A shift does occur because the reaction quotient changes even though K stays the same, and the exothermic direction responds to temperature, not to pressure.

Correct answer: Anodic metal oxidation paired with a cathodic reduction

Galvanic corrosion is an electrochemical cell: the metal oxidizes and dissolves at the anode while a species such as oxygen or water is reduced at the cathode. Dry high-temperature attack is a direct chemical reaction with no electrolyte, and loss from abrasive impact is erosion, a mechanical process. Metal dissolves at the anode, not the cathode, which is the protected electrode.

Correct answer: A sacrificial-anode protection system

Attaching magnesium blocks that corrode in place of the steel is cathodic protection with sacrificial anodes: the more active metal becomes the anode and the pipe becomes the cathode. Impressed-current protection also makes the pipe a cathode, but it uses an outside power supply and inert anodes rather than metal that is consumed. A barrier coating only isolates the surface, and inhibitors are chemicals dosed into the environment, so neither uses an attached metal.

Correct answer: Aliasing

The error is aliasing: content above half the sampling rate, the Nyquist frequency, folds back and shows up as false lower frequencies, which is why an anti-aliasing filter goes before the converter. Spectral leakage smears energy into nearby bins when a record holds a non-integer number of cycles, but it creates no false low frequency. Quantization is amplitude rounding, and jitter is timing error in the sample clock.

Correct answer: 4096 levels, about 1.22 mV each

A 12-bit converter has 2^12 = 4096 levels, and 5 V divided by 4096 is about 1.22 mV per least significant bit. The 2048-level answer is an 11-bit converter, one bit short. The 1024-level answer is 10 bits, and 256 levels is an 8-bit converter, so each of those gives a larger step than this device.

Correct answer: A digital-to-analog converter

A digital-to-analog converter is the device that turns the microcontroller's numeric codes into a matching continuous output voltage for the actuator. A sample-and-hold amplifier only freezes an analog value that already exists, so it cannot build a voltage from digital codes. A low-pass anti-aliasing filter limits signal bandwidth ahead of sampling, and a hysteresis comparator outputs only a high or low level, never a smoothly varying one.

Correct answer: 4 kHz

The Nyquist criterion requires a sampling rate greater than twice the highest frequency present, and twice 2 kHz is 4 kHz. Sampling at 2 kHz equals the signal frequency itself and aliases badly, while 1 kHz confuses the minimum rate with half the signal frequency. A rate of 8 kHz would work, but it is four times the signal content, so it is not the minimum the criterion sets.

Correct answer: An XOR gate

An XOR (exclusive-OR) gate outputs logic 1 only when its two inputs differ and outputs 0 when they match. An XNOR gate does the reverse, giving 1 when the inputs are equal. A NAND gate outputs 1 unless both inputs are high, so it is also 1 when both are low, and a NOR gate outputs 1 only when both inputs are low.

Correct answer: A NOT gate

A triangle with a bubble on its output is the NOT gate, or inverter, which outputs the complement of its single input. A buffer uses the same triangle but has no bubble, so it passes the input through unchanged. NAND and XNOR gates are two-input gates drawn with different body shapes, so neither matches a single-input triangle.

Correct answer: An OR gate, low only if both inputs are low

An OR gate is low only when both inputs are low, so a trip on either detector, or on both, drives the output high and sounds the alarm. An AND gate goes high only when both detectors trip together, so a single tripped detector would stay silent. A NAND gate holds the alarm on in normal operation, and an XOR gate falls silent when both detectors trip at once, which defeats the interlock.

Correct answer: It reads 100 ohms at 0 degrees Celsius

The 100 in Pt100 is the element's nominal resistance, 100 ohms at 0 degrees Celsius, and the resistance then rises in a known way with temperature. Platinum RTDs routinely work far above 100 °C, so the number is not a service limit. Excitation current is kept near one milliamp to limit self-heating, and response time is a separate datasheet value that depends on the probe's construction.

Correct answer: 250 kPa

The 4 to 20 mA signal has a 16 mA span with a 4 mA live zero, so 12 mA sits at (12 - 4) / 16 = 0.5 of span, and 0.5 x 500 kPa = 250 kPa. Taking 12 / 20 of the span without removing the live zero gives 300 kPa. Dividing 8 mA by 20 mA gives 200 kPa, and dividing 12 mA by the 16 mA span without subtracting the offset gives 375 kPa.

Correct answer: A glass-membrane cell that develops a millivolt potential

A pH sensor is a potentiometric element: its glass membrane develops a small millivolt potential that depends on hydrogen-ion activity, and that voltage is converted to pH. A conductivity cell responds to all dissolved ions together, so it cannot isolate hydrogen ions. A thermistor responds to temperature, which pH meters measure only for compensation, and a capacitive probe senses permittivity rather than ion activity.

Correct answer: removes the impartial check a third-party review exists to give

The objection is a conflict of interest: an engineer who certifies their own design cannot give the independent, impartial review a third-party check exists to provide. Fee splitting is a separate rule about sharing fees for referrals, and no fee arrangement is described here. No outside party sees the client's data when the designer reviews their own work, and nothing in the scenario puts the engineer outside their competence.

Correct answer: Public safety comes first, so the engineer must alert the proper authority

Codes of ethics make protection of public safety, health, and welfare paramount, so an imminent danger can require the engineer to notify the appropriate authority even under a confidentiality agreement. Telling only the client leaves the public exposed if the client does nothing. Waiting for written consent lets the client veto a duty owed to the public, and releasing the full project file goes far beyond reporting the hazard and breaches confidentiality needlessly.

Correct answer: Required standards govern, whatever the regional practice is

The engineer must meet required standards and protect the public, and a shortcut does not become acceptable because it is common. Regional habit does not lower a required inspection, so it cannot define an acceptable standard of care here. A contractor's offer to carry liability does not release the engineer from professional duties, and a clean safety record only means harm has not happened yet.

Correct answer: Bid rigging, a form of collusion between competing firms

Secretly rotating deliberately high bids so a chosen firm wins is bid rigging, a collusive and deceptive practice the codes forbid. Bid shopping is a contractor pressuring subcontractors to undercut their quotes, which is not what these firms are doing. A joint venture is openly declared to the client, and value engineering is an honest review of function and cost, so neither describes hidden coordination of bids.

Correct answer: Reason from the core canons of the code, with public welfare first

When no specific rule fits, the engineer should reason from the code's fundamental canons, which place public safety, health, and welfare above all else. The client's lowest cost and the employer's schedule are legitimate interests, but the canons rank both below the public. Job security is a personal interest, and putting it first is the kind of conflict the codes tell engineers to avoid.

Correct answer: Was in responsible charge throughout the work

A seal certifies that the licensee was in responsible charge, meaning they personally prepared the work or directly supervised it, which ties accountability to a qualified engineer. Working for the same firm as the designer gives no such control over the work. A clean disciplinary record and liability insurance are not what a seal attests, and neither authorizes sealing work the engineer never directed.

Correct answer: Prohibited, since the practice act reserves the title for licensees

Licensure laws protect the title Professional Engineer and the PE designation, so using them without a license is unlawful, and boards act to stop it. An accredited degree is only one step toward licensure and confers no right to the title. Practice acts regulate both the practice and the title, and the violation exists from the moment the title is used, whether or not anyone is harmed.

Correct answer: The state engineering practice act, which creates the board

A state board's power to adopt conduct rules and discipline licensees comes from the state engineering practice act, the statute that creates and empowers it. The NCEES Model Law is a template states may adapt, and it has no binding force until a legislature enacts it. A state engineering society is a voluntary association whose bylaws bind only its members, and an employment contract governs only the parties to it.

Correct answer: Conduct that falls below the accepted professional standard of care

Boards reserve discipline for negligence, incompetence, or misconduct, meaning conduct that falls below the accepted standard of care, such as repeatedly ignoring known code requirements. An honest, reasonable judgment that a competent engineer might also make is not sanctionable. Publicity has no bearing on whether conduct meets the standard, and the standard is the same for a new engineer as for a senior one.

Correct answer: Withdrawal held at or below the rate that nature renews it

Sustainability means using a renewable resource no faster than nature replenishes it, so holding withdrawal at or below the renewal rate keeps the aquifer available for future users. Pacing withdrawal to pay back capital is an economic goal, not a sustainability one. Sizing to the worst drought sets a design demand but does not keep use within renewal, and pumping at whatever rate demand sets until the aquifer runs dry is depletion, the opposite of sustainable use.

Correct answer: Weigh trade-offs across several impact categories

Life-cycle analysis reports several impact categories, such as greenhouse gases and water use, and a design rarely wins on all of them, so the engineer must weigh the trade-offs. Collapsing the result to a single carbon score throws away the water finding that separates the designs. Purchase price is not an environmental impact, and the designs are clearly not equal, since each one wins on a different measure.

Correct answer: A circular economy model

Looping one operation's byproduct back in as another's feedstock keeps materials in use, which is the core idea of a circular economy. A cradle-to-grave view follows a product from extraction to final disposal, so it still ends in waste. End-of-pipe control treats waste after it is made rather than designing it out, and take-make-dispose is the linear model a circular economy replaces.

Correct answer: Resilience

Resilience is a system's ability to withstand, adapt to, and recover from disruptions over its life, which is exactly the focus on sea-level rise and storms described here. Durability is resistance to wear and decay, not recovery after a disruption. Redundancy is one tactic for building resilience rather than the goal itself, and reliability is the probability of operating without failure, which says nothing about recovering afterward.

Correct answer: Negative externalities

Pollution-related health and environmental harm imposed on society and left out of a project's budget are negative externalities. An opportunity cost is the value of the best alternative the project gives up, not a cost pushed onto outsiders. Indirect overhead is a real project expense carried in the budget, and a contingency allowance is money the project sets aside for its own risks.

Correct answer: The safety data sheet

Under the Hazard Communication Standard the employer must keep a safety data sheet for each hazardous chemical readily accessible, and it gives the detailed handling, first-aid, and exposure information. The chemical inventory list only names the chemicals on site. The written hazard communication program describes how the employer runs labeling, SDS access, and training, and the OSHA 300 log records injuries and illnesses, not chemical data.

Correct answer: Section 4

In the 16-section GHS safety data sheet, first-aid measures are in Section 4. Section 2 is hazard identification, which lists the hazards and the label elements but not the response steps. Section 8 covers exposure controls and personal protection, and Section 11 gives toxicological information, so neither tells a responder what to do immediately after an exposure.

Correct answer: Danger

The Hazard Communication Standard allows only two signal words, and Danger marks the more severe hazard categories. Warning is the other approved word and marks the less severe categories. Caution and Notice are not signal words under the standard; they belong to other sign and tag systems.

Correct answer: Acute toxicity that can be fatal

The skull and crossbones pictogram communicates acute toxicity severe enough to be fatal or toxic at low doses. Carcinogenic and mutagenic effects use the health hazard pictogram, a silhouette with a star on the chest. Severe burns and eye damage use the corrosion pictogram, and skin sensitization or irritation uses the exclamation mark.

Correct answer: 10 units

Twenty-four days is three half-lives of 8 days, so the activity halves three times: 80 to 40 to 20 to 10 units. Twenty units counts only two half-lives. About 26.7 units comes from dividing 80 by three, as if decay were linear, and about 4.0 units comes from using e^-3 instead of (1/2)^3.

Correct answer: The median lethal dose for a group of test animals

LD50 is the median lethal dose, the dose expected to kill 50 percent of a test population, and a lower LD50 means a more toxic substance. The lowest dose producing an observed adverse effect is the LOAEL, a different toxicology value. A reference dose is an estimated daily intake a person can take over a lifetime without appreciable harm, and the concentration at which a vapor ignites is its lower flammable limit.

Correct answer: Time, distance, and shielding

Ionizing radiation dose is controlled by limiting time near the source, increasing distance from it, and placing shielding between the source and the worker. Monitoring and dosimetry measure the dose a worker received and training explains the hazard, but none of them reduces the dose itself. Ventilation and dilution address airborne contaminants, and grounding and bonding address static and electrical hazards.

Correct answer: 1/16

Each half-life halves the remaining material, so after four half-lives the fraction is (1/2)^4 = 1/16. One-eighth is the result after three half-lives, and one-thirty-second is the result after five. One-quarter comes from dividing by the number of half-lives instead of halving once for each one.

Correct answer: The mean level averaged over one full 8-hour shift

A time-weighted average limit is the mean airborne concentration a worker may be exposed to over a normal 8-hour shift. A peak that must never be passed is a ceiling limit, which is a different value. The level allowed over any 15-minute period is the short-term exposure limit, and the level that triggers periodic monitoring is the action level, which sits below the permissible limit.

Correct answer: The ceiling limit

The ceiling limit is the concentration that must not be exceeded at any instant during the shift. The short-term exposure limit is a 15-minute average, so brief moments above it can still average out. The 8-hour TWA is a full-shift mean, and the action level is a lower trigger for monitoring and medical surveillance, not an instantaneous maximum.

Correct answer: It is over the limit, so the employer must reduce it

At 60 micrograms per cubic meter the exposure is above the 50 microgram permissible exposure limit, so it is out of compliance and the employer must add controls to bring it down. Being close to a limit does not make an exposure that exceeds it acceptable. A reading 20 percent over the limit cannot be dismissed as sampling error, and the lead action level of 30 micrograms is lower still, so the exposure is far above it.

Correct answer: Limited means of entry and exit, and not designed for steady occupancy

A confined space has limited or restricted means of entry and exit and is not designed for continuous worker occupancy, which is what the single hatch and occasional maintenance entry show. Being below grade is not part of the definition, since tanks and silos above ground also qualify. How long or how often workers stay inside does not create the classification, and poor lighting is a hazard to manage rather than a defining feature.

Correct answer: To dilute contaminants and keep oxygen in a safe range

Confined-space ventilation exists mainly to dilute and remove toxic or flammable atmospheres and to keep oxygen within a safe range for entrants. A tank is not ventilated to hold its shape, and blowers deliver too little pressure to prevent buckling. Cooling and drying can be side benefits, but neither is the hazard the ventilation is required to control.

Correct answer: Oxygen, then flammables, then toxics

Testing goes oxygen first, then flammable gases, then toxic gases. Combustible-gas sensors need enough oxygen to read correctly, so oxygen is confirmed before flammability, and fire or explosion is the more immediate threat, so flammables come before toxics. Testing flammables before oxygen risks a false low reading, and putting toxics ahead of flammables reverses the order of the more immediate danger.

Correct answer: It deadens the sense of smell as its concentration climbs

Hydrogen sulfide smells like rotten eggs at low levels, but at higher, dangerous concentrations it quickly paralyzes the sense of smell, so the odor seems to fade just as the hazard grows. It is not odorless, since its smell is noticed at very low concentrations. Its odor is foul rather than sweet, and it is detectable by smell far below lethal levels, which is why its disappearance is so misleading.

Correct answer: It has no color or odor and it binds to hemoglobin

Carbon monoxide gives no warning by sight or smell and binds to hemoglobin far more strongly than oxygen, starving tissue of oxygen even at modest concentrations. It is slightly lighter than air, so it mixes through an enclosed area rather than pooling at the floor. It does not irritate the airways, which is part of why it goes unnoticed, and it is a chemical asphyxiant, not a simple one that works only by displacing oxygen.

Correct answer: A dollar today is worth more than a dollar received later

The time value of money holds that a dollar today is worth more than a dollar received later, because today's dollar can be invested to earn interest. Inflation can add to the effect, but money has time value even with no inflation at all. A dollar's worth does depend on when it arrives, and a higher interest rate lowers, not raises, the present worth of a future dollar.

Correct answer: $6,806

The present worth is P = F / (1 + i)^n = 10,000 / (1.08)^5, and (1.08)^5 is about 1.4693, so P is about $6,806. Discounting for six years instead of five gives about $6,302. Simple interest, 10,000 / (1 + 0.08 x 5), gives about $7,143, and subtracting 8 percent of the face value for each of five years gives $6,000.

Correct answer: The rate at which the net present worth of the cash flows is zero

The internal rate of return is the interest rate at which the net present worth of all the project's cash flows equals zero. The minimum rate a firm will accept is the MARR, the hurdle the IRR is compared against. A lender's financing rate is a cost of borrowing, not a property of the project's cash flows, and dividing annual income by first cost gives a simple return that ignores the timing of money.

Correct answer: $5,000

Straight-line depreciation is (cost - salvage) / life = (50,000 - 5,000) / 9 = $5,000 per year. Ignoring salvage gives 50,000 / 9, about $5,556, and adding salvage to cost instead of subtracting it gives 55,000 / 9, about $6,111. Dividing the depreciable basis by 10 years instead of 9 gives $4,500.

Correct answer: -$105

Discounting each $8,000 return at 10 percent gives a present worth of about 8,000 x 2.4869 = $19,895, and subtracting the $20,000 investment leaves an NPV of about -$105. Ignoring discounting entirely gives 24,000 - 20,000 = $4,000. Discounting all three returns by only one year gives about $1,818, and discounting all three as if they arrived in year 3 gives about -$1,968.

Correct answer: 4,000 units

Break-even quantity is fixed cost divided by unit contribution margin: 60,000 / (40 - 25) = 4,000 units. Dividing by the $40 price alone gives 1,500 units, and dividing by the $25 variable cost alone gives 2,400 units. The 6,400 figure comes from finding break-even revenue of $160,000 and then dividing it by the variable cost instead of the selling price.

Correct answer: A ratio of at least 1.0

A public project is economically justified when its benefit-cost ratio is at least 1.0, meaning equivalent benefits at least equal equivalent costs. A ratio below 1.0 means costs exceed benefits. The ratio is not compared with the MARR, because the interest rate is already used to convert benefits and costs to equivalent worth, and any project with positive benefits has a ratio above zero, which proves nothing.

Correct answer: Larger charges in the early years and smaller ones later

Declining-balance methods front-load depreciation, recording larger charges in the early years and smaller ones later compared with straight-line. Equal charges every year describe straight-line itself. Depreciation still stops once book value reaches salvage, so the total write-off does not exceed cost minus salvage, and the method still needs a useful-life estimate to set its rate.

Correct answer: $7,163

Future worth is F = P(1 + i)^n = 4,000 x (1.06)^10, and (1.06)^10 is about 1.7908, so F is about $7,163. Simple interest, 4,000 x (1 + 0.06 x 10), gives only $6,400 because it ignores interest on interest. Compounding for nine years gives about $6,758, and compounding for eleven years gives about $7,593.

Correct answer: Accept A and reject B, since B earns below the MARR

At the MARR, a positive NPV means Alternative A earns more than the required return and a negative NPV means Alternative B earns less, so A is acceptable and B is not. Being tested at the same rate does not make every alternative acceptable; the sign of the result decides. A negative NPV signals an inadequate return, not a lower cost, and NPV at the MARR is a standard basis for comparing projects.

Correct answer: Reject it, since the IRR falls short of the MARR

A project is acceptable only when its IRR meets or exceeds the MARR, and 12 percent is below the 15 percent MARR, so it is rejected. The NPV evaluated at 12 percent is zero by the definition of IRR, so that reason is false even though the decision matches. A positive return alone does not clear the hurdle, and bank interest is irrelevant because the firm has set 15 percent as its minimum.

Correct answer: $8,200

Present worth of a uniform series is P = A(P/A, 7%, 5) = 2,000 x 4.1002, about $8,200. Adding the five payments without discounting gives $10,000. Discounting that full $10,000 as one lump at year 5 gives about $7,130, and discounting only a single $2,000 payment from year 5 gives about $1,426.

Correct answer: It rises, because each unit contributes less margin

Break-even quantity is fixed cost divided by the contribution margin, price minus variable cost, so a higher variable cost shrinks the margin and raises the quantity needed. Higher unit cost does not lower the quantity; it means each sale covers less of the fixed cost. Unchanged fixed costs do not hold the quantity steady when the margin shrinks, and break-even becomes undefined only if variable cost passes the price, which the stem does not say.

Correct answer: 12.68% per year

The effective rate is $(1 + 0.12/12)^{12} - 1 = (1.01)^{12} - 1 \approx 0.1268$, or 12.68% per year. Quoting 12.00% treats the nominal rate as if interest compounded only once a year. The 12.55% figure is what quarterly compounding gives, $(1.03)^4 - 1$, and 12.75% is the continuous-compounding result $e^{0.12} - 1$, so neither matches monthly periods.

Correct answer: Applied loads, couples, and the reaction at every support

A free body diagram isolates the body and shows the applied loads, the couples and the reaction at every support, because the equilibrium equations are written from exactly those quantities. Internal member forces do not belong on it, since they cancel in pairs inside an isolated rigid body, and leaving the support reactions off makes the diagram incomplete. Inertia forces belong to a dynamic analysis, and the weight alone ignores the other loads and the supports.

Correct answer: One force acting normal to the surface it rolls on

A frictionless roller is drawn as one force acting normal to the surface it rolls on, since it cannot resist sliding along that surface or rotation. A horizontal and a vertical force with no moment describes a pin, and two forces plus a resisting end moment describes a fixed support. A force along the length of the beam would need friction or a restraint parallel to the surface, which a frictionless roller does not provide.

Correct answer: 400 N

Taking moments about the right support, $R_L(6) = 600(4)$, so $R_L = 400$ N. A value of 600 N assigns the whole load to one support and leaves nothing for the roller. The 900 N result inverts the lever ratio, 600(6/4), and 1,800 N uses the 2 m distance with the ratio inverted, 600(6/2), so both violate vertical equilibrium, which needs the two reactions to sum to 600 N.

Correct answer: 3 scalar equations

A planar rigid body has 3 independent scalar equations: force sums in two perpendicular directions and one moment sum. Two equations is the count for a single particle or a concurrent force system, where the moment equation adds nothing. There is no fourth independent equation in the plane, and 6 is the count for a rigid body in three dimensions.

Correct answer: Three: a horizontal force, a vertical force, a moment

A fixed support prevents horizontal translation, vertical translation and rotation, so it supplies three reactions: a horizontal force, a vertical force and a moment. A horizontal and a vertical force with no moment is a pin, which lets the beam rotate. A single normal force is a roller. A vertical force and a moment with no horizontal force would leave the end free to slide, which a built-in end does not allow.

Correct answer: The area-weighted mean position of all of its elements

Each centroidal coordinate is the first moment of area divided by the total area, so the centroid is the area-weighted mean position of all of its elements. Bending stress is zero, not at its peak, on a neutral axis through the centroid. The midpoint of a bounding rectangle matches the centroid only for symmetric shapes, and the polar moment about the centroid is the smallest of any point, not the greatest.

Correct answer: x = 5.0 cm

The composite centroid is $\bar{x} = (20 \cdot 2 + 30 \cdot 7)/(20 + 30) = 250/50 = 5.0$ cm. A value of 7.0 cm is just the centroid of the larger rectangle and ignores the smaller one. Dividing the 250 cm³ first moment by only the 30 cm² area gives 8.3 cm, and dividing by only the 20 cm² area gives 12.5 cm, both of which fall outside the span between the two part centroids.

Correct answer: $\bar{x} = b/3$

A triangle's centroid sits one-third of the way from each side toward the opposite vertex, so measured from the right-angle vertex along the base it is $b/3$. The value $b/2$ is the centroid of a rectangle, not a triangle. The value $2b/3$ is the same point measured from the acute vertex at the far end of the base, and $3b/4$ belongs to a parabolic spandrel.

Correct answer: How widely the area is distributed away from the axis

The area moment of inertia is $\int y^2 \, dA$, so it measures how widely the area is distributed away from the axis, with remote area counting heavily. How much area falls on one side of an axis is not what it measures, and the centroid's position comes from the first moment of area, not the second. Material density never enters, because the area moment of inertia is purely geometric.

Correct answer: $I_x = bh^3/12$

About the horizontal centroidal axis the rectangle gives $I_x = bh^3/12$, with the height cubed because it is measured perpendicular to that axis. The form $bh^3/3$ is the moment of inertia about the base, not the centroid. Swapping to $hb^3/12$ gives the vertical centroidal axis, and $bh^2/6$ is the elastic section modulus, which has units of length cubed.

Correct answer: $72 \ \text{cm}^4$

With the axis parallel to the 4 cm width, $I = bh^3/12 = 4(6^3)/12 = 72 \ \text{cm}^4$. Cubing the width instead, $6(4^3)/12 = 32 \ \text{cm}^4$, gives the axis parallel to the height. The value $288 \ \text{cm}^4$ uses $bh^3/3$, which is about the base, and $864 \ \text{cm}^4$ drops the division by 12 altogether.

Correct answer: The second moment about an axis offset from the centroid

The parallel axis theorem, $I = I_c + Ad^2$, gives the second moment of area about an axis offset from the centroid and parallel to the centroidal axis. A composite centroid is located with first moments of area, not with this theorem. Shear flow at a glued seam comes from $VQ/I$, and the product of inertia on a rotated axis comes from the rotation (Mohr's circle) equations rather than from a parallel shift.

Correct answer: $140 \ \text{cm}^4$

The theorem gives $I = I_c + Ad^2 = 50 + 10(3^2) = 140 \ \text{cm}^4$. Using $Ad$ instead of $Ad^2$ gives $50 + 30 = 80 \ \text{cm}^4$, and $90 \ \text{cm}^4$ is the transfer term alone without the centroidal value. Adding $(Ad)^2 = 900$ instead of $Ad^2$ gives $950 \ \text{cm}^4$, which squares the area as well as the distance.

Correct answer: The perpendicular gap between the centroidal and new axes

In $Ad^2$, d is the perpendicular gap between the centroidal axis and the new axis, and the theorem only holds when one of the two passes through the centroid. The distance to the outermost fiber is the c used in bending stress, not a transfer distance. A radial distance from an arbitrary global origin is not measured perpendicular to the axis, and the diagonal of the section is a property of its shape, not of where the new axis lies.

Correct answer: $k^2 = I/A$

The radius of gyration is defined by $I = Ak^2$, so $k^2 = I/A$ and $k = \sqrt{I/A}$. Inverting the ratio to $A/I$ gives units of one over length squared, and the product $IA$ gives length to the sixth. Dividing by $A^2$ leaves a dimensionless ratio, so only $I/A$ yields a squared length.

Correct answer: 4.00 cm

The radius of gyration is $k = \sqrt{I/A} = \sqrt{800/50} = \sqrt{16} = 4.00$ cm. The value 16.00 cm stops at $I/A$ and never takes the square root. Inverting the ratio, $\sqrt{50/800}$, gives 0.25 cm, and $\sqrt{800} \approx 28.28$ cm forgets to divide by the area.

Correct answer: Two: a force sum along each orthogonal axis

Each joint of a planar truss gives two equations, a force sum along each of two orthogonal axes. The member forces at a joint are concurrent, so a moment sum about the joint is automatically zero and adds no information. One equation per member is not how joint equilibrium works, and six equations is the count for a rigid body in three dimensions.

Correct answer: Members are pinned links in pure tension or compression

The method of joints treats every member as a pinned two-force link in pure tension or compression, because ideal truss joints are pins and loads are applied only at the joints. Welded joints that transmit bending moments, or rigid frame parts resisting joint moments, describe frame analysis rather than an ideal truss. Since loads act at the joints, members do not resist bending from loads applied along their spans.

Correct answer: 100.0 N

Vertical equilibrium needs $F \sin 30^\circ = 50$ N, so $F = 50/0.5 = 100.0$ N. Multiplying instead of dividing, $50 \sin 30^\circ$, gives 25.0 N. Using cosine for the vertical component, $50/\cos 30^\circ$, gives 57.7 N, and 86.6 N is the horizontal component of the correct 100 N force, which is what the horizontal member must balance.

Correct answer: 60 N

Maximum static friction is $\mu_s N = 0.30 \times 200 = 60$ N. The 140 N figure subtracts the friction from the normal force, which has no physical meaning here. The 200 N figure is the normal force itself, and 667 N comes from dividing the normal force by the coefficient instead of multiplying.

Correct answer: $\theta = \tan^{-1} \mu_s$

At impending slip, $W \sin\theta = \mu_s W \cos\theta$, so $\tan\theta = \mu_s$ and $\theta = \tan^{-1} \mu_s$, the angle of repose. The inverse sine would follow only if friction were $\mu_s W$, ignoring the reduced normal force on the slope. The inverse cosine has no basis in the force balance, and the inverse cotangent inverts the ratio, giving the complement of the correct angle.

Correct answer: $F_f = 30$ N

Because the block stays at rest, static friction takes whatever value keeps it in equilibrium, which is the 30 N applied force. The 40 N figure, $0.40 \times 100$, is only the maximum available and is reached only when sliding impends. A friction force of 0 N would leave the 30 N push unbalanced, and 12 N multiplies the coefficient by the push instead of using equilibrium.

Correct answer: A pure turning effect, equal wherever moments are taken

A couple is two equal, opposite, parallel forces, so its resultant force is zero and it produces a pure turning effect that is equal wherever moments are taken. Because the forces cancel, a couple cannot produce a net force through the centroid. Its moment does not change with the reference point, and an off-center push differs from a couple precisely because it has a nonzero resultant force.

Correct answer: Velocity has a direction, and speed is its magnitude

Velocity is a vector with a direction, and speed is its magnitude, a scalar. The instantaneous speed always equals the magnitude of the velocity, so velocity does not exceed speed when the path curves. Velocity is the rate of change of position, not of distance traveled, and the two quantities are not separated by their units, since both can be expressed in m/s or km/h.

Correct answer: v = 12 m/s

From rest with constant acceleration, $v = at = 3 \times 4 = 12$ m/s. Dividing acceleration by time gives 0.75 m/s, which is not even a velocity. Halving the product gives 6 m/s, the average velocity over the interval rather than the final value, and 7 m/s simply adds the two numbers.

Correct answer: s = 25 m

From rest, $s = \tfrac{1}{2}at^2 = \tfrac{1}{2}(2)(5^2) = 25$ m. Forgetting to square the time gives $\tfrac{1}{2}(2)(5) = 5$ m. The value 10 m is the final velocity, $at$, read as a distance, and 50 m drops the one-half factor.

Correct answer: 20.4 m

At the peak the velocity is zero, so $h = v_0^2/(2g) = 400/19.62 \approx 20.4$ m. Leaving the speed unsquared gives $20/19.62 \approx 1.02$ m, and 2.04 m is the time to the peak in seconds, $v_0/g$, mistaken for a height. Dividing by g instead of 2g gives 40.8 m, twice the true rise.

Correct answer: Horizontal part is fixed, and the vertical part changes at g

Gravity acts only vertically, so the horizontal part of the velocity is fixed and the vertical part changes at the constant rate g for the whole flight. With no horizontal force there is no horizontal acceleration, so the horizontal part cannot shrink to zero at the peak and can never change at g. The vertical part does not wait until the peak to start changing, and it keeps changing after the peak rather than becoming fixed.

Correct answer: Net work equals the change in kinetic energy

The work-energy theorem says the net work done on a particle equals the change in its kinetic energy, which follows from integrating Newton's second law over displacement. Change in linear momentum is matched by impulse, not work, and power is a rate of doing work rather than a momentum quantity. Impulse is not related to potential energy at all.

Correct answer: 16 J

Net work equals the change in kinetic energy: $\tfrac{1}{2}(2)(5^2 - 3^2) = 25 - 9 = 16$ J. Squaring the difference in speed, $\tfrac{1}{2}(2)(5 - 3)^2$, gives 4 J, which is not the change in kinetic energy. The value 25 J is only the final kinetic energy, and 34 J adds the two kinetic energies instead of subtracting them.

Correct answer: 200 J

For a constant force along the displacement, $W = Fd = 50 \times 4 = 200$ J. Dividing the force by the distance gives 12.5 J, which is not a work quantity. Adding the two numbers gives 54 J, and 100 J brings in a one-half factor that belongs to kinetic energy, not to work by a constant force.

Correct answer: −200 kJ

The braking work equals the change in kinetic energy, $0 - \tfrac{1}{2}(1000)(20^2) = -200{,}000$ J, or −200 kJ. Leaving out the one-half gives −400 kJ. The −20 kJ value uses $mv$, which is momentum, not energy, and −10 kJ uses $\tfrac{1}{2}mv$, which skips squaring the speed.

Correct answer: The time rate at which the work is done

Power is the time rate at which work is done, $P = dW/dt$, equal to force times velocity for a moving point of application. The time rate at which momentum changes is force, by Newton's second law. A force times the distance moved is work itself, and mass times speed is the magnitude of linear momentum.

Correct answer: 1,962 W

At steady speed the lifting force equals the weight, $100 \times 9.81 = 981$ N, so $P = Fv = 981 \times 2 = 1{,}962$ W. The 981 W figure is the weight with the speed left out, and 200 W multiplies mass by speed without gravity. Squaring the speed gives 3,924 W, which is not how power relates force and velocity.

Correct answer: Mass times velocity, a vector along the path

Linear momentum is mass times velocity, $p = mv$, a vector that points along the path of motion. Half the mass times the speed squared is kinetic energy. Force times duration is impulse, which equals a change in momentum rather than the momentum itself, and mass times acceleration is the net force from Newton's second law.

Correct answer: The change in its linear momentum vector

Integrating the net force over time gives the impulse, and by Newton's second law that equals the change in the particle's linear momentum vector. A change in overall kinetic energy equals the net work done, which is the work-energy theorem, not the impulse-momentum principle. The rate at which momentum changes is the net force itself, not its time integral.

Correct answer: 30 N·s

For a constant force, the impulse is $Ft = 10 \times 3 = 30$ N·s. Dividing the force by the time gives 3.3, which is not an impulse, and 13 simply adds the two numbers. The value 15 N·s brings in a one-half factor that applies only when a force ramps linearly from zero, not to a constant force.

Correct answer: 4 kg·m/s

Taking the incoming direction as positive, momentum goes from $+0.5(4) = +2$ to $-2$ kg·m/s, so the magnitude of the change is 4 kg·m/s. Zero would follow only if direction were ignored, since the speed is unchanged but the vector reverses. The value 2 kg·m/s is the momentum before impact, not the change, and 8 kg·m/s doubles the correct change a second time.

Correct answer: Perfectly inelastic, conserving momentum but not kinetic energy

Bodies that stick together undergo a perfectly inelastic collision: momentum is conserved because no external impulse acts, but kinetic energy is lost to deformation and heat. A perfectly elastic collision conserves kinetic energy and the bodies separate, so it cannot describe sticking. A superelastic event gains kinetic energy from stored energy, which merging bodies do not release, and momentum is conserved in every collision free of external impulse.

Correct answer: 3.0 m/s

Momentum is conserved: $3(4) + 1(0) = (3 + 1)v$, so $v = 12/4 = 3.0$ m/s. Averaging the two speeds gives 2.0 m/s, which ignores the masses. The value 4.0 m/s ignores the added 1 kg cart, and 12.0 m/s is the total momentum in kg·m/s mistaken for a velocity.

Correct answer: How much the body opposes an angular acceleration

The mass moment of inertia, $\int r^2 \, dm$, is the rotational counterpart of mass, so it measures how much the body opposes an angular acceleration about the axis through $\sum M = I\alpha$. The kinetic energy held while spinning is $\tfrac{1}{2} I\omega^2$, which also needs a spin rate, so it is not the same property. Resistance of a section to bending depends on the area moment of inertia, and the load the body places on supports does not depend on how the mass is spread about an axis.

Correct answer: $I = \tfrac{1}{2} m r^2$

Integrating over a uniform solid disk about its central axis gives $I = \tfrac{1}{2} m r^2$. The form $m r^2$ is a thin ring, where all mass sits at the rim. The one-quarter coefficient is the same disk about a diameter, and $\tfrac{2}{5} m r^2$ is a solid sphere.

Correct answer: 0.50 kg·m²

For a solid disk, $I = \tfrac{1}{2} m r^2 = \tfrac{1}{2}(4)(0.5^2) = 0.50$ kg·m². The value 0.25 uses the one-quarter coefficient that applies about a diameter, not the central axis. Dropping the one-half gives 1.00, the thin-ring result, and 2.00 multiplies mass by radius without squaring it.

Correct answer: $\sum M = I\alpha$

For rotation about a fixed axis, $\sum M = I\alpha$, the rotational form of Newton's second law. Using mass m in place of the mass moment of inertia ignores how far the mass sits from the axis. The product $I\omega$ is angular momentum, and $I\omega^2/2$ is rotational kinetic energy, so neither relates torque to angular acceleration.

Correct answer: $\alpha = 4$ rad/s²

From $\sum M = I\alpha$, $\alpha = 20/5 = 4$ rad/s². Subtracting the inertia from the torque gives 15 and adding them gives 25, neither of which is dimensionally valid. Multiplying torque by inertia gives 100, which inverts the relationship.

Correct answer: $\omega_n = \sqrt{k/m}$

Free vibration of $m\ddot{x} + kx = 0$ gives $\omega_n = \sqrt{k/m}$ in rad/s, so a stiffer spring raises the frequency and a larger mass lowers it. The form $\sqrt{m/k}$ is inverted and has units of time. Leaving out the square root, $k/m$, gives $\omega_n^2$, and the $1/(2\pi)$ factor converts to the natural frequency in hertz, not radians per second.

Correct answer: Elongation per unit of the original gauge length

Engineering strain is the elongation divided by the original gauge length, a dimensionless ratio. Dividing by the stretched length gives a different measure related to true strain, not engineering strain. Load per unit area is the definition of stress, and elongation per unit force is the flexibility of the member, which carries units of length per force.

Correct answer: $4.0 \times 10^{-4}$

The strain is $4.0 \times 10^{-4}$: converting 1.0 mm to 0.001 m and dividing by 2.5 m gives 0.0004. The value $4.0 \times 10^{-1}$ divides millimeters by meters without converting, and $4.0 \times 10^{-3}$ slips one decimal place. The value $2.5 \times 10^{-3}$ divides the length by 1,000 instead of dividing the stretch by the length.

Correct answer: $5.0 \times 10^{-4}$

The strain is $5.0 \times 10^{-4}$. Hooke's law gives strain = stress / E = 55 MPa / 110,000 MPa = 0.0005. The value $5.0 \times 10^{-1}$ divides 55 by 110 and ignores that one is in MPa and the other in GPa. The value $2.0 \times 10^{-3}$ inverts the ratio before scaling, and $5.0 \times 10^{-3}$ misplaces the GPa to MPa conversion by a factor of ten.

Correct answer: Half as much elongation as bar B

Bar A elongates half as much. Axial elongation is PL/(AE), so with the same load, length and modulus it is inversely proportional to area, and doubling the area halves it. Twice as much reverses that inverse relation, and one fourth would apply if the diameter rather than the area had doubled. Equal elongation would require equal stress, but bar A carries the same force over twice the area.

Correct answer: Zero, with the shear changing sign there

The shear is zero at the section of maximum moment, because shear is the slope of the moment diagram and the moment peaks where that slope crosses zero, at midspan for this beam. The value wL/2 is the shear just inside each support, where the moment is zero. The value wL/4 is the shear at the quarter points, not at midspan, and wL/8 confuses shear with the coefficient in the maximum moment wL^2/8.

Correct answer: Shear linear and moment parabolic

Under a uniform load the shear is linear and the moment is parabolic. The shear changes at the constant rate w, so it is a straight sloped line, and integrating that line gives a second-degree moment curve. Constant shear with linear moment describes an unloaded segment between point loads, and parabolic shear with cubic moment belongs to a linearly varying load. Constant shear can never integrate to a cubic moment, so that pairing fits no loading at all.

Correct answer: 0.6 kN·m

The maximum moment is 600 N·m, or 0.6 kN·m. By symmetry each support reaction is 600 N, and the moment at either load is 600 N times its 1 m distance from the support, which stays constant between the loads. The value 0.3 kN·m halves the reaction, 1.2 kN·m multiplies both loads by 1 m, and 1.5 kN·m applies one reaction over half the 5 m span as if the loads were not there.

Correct answer: An abrupt vertical step equal to the load

A concentrated load causes an abrupt vertical step in the shear diagram equal to the load. The load acts over essentially zero length, so the shear changes by its full value at one section. A sloped straight run belongs to a distributed load, a kink in an otherwise smooth line is what the point load produces in the moment diagram, and a parabolic shear needs a linearly varying load.

Correct answer: Center at 40 MPa

The center is at 40 MPa, the average of the two normal stresses, (70 + 10)/2. The value 30 MPa is half their difference, which is the radius of the circle rather than its center. The value 35 MPa halves only the x stress and ignores the y stress, and 80 MPa is the sum without dividing by two.

Correct answer: τmax = 25 MPa

The maximum in-plane shear is 25 MPa. The Mohr's circle radius is $\sqrt{((50 - 50)/2)^2 + 25^2} = 25$ MPa, because equal normal stresses add nothing. The value 35 MPa uses 25 MPa in place of the half-difference term, which should be zero. The value 50 MPa is the normal stress itself, and 75 MPa adds the normal and shear stresses directly.