A licensed engineer discovers that a colleague's design report contains a calculation error that could compromise the structural safety of a pedestrian bridge already under construction. According to standard engineering codes of ethics, what is the engineer's primary obligation in this situation?

Report the safety concern to the appropriate parties, holding paramount the safety, health, and welfare of the public

Say nothing, since the report was not authored by this engineer

Correct the report quietly without informing anyone, to avoid conflict with the colleague

Wait until the bridge is complete to see whether the error causes a visible problem

Determine the value of the definite integral of (3x² − 4x + 1) evaluated from x = 0 to x = 2.

2, from correctly forming the antiderivative x³ − 2x² + x and evaluating it at the limits

10, from an arithmetic slip while combining the evaluated terms

18, from a sign error that turns −4x into +4x in the antiderivative

0, from omitting the constant term when forming the antiderivative

Two force vectors are given as A = 3i + 4j (N) and B = 4i − 3j (N). What is the angle between vectors A and B?

90°, from finding that the dot product A·B equals zero

45°, from averaging the directions of the two vectors without computing the dot product

0°, from assuming the vectors are parallel because they share the same component magnitudes

180°, from assuming the vectors point in exactly opposite directions

During a project, an engineer identifies that specifying a cheaper alloy would save the client money but would reduce the safety margin of a rotating component below accepted design practice. The client insists on the cheaper alloy. What should the engineer do?

Refuse to proceed with the unsafe specification and explain the safety risk to the client, escalating if necessary

Resign from the project without informing the client of the underlying safety reason

Specify the cheaper alloy but omit the reduced safety margin from the calculations

Comply with the client's wishes, since the client is paying for the work

A mechanical engineer serves as an expert witness in a product liability case. The engineer's honest analysis would not favor the party paying for the testimony. What does professional ethics require of the engineer?

Present the findings objectively and truthfully, regardless of which party they favor

Decline to testify unless the analysis can be adjusted to favor the paying party

Delay the testimony indefinitely to avoid the conflict

Present only the portions of the analysis that support the client's position

A piece of production equipment costs $50,000 today and is expected to generate $15,000 per year in savings for 5 years. Using an interest rate of 10% per year and a uniform-series present-worth factor (P/A,10%,5) of 3.7908, what is the present worth of the savings?

$56,862, from multiplying the annual savings of $15,000 by the given present-worth factor

$37,908, from applying the present-worth factor to an assumed $10,000 annual amount instead of the actual $15,000

$75,000, from multiplying the annual savings by 5 years without discounting

$50,000, mistaking the present worth of savings for the equipment's purchase cost

A piece of equipment is purchased for $80,000 and has an estimated salvage value of $10,000 after a 7-year useful life. Using straight-line depreciation, what is the annual depreciation charge?

$10,000/year, from dividing the depreciable amount of $70,000 by the 7-year life

$8,750/year, from dividing the depreciable amount by an incorrect 8-year life

$70,000/year, from failing to divide the depreciable amount by the useful life at all

$11,429/year, from dividing the full purchase price by the useful life without subtracting salvage value

A resistive heating element draws 4 A of current when connected to a 120 V supply. What is the power dissipated by the element?

480 W, from multiplying the voltage by the current using P = VI

14,400 W, from squaring the voltage without dividing by resistance

124 W, from adding the voltage and current instead of multiplying them

30 W, from dividing the voltage by the current instead of multiplying

A capacitor with a capacitance of 10 μF is charged to 100 V. How much energy is stored in the capacitor, using E = (1/2)CV²?

0.05 J, from evaluating (1/2)(10×10⁻⁶)(100)²

50 J, from mistaking the 10 μF capacitance for 10 mF

0.10 J, from omitting the factor of one-half in the energy formula

0.0005 J, from using V instead of V² in the calculation

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PRACTICE ENGINE · FE MECHANICAL

FE Mechanical Practice Exam.
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QUESTION 1 / 61Mathematics, Probability & StatisticsMedium0/0

A first-order linear ODE governing a cooling process is dT/dt + 0.2T = 0, with initial condition T(0) = 100°C. What is the temperature predicted by this model at t = 5 seconds?

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1. A first-order linear ODE governing a cooling process is dT/dt + 0.2T = 0, with initial condition T(0) = 100°C. What is the temperature predicted by this model at t = 5 seconds?
- A. 80.0°C, from a linear (rather than exponential) decay approximation
- B. 36.8°C, from evaluating T(0)·e^(−0.2·5)
- C. 13.5°C, from doubling the decay-rate constant before evaluating the exponential
- D. 50.0°C, from assuming the temperature halves over the given time interval
Show answer & explanation
Answer: B
The ODE dT/dt = −0.2T has the standard solution T(t) = T(0)e^(−kt), so plugging in k = 0.2, t = 5 gives T(0)e^(−1) ≈ 0.368·100 = 36.8°C. The linear-decay distractor is tempting because it is simpler to compute, but first-order decay is exponential, not linear, so it understates the temperature at short times and overstates the rate of loss.
2. A licensed engineer discovers that a colleague's design report contains a calculation error that could compromise the structural safety of a pedestrian bridge already under construction. According to standard engineering codes of ethics, what is the engineer's primary obligation in this situation?
- A. Report the safety concern to the appropriate parties, holding paramount the safety, health, and welfare of the public
- B. Correct the report quietly without informing anyone, to avoid conflict with the colleague
- C. Say nothing, since the report was not authored by this engineer
- D. Wait until the bridge is complete to see whether the error causes a visible problem
Show answer & explanation
Answer: A
Engineering codes of ethics consistently hold the safety, health, and welfare of the public as the profession's paramount obligation, which requires promptly raising a discovered safety concern to the parties who can act on it, regardless of who made the original error. Staying silent, waiting for visible failure, or quietly patching the report without disclosure all fail to protect public safety and leave others unable to make informed decisions about the structure.
3. Determine the value of the definite integral of (3x² − 4x + 1) evaluated from x = 0 to x = 2.
- A. 18, from a sign error that turns −4x into +4x in the antiderivative
- B. 10, from an arithmetic slip while combining the evaluated terms
- C. 0, from omitting the constant term when forming the antiderivative
- D. 2, from correctly forming the antiderivative x³ − 2x² + x and evaluating it at the limits
Show answer & explanation
Answer: D
The antiderivative of 3x² − 4x + 1 is x³ − 2x² + x. Evaluating at x = 2 gives 8 − 8 + 2 = 2, and evaluating at x = 0 gives 0, so the definite integral equals 2. Dropping the constant term or flipping the sign of the middle term changes the antiderivative itself, which is why those shortcuts land on values far from the correct result.
4. Two force vectors are given as A = 3i + 4j (N) and B = 4i − 3j (N). What is the angle between vectors A and B?
- A. 90°, from finding that the dot product A·B equals zero
- B. 0°, from assuming the vectors are parallel because they share the same component magnitudes
- C. 45°, from averaging the directions of the two vectors without computing the dot product
- D. 180°, from assuming the vectors point in exactly opposite directions
Show answer & explanation
Answer: A
The dot product A·B = (3)(4) + (4)(−3) = 12 − 12 = 0, and a zero dot product between two nonzero vectors means they are perpendicular, so the angle between them is 90°. Assuming the vectors are parallel or opposite based only on their component magnitudes ignores that the actual directions differ, which the dot-product calculation reveals.
5. During a project, an engineer identifies that specifying a cheaper alloy would save the client money but would reduce the safety margin of a rotating component below accepted design practice. The client insists on the cheaper alloy. What should the engineer do?
- A. Specify the cheaper alloy but omit the reduced safety margin from the calculations
- B. Comply with the client's wishes, since the client is paying for the work
- C. Resign from the project without informing the client of the underlying safety reason
- D. Refuse to proceed with the unsafe specification and explain the safety risk to the client, escalating if necessary
Show answer & explanation
Answer: D
When a client's cost preference would push a design below an accepted safety margin, the engineer's obligation to protect public safety takes priority over simply following instructions, which means raising the concern clearly and escalating rather than complying silently. Quietly omitting the issue from calculations conceals a known hazard, and resigning without explanation removes the engineer from the situation without addressing the risk to anyone who might rely on the design.
6. A mechanical engineer serves as an expert witness in a product liability case. The engineer's honest analysis would not favor the party paying for the testimony. What does professional ethics require of the engineer?
- A. Delay the testimony indefinitely to avoid the conflict
- B. Present the findings objectively and truthfully, regardless of which party they favor
- C. Present only the portions of the analysis that support the client's position
- D. Decline to testify unless the analysis can be adjusted to favor the paying party
Show answer & explanation
Answer: B
Professional ethics requires expert testimony to reflect an objective, truthful technical analysis, independent of who is paying for the work, because the value of expert testimony depends on its impartiality. Selectively presenting only favorable findings, or refusing to testify unless the conclusions can be changed to suit the paying party, both substitute advocacy for the honest technical judgment that the role requires.
7. A piece of production equipment costs $50,000 today and is expected to generate $15,000 per year in savings for 5 years. Using an interest rate of 10% per year and a uniform-series present-worth factor (P/A,10%,5) of 3.7908, what is the present worth of the savings?
- A. $75,000, from multiplying the annual savings by 5 years without discounting
- B. $56,862, from multiplying the annual savings of $15,000 by the given present-worth factor
- C. $37,908, from applying the present-worth factor to an assumed $10,000 annual amount instead of the actual $15,000
- D. $50,000, mistaking the present worth of savings for the equipment's purchase cost
Show answer & explanation
Answer: B
The present worth of a uniform annual series is found by multiplying the annual amount by the present-worth factor for the given interest rate and number of years, so $15,000 × 3.7908 ≈ $56,862. Simply multiplying the annual savings by the number of years ignores the time value of money entirely, and applying the factor to the wrong cash-flow amount produces a present worth that does not correspond to the actual $15,000 savings stream.
8. A piece of equipment is purchased for $80,000 and has an estimated salvage value of $10,000 after a 7-year useful life. Using straight-line depreciation, what is the annual depreciation charge?
- A. $8,750/year, from dividing the depreciable amount by an incorrect 8-year life
- B. $10,000/year, from dividing the depreciable amount of $70,000 by the 7-year life
- C. $11,429/year, from dividing the full purchase price by the useful life without subtracting salvage value
- D. $70,000/year, from failing to divide the depreciable amount by the useful life at all
Show answer & explanation
Answer: B
Straight-line depreciation divides the depreciable base, which is the purchase price minus salvage value ($80,000 − $10,000 = $70,000), by the useful life, giving $70,000 / 7 = $10,000 per year. Forgetting to subtract the salvage value before dividing, or dividing by the wrong number of years, both shift the depreciation charge away from the value that correctly spreads the asset's loss in value over its service life.
9. A resistive heating element draws 4 A of current when connected to a 120 V supply. What is the power dissipated by the element?
- A. 30 W, from dividing the voltage by the current instead of multiplying
- B. 480 W, from multiplying the voltage by the current using P = VI
- C. 124 W, from adding the voltage and current instead of multiplying them
- D. 14,400 W, from squaring the voltage without dividing by resistance
Show answer & explanation
Answer: B
Electrical power delivered to a resistive load is the product of voltage and current, so P = VI = 120 × 4 = 480 W. Dividing instead of multiplying, or adding the two quantities, both misapply the basic power relationship, and squaring the voltage alone ignores that the P = V²/R form requires dividing by resistance, not using voltage squared on its own.
10. A capacitor with a capacitance of 10 μF is charged to 100 V. How much energy is stored in the capacitor, using E = (1/2)CV²?
- A. 0.05 J, from evaluating (1/2)(10×10⁻⁶)(100)²
- B. 0.0005 J, from using V instead of V² in the calculation
- C. 50 J, from mistaking the 10 μF capacitance for 10 mF
- D. 0.10 J, from omitting the factor of one-half in the energy formula
Show answer & explanation
Answer: A
The stored energy is (1/2)CV² = 0.5 × (10×10⁻⁶ F) × (100 V)² = 0.5 × 10×10⁻⁶ × 10,000 = 0.05 J. Dropping the one-half factor doubles the result, using voltage instead of voltage squared understates it drastically, and confusing microfarads with millifarads inflates the capacitance by a factor of 1,000, each producing an energy value inconsistent with the actual charge stored.
11. An RC circuit has a resistance of 2 kΩ and a capacitance of 500 μF. What is the time constant of this circuit?
- A. 2.0 s, from using double the given capacitance value
- B. 1.0 s, from evaluating τ = RC with R and C converted to base SI units
- C. 0.001 s, from failing to convert the resistance from kΩ to ohms
- D. 0.5 s, from using half the given resistance value
Show answer & explanation
Answer: B
The time constant of an RC circuit is τ = RC, and converting to base units gives 2,000 Ω × 500×10⁻⁶ F = 1.0 s. Using an incorrect resistance or capacitance value changes the product directly, and forgetting to convert kilohms to ohms understates the resistance by a factor of 1,000, which understates the time constant by the same factor.
12. A straight conductor carrying a current of 5 A is placed perpendicular to a uniform magnetic field of 0.2 T. If the conductor length within the field is 0.4 m, what is the magnitude of the magnetic force on the conductor, using F = BIL?
- A. 0.4 N, from evaluating F = (0.2)(5)(0.4)
- B. 1.0 N, from using a conductor length of 1 m instead of the given 0.4 m
- C. 4.0 N, from a decimal-placement error in multiplying the three given values
- D. 0.04 N, from a decimal-placement error in the opposite direction
Show answer & explanation
Answer: A
For a current-carrying conductor perpendicular to a magnetic field, the force is F = BIL = 0.2 T × 5 A × 0.4 m = 0.4 N. Misplacing a decimal point while multiplying the three quantities, or substituting an incorrect conductor length, both shift the result away from the value that the given field strength, current, and length actually produce.
13. A component on a rotating machine travels in a circular path of radius 0.3 m at a constant angular velocity of 100 rad/s. What is the magnitude of the centripetal acceleration of the component?
- A. 3000 m/s², from evaluating a = ω²r with the given values
- B. 30 m/s², from using a = ωr instead of a = ω²r
- C. 33,333 m/s², from mistakenly dividing ω² by r instead of multiplying
- D. 300 m/s², from a decimal-placement error in the final multiplication
Show answer & explanation
Answer: A
Centripetal acceleration for circular motion at constant angular velocity is a = ω²r, so a = (100)²(0.3) = 3000 m/s². Using ω instead of ω² understates the result because it omits one full factor of the angular velocity, while dividing by the radius instead of multiplying inverts the role that radius actually plays in the relationship.
14. A composite flat bracket consists of a rectangular plate 6 cm wide by 2 cm tall (area 12 cm², centroid 1 cm from the base) bonded directly beneath a second rectangular plate 6 cm wide by 4 cm tall (area 24 cm², centroid 4 cm from the base). What is the y-coordinate of the centroid of the composite section, measured from the base?
- A. 3.0 cm, from computing the area-weighted average of the two centroid locations
- B. 2.5 cm, from simply averaging the two individual centroid locations without weighting by area
- C. 3.33 cm, from using an incorrect total area of 32 cm² in the denominator
- D. 2.0 cm, from swapping the two areas when applying the weighting
Show answer & explanation
Answer: A
The centroid of a composite area is the area-weighted average of the individual centroids: (12×1 + 24×4)/(12+24) = 108/36 = 3.0 cm. Averaging the two centroid locations without weighting by area, swapping which area goes with which centroid, or dividing by an incorrect total area all distort the weighting and shift the computed centroid away from its true location.
15. A force of 200 N is applied at the end of a 0.5 m wrench handle, directed perpendicular to the handle. What moment does this force create about the bolt at the other end of the handle?
- A. 100 N·m, from multiplying the force by the perpendicular lever arm
- B. 400 N·m, from dividing the force by the lever arm instead of multiplying
- C. 10 N·m, from a decimal-placement error in the multiplication
- D. 25 N·m, from squaring the lever arm and dividing by an unrelated factor
Show answer & explanation
Answer: A
When a force acts perpendicular to a lever arm, the moment it creates equals the force multiplied by the length of that arm, so M = 200 N × 0.5 m = 100 N·m. Dividing the force by the arm length instead of multiplying, or misplacing a decimal point in the calculation, both produce a moment that does not correspond to the actual leverage the handle provides.
16. A simply supported beam 6 m long carries a single concentrated load of 12 kN located 2 m from the left support. What is the reaction force at the right support?
- A. 12 kN, from assuming the right support alone carries the full applied load
- B. 6 kN, from splitting the load evenly between the two supports
- C. 8 kN, which is actually the reaction at the left support rather than the right
- D. 4 kN, from summing moments about the left support and solving for the right reaction
Show answer & explanation
Answer: D
Taking moments about the left support, the right reaction times the 6 m span must balance the 12 kN load acting at its 2 m arm, so R_right = (12 × 2)/6 = 4 kN, leaving 8 kN at the left support. Splitting the load evenly ignores that the load is not centered on the span, and swapping which reaction gets which value confuses the support closer to the load with the one farther away.
17. A projectile is launched horizontally from a height of 20 m above the ground with an initial speed of 15 m/s. Using g = 9.81 m/s² and neglecting air resistance, what is the horizontal distance traveled when the projectile lands?
- A. 40.0 m, from using an incorrect fall-time formula that overstates the time of flight
- B. 60.0 m, from assuming a fall time that is double the correct value
- C. 30.3 m, from finding the correct fall time of about 2.02 s and multiplying by the horizontal speed
- D. 15.0 m, from assuming the fall time is exactly 1 second
Show answer & explanation
Answer: C
The time to fall a height h under gravity, starting with no vertical velocity, is t = sqrt(2h/g) = sqrt(40/9.81) ≈ 2.02 s, and the horizontal distance is the horizontal speed times this time, 15 × 2.02 ≈ 30.3 m. Assuming a fall time of exactly 1 second, or doubling the correct fall time, both replace the actual kinematic result with a value that does not match the given height and gravitational acceleration.
18. A 2 kg ball moving at 6 m/s strikes a wall and rebounds at 4 m/s in the opposite direction. What is the magnitude of the impulse delivered to the ball by the wall?
- A. 4 N·s, from subtracting the two speeds without accounting for the reversal in direction
- B. 20 N·s, from accounting for the reversal in direction when computing the change in momentum
- C. 12 N·s, from using only the incoming momentum and ignoring the rebound
- D. 8 N·s, from using only the rebound momentum and ignoring the incoming motion
Show answer & explanation
Answer: B
Because the ball reverses direction, the change in velocity is the sum of the incoming and outgoing speeds rather than their difference, giving Δv = 6 + 4 = 10 m/s and an impulse magnitude of m·Δv = 2 × 10 = 20 N·s. Subtracting the speeds treats the rebound as if the ball kept moving the same way, and using only one of the two momentum values ignores half of what actually changed during the collision.
19. A machine component of mass 4 kg is mounted on a spring with stiffness 1600 N/m. What is the undamped natural frequency of vibration of this system, in rad/s, using ωn = sqrt(k/m)?
- A. 14.1 rad/s, from using twice the given mass in the calculation
- B. 0.05 rad/s, from inverting the ratio of stiffness to mass before taking the square root
- C. 400 rad/s, from omitting the square root in the calculation
- D. 20 rad/s, from evaluating sqrt(1600/4)
Show answer & explanation
Answer: D
The undamped natural frequency is the square root of stiffness divided by mass, so ωn = sqrt(1600/4) = sqrt(400) = 20 rad/s. Skipping the square root leaves the ratio itself rather than a frequency, and inverting the ratio or substituting an incorrect mass both change the quantity under the root to something that no longer represents this system's actual stiffness-to-mass relationship.
20. A bolt with a diameter of 20 mm carries a direct shear load of 31.4 kN across its cross-section. What is the average shear stress in the bolt?
- A. 100 MPa, from dividing the shear load by the correct circular cross-sectional area
- B. 200 MPa, from a factor-of-two error in computing the cross-sectional area
- C. 78.5 MPa, from using d² instead of the correct circular area πd²/4 in the denominator
- D. 25 MPa, from mistakenly treating the diameter as a radius when computing the area
Show answer & explanation
Answer: A
The bolt's circular cross-sectional area is πd²/4 = π(20)²/4 ≈ 314.2 mm², so the average shear stress is 31,400 N / 314.2 mm² ≈ 100 MPa. Using d² in place of the full circular-area formula overstates the effective area's denominator relationship, and doubling the diameter as if it were a radius produces an area four times too large, both of which shift the computed stress well away from the correct value.
21. A rectangular beam cross-section is 100 mm wide and 200 mm deep, subjected to a bending moment of 30 kN·m. What is the maximum bending stress in the beam, using σ = Mc/I with I = bh³/12?
- A. 45 MPa, from computing I = bh³/12 and using c equal to half the section depth
- B. 180 MPa, from swapping the base and height values when computing the moment of inertia
- C. 22.5 MPa, from using bh³/6 in place of the correct bh³/12 for the moment of inertia
- D. 90 MPa, from using the full section depth as c instead of half the depth
Show answer & explanation
Answer: A
With b = 100 mm and h = 200 mm, I = bh³/12 = 66.67×10⁶ mm⁴, and the distance from the neutral axis to the outer fiber is c = h/2 = 100 mm, so σ = Mc/I = (30×10⁶ N·mm)(100)/(66.67×10⁶) = 45 MPa. Using the full depth as c instead of half of it, using an incorrect denominator of 6 instead of 12, or swapping which dimension is cubed all change the section's computed bending resistance away from its actual value.
22. A machine part is designed with an allowable stress of 150 MPa. If the material's ultimate tensile strength is 450 MPa, what is the factor of safety based on ultimate strength?
- A. 0.33, from inverting the ratio of allowable stress to ultimate strength
- B. 3, from dividing the ultimate strength by the allowable stress
- C. 1.5, from an arithmetic slip that halves the correct ratio
- D. 300, from subtracting the two stresses instead of dividing them
Show answer & explanation
Answer: B
Factor of safety based on ultimate strength is the ultimate strength divided by the allowable (design) stress, so FS = 450/150 = 3. Inverting the ratio produces a value less than one, which would incorrectly suggest the design stress exceeds the material's strength, and subtracting the two stresses instead of dividing them changes the comparison from a ratio to a difference, which is not what factor of safety measures.
23. During a tensile test, a specimen experiences a stress of 210 MPa and a corresponding strain of 0.00105 within the linear elastic region. What is the modulus of elasticity of this material, and which common engineering material does that value most closely correspond to?
- A. 70 GPa, consistent with aluminum, from using an incorrect strain value in the division
- B. 400 GPa, exceeding typical structural metals, from a factor-of-two error in the division
- C. 2 GPa, consistent with a polymer, from a decimal-placement error in the division
- D. 200 GPa, consistent with steel, from dividing the given stress by the given strain
Show answer & explanation
Answer: D
Within the linear elastic region, modulus of elasticity is stress divided by strain, so E = 210 MPa / 0.00105 = 200,000 MPa = 200 GPa, a value typical of steel. Substituting a different strain value or introducing a decimal or factor-of-two slip in the division produces a modulus that does not match either the given data or a realistic structural material.
24. A medium-carbon steel part is heated above its critical transformation temperature and then rapidly quenched in water, rather than being furnace-cooled slowly. What is the primary effect of this rapid quenching on the steel's microstructure and resulting mechanical properties?
- A. It causes the steel to fully anneal, softening the material uniformly
- B. It has no effect on microstructure, since cooling rate does not influence phase transformation
- C. It promotes formation of coarse pearlite, increasing ductility and reducing hardness
- D. It promotes formation of martensite, increasing hardness and strength but reducing ductility
Show answer & explanation
Answer: D
Rapid quenching from above the critical transformation temperature suppresses the diffusion-controlled formation of pearlite and instead produces the hard, brittle martensitic microstructure, raising hardness and strength while lowering ductility. Slow furnace cooling, by contrast, allows coarser, softer microstructures such as pearlite to form, which is the opposite of what quenching produces, and cooling rate is in fact a primary control variable in heat treatment rather than something without effect.
25. A cast iron component fractures suddenly with little visible plastic deformation prior to failure, while a low-carbon steel component of similar geometry necks visibly before fracturing under similar loading. What does this comparison best illustrate?
- A. Cast iron has a higher ductility than low-carbon steel
- B. Both materials exhibit identical fracture toughness because they are both iron-based alloys
- C. Cast iron behaves in a brittle manner while the low-carbon steel behaves in a ductile manner
- D. The steel component failed due to brittle fracture while the cast iron exhibited ductile yielding
Show answer & explanation
Answer: C
The absence of visible plastic deformation before fracture is the hallmark of brittle behavior, which describes the cast iron in this comparison, while the visible necking before fracture in the steel is characteristic of ductile behavior. Reversing which material displays which behavior, or assuming that sharing an iron base guarantees identical fracture response, both contradict the described failure behavior of each part.
26. Two identical-geometry connecting rods are produced, one by sand casting and one by hot forging. Assuming no other defects, which statement best reflects the expected difference in mechanical properties between the two parts?
- A. The cast rod is stronger, because casting always produces a finer grain size than forging
- B. Both parts have identical mechanical properties because both are made from the same base alloy
- C. The cast rod generally has superior fatigue resistance because casting produces a more uniform grain structure than forging
- D. The forged rod generally has superior strength and fatigue resistance due to grain refinement and reduced internal porosity
Show answer & explanation
Answer: D
Hot forging plastically deforms the metal, refining and aligning its grain structure along the load path and eliminating the shrinkage porosity that can occur during casting solidification, which typically gives forged parts better strength and fatigue resistance than cast parts of the same alloy and geometry. Claiming the cast part has finer grains, more uniform structure, or identical properties overlooks that the two processes produce fundamentally different internal microstructures even from the same starting alloy.
27. Water flows through a pipe that narrows from a diameter of 100 mm to 50 mm. If the velocity in the larger section is 2 m/s, what is the velocity in the smaller section, using the continuity equation A1V1 = A2V2?
- A. 2 m/s, from assuming velocity stays constant because volumetric flow rate is constant
- B. 16 m/s, from scaling velocity by the cube of the diameter ratio instead of the square
- C. 8 m/s, from scaling velocity by the square of the diameter ratio, matching the ratio of cross-sectional areas
- D. 4 m/s, from scaling velocity by the ratio of diameters instead of the ratio of areas
Show answer & explanation
Answer: C
Continuity requires the volumetric flow rate to stay constant, so velocity scales with the inverse ratio of cross-sectional areas, which for a pipe scales with the square of the diameter ratio: (100/50)² = 4, giving V2 = 2 × 4 = 8 m/s. Scaling by the diameter ratio directly, rather than its square, understates how much the flow must speed up, and assuming velocity stays constant ignores that a smaller area requires a proportionally faster flow to carry the same volume per second.
28. Water flows through a horizontal pipe at 3 m/s where the pressure is 200 kPa. The pipe narrows to a section where the velocity increases to 6 m/s at the same elevation. Using Bernoulli's equation and a water density of 1000 kg/m³, what is the pressure in the narrower section?
- A. 200 kPa, from assuming pressure is unaffected by the change in velocity
- B. 186.5 kPa, from applying Bernoulli's equation and accounting for the increase in velocity head
- C. 213.5 kPa, from a sign error that adds the velocity-head term instead of subtracting it
- D. 173 kPa, from omitting the one-half factor in the velocity-head term
Show answer & explanation
Answer: B
For horizontal flow with no elevation change, Bernoulli's equation gives P2 = P1 + (1/2)ρ(V1² − V2²) = 200,000 + 0.5(1000)(9 − 36) = 200,000 − 13,500 = 186,500 Pa ≈ 186.5 kPa. Adding the velocity-head term instead of subtracting it reverses the effect that speeding up should have on pressure, assuming pressure is unaffected ignores Bernoulli's trade-off between velocity and pressure entirely, and omitting the one-half factor doubles the size of that trade-off beyond what the equation actually predicts.
29. Water (density 1000 kg/m³, dynamic viscosity 0.001 Pa·s) flows through a 50 mm diameter pipe at an average velocity of 2 m/s. What is the Reynolds number of this flow, and is the flow laminar or turbulent?
- A. Re ≈ 100,000; laminar, from computing the correct Reynolds number but drawing the wrong flow-regime conclusion
- B. Re ≈ 100; laminar, from omitting the density term when computing Reynolds number
- C. Re ≈ 100,000; turbulent, from evaluating Re = ρVD/μ and comparing it to the typical transition threshold
- D. Re ≈ 10,000; turbulent, from a decimal-placement error of a factor of 10 in the calculation
Show answer & explanation
Answer: C
Reynolds number is Re = ρVD/μ = (1000)(2)(0.05)/0.001 = 100,000, and since this value is far above the typical transition range of about 2,100 to 4,000, the flow is turbulent. Leaving out the density term drops a factor of 1,000 from the calculation, a decimal error shifts the result by a factor of 10, and correctly computing the Reynolds number but then labeling it laminar ignores that a value this high is well beyond the laminar regime.
30. A pump delivers water at a flow rate of 0.02 m³/s against a total head of 30 m. Using a water density of 1000 kg/m³ and g = 9.81 m/s², what is the hydraulic power required, using P = ρgQH?
- A. 5.89 kW, from evaluating P = ρgQH with the given values
- B. 11.8 kW, from using a total head of 60 m instead of the given 30 m
- C. 58.9 kW, from a decimal-placement error of a factor of 10
- D. 0.6 kW, from omitting the gravitational acceleration term entirely
Show answer & explanation
Answer: A
Hydraulic pump power is P = ρgQH = (1000)(9.81)(0.02)(30) ≈ 5886 W ≈ 5.89 kW. Leaving out the gravitational acceleration term removes a necessary factor of about 9.81 from the product, a decimal-placement slip shifts the result by a factor of 10, and doubling the head value directly doubles the computed power, none of which reflect the actual flow rate and head given.
31. A rigid tank contains an ideal gas at 300 K and 200 kPa. If the tank is heated at constant volume until the pressure reaches 250 kPa, what is the final temperature of the gas?
- A. 375 K, from scaling the initial temperature by the ratio of final to initial pressure
- B. 240 K, from inverting the pressure ratio used to scale the initial temperature
- C. 300 K, from assuming temperature stays constant simply because volume is held constant
- D. 350 K, from adding a fixed increment rather than applying a proportional scaling
Show answer & explanation
Answer: A
At constant volume for an ideal gas, pressure and absolute temperature are directly proportional, so T2 = T1(P2/P1) = 300 × (250/200) = 375 K. Inverting the pressure ratio would imply the temperature dropped as pressure rose, which contradicts the direct proportionality, and assuming the temperature is unchanged confuses constant volume with constant temperature, which are different conditions entirely.
32. A power plant operating on a Rankine cycle rejects 600 kJ/kg of heat in the condenser while the boiler adds 1000 kJ/kg of heat to the working fluid. What is the thermal efficiency of the cycle?
- A. 400%, from using the net work value directly as a percentage without dividing by the heat added
- B. 40%, from computing net work as heat added minus heat rejected and dividing by heat added
- C. 167%, from inverting the ratio of heat added to net work
- D. 60%, from reporting the fraction of heat rejected as if it were the thermal efficiency
Show answer & explanation
Answer: B
Thermal efficiency is net work output divided by heat input, and net work equals heat added minus heat rejected, so η = (1000 − 600)/1000 = 0.40, or 40%. Reporting the heat-rejected fraction (600/1000 = 0.60) mistakes what fraction of energy is wasted for what fraction is converted to useful work, and failing to divide the net work value by the heat input leaves a raw energy number rather than the efficiency ratio the question asks for.
33. A flat wall 0.2 m thick has a thermal conductivity of 0.8 W/(m·K) and a cross-sectional area of 10 m². If the temperature difference across the wall is 25°C, what is the rate of heat conduction through the wall, using Fourier's law Q = kAΔT/L?
- A. 1000 W, from evaluating Q = kAΔT/L with the given values
- B. 40 W, from multiplying by the wall thickness instead of dividing by it
- C. 500 W, from using a wall thickness of 0.4 m instead of the given 0.2 m
- D. 100 W, from omitting the area term from the calculation
Show answer & explanation
Answer: A
Fourier's law for steady one-dimensional conduction gives Q = kAΔT/L = (0.8)(10)(25)/0.2 = 200/0.2 = 1000 W. Leaving out the area term removes a factor of 10 from the numerator, multiplying by the thickness instead of dividing by it inverts the role that a thicker wall should play in reducing heat flow, and doubling the thickness value directly halves the computed heat-transfer rate relative to the correct case.
34. A hot plate with a surface area of 2 m² and a surface temperature of 80°C is cooled by air at 20°C, with a convective heat transfer coefficient of 15 W/(m²·K). What is the rate of convective heat loss from the plate, using Q = hAΔT?
- A. 120 W, from omitting the convection coefficient from the calculation
- B. 900 W, from using half of the actual temperature difference
- C. 2400 W, from using the plate's absolute surface temperature instead of the temperature difference
- D. 1800 W, from evaluating Q = hAΔT with the correct temperature difference of 60°C
Show answer & explanation
Answer: D
The temperature difference driving convection is 80 − 20 = 60°C, so Q = hAΔT = (15)(2)(60) = 1800 W. Using half of that temperature difference understates the driving potential for heat transfer, dropping the convection coefficient removes the factor that links surface conditions to the actual heat-loss rate, and substituting the plate's absolute surface temperature for the temperature difference confuses an absolute condition with the relative difference that convection actually depends on.
35. Air with a mass flow rate of 2 kg/s is heated in a duct from 15°C to 35°C. Using a specific heat of air of 1.005 kJ/(kg·K), what is the rate of heat addition required?
- A. 40.2 kW, from evaluating Q = m·cp·ΔT with the given values
- B. 2.01 kW, from a decimal-placement error of a factor of 10
- C. 80.4 kW, from using double the actual temperature rise
- D. 20.1 kW, from using half of the actual temperature rise
Show answer & explanation
Answer: A
The rate of sensible heat addition is Q = m·cp·ΔT = (2)(1.005)(20) = 40.2 kW, using the actual temperature rise of 35 − 15 = 20°C. Using half or double that temperature rise scales the result proportionally in the wrong direction, since the relationship between heat rate and temperature change is linear, and a decimal-placement slip changes the magnitude without reflecting the actual flow rate or specific heat given.
36. A pressure transducer has a stated accuracy of ±0.5% of its full-scale range of 200 kPa. What is the maximum expected measurement error, in kPa, for a reading taken anywhere within this transducer's range?
- A. 1.0 kPa, from applying the stated percentage to the full-scale range
- B. 2.0 kPa, from doubling the correctly computed error value
- C. 0.5 kPa, from an arithmetic slip while applying the percentage to the full-scale range
- D. 10 kPa, from treating the stated 0.5% accuracy as if it were 5%
Show answer & explanation
Answer: A
An accuracy specification given as a percentage of full-scale range applies that percentage to the range itself, not to the individual reading, so the maximum error is 0.5% × 200 kPa = 1.0 kPa regardless of where in the range the reading falls. Misreading the stated percentage as ten times larger, or making an arithmetic slip in applying it, both produce an error bound that overstates or understates what the manufacturer's specification actually guarantees.
37. A batch of machined shafts has a 4% probability of failing a dimensional inspection, independent of one another. In a sample of 5 shafts, what is the probability that exactly one shaft fails inspection?
- A. 0.200, using only 5 × 0.04 and omitting the probability that the other four pass
- B. 0.037, from mistakenly using (0.04)^1(0.96)^0 without the combinatorial and remaining-shaft terms
- C. 0.170, using the binomial probability C(5,1)(0.04)^1(0.96)^4
- D. 0.815, the probability that zero shafts fail, mistaken for the probability that exactly one fails
Show answer & explanation
Answer: C
The binomial formula requires multiplying the number of ways to choose which shaft fails, C(5,1) = 5, by the probability that shaft fails (0.04) and the probability the other four pass (0.96^4 ≈ 0.849), giving 5 × 0.04 × 0.849 ≈ 0.170. Simply multiplying the sample size by the failure rate ignores that the remaining shafts must all pass, which is why that shortcut overstates the true probability.
38. Five tensile test specimens yield ultimate strengths of 60, 62, 58, 64, and 61 ksi. What is the sample standard deviation of these results, using n−1 in the denominator?
- A. 1.58 ksi, from dividing the sum of squared deviations by 2n before the square root
- B. 2.24 ksi, from dividing the sum of squared deviations by n−1 = 4 and taking the square root
- C. 4.47 ksi, from taking the square root of the sum of squared deviations without dividing by n−1
- D. 2.00 ksi, from dividing the sum of squared deviations by n instead of n−1 before the square root
Show answer & explanation
Answer: B
The mean of the five values is 61 ksi, and the squared deviations from the mean sum to 20 ksi². Dividing by n−1 = 4 gives a variance of 5 ksi², and the square root of 5 is approximately 2.24 ksi. Skipping the division before taking the square root, or dividing by the wrong denominator, produces a value that no longer represents the sample's actual spread.
39. A linear regression relating applied load (x) to measured deflection (y) for a beam yields a correlation coefficient r = 0.98. What does this value indicate about the relationship between load and deflection in the tested range?
- A. Only 2% of the variation in deflection is explained by the load
- B. The relationship is strong but must be quadratic rather than linear
- C. There is a strong, nearly linear positive relationship between load and deflection
- D. Deflection is the cause of the applied load increasing, rather than the reverse
Show answer & explanation
Answer: C
A correlation coefficient close to +1, such as 0.98, indicates that the two variables move together in a strong, nearly linear positive fashion across the tested range. Confusing r with the unexplained variance, or assuming correlation implies a specific causal direction or a nonlinear form, misreads what the statistic actually measures.
40. An engineer is asked by a client to sign and seal drawings for a mechanical system outside the engineer's area of competence, in exchange for a bonus fee. What is the ethically and professionally correct response?
- A. Accept the bonus and sign, since the client assumes all liability for the design
- B. Sign the drawings but add a disclaimer limiting personal liability
- C. Delegate the sealing to a junior engineer without reviewing the work
- D. Decline to sign, since engineers should only practice and attest to work in areas where they are qualified by education and experience
Show answer & explanation
Answer: D
Professional practice standards require engineers to perform and certify only work that falls within their qualifications, so accepting a fee to seal unfamiliar work is improper regardless of any disclaimer or how liability is framed. Adding a disclaimer does not restore the missing competence, and passing the responsibility to an unreviewed junior engineer does not relieve the sealing engineer of accountability for the attestation.
41. A manufacturer is choosing between two machines. Machine A has a fixed cost of $20,000/year and a variable cost of $8/unit. Machine B has a fixed cost of $35,000/year and a variable cost of $5/unit. At what annual production volume do the two machines have equal total cost?
- A. 3,000 units, from using a variable-cost difference of $5 instead of the actual $3 difference
- B. 5,000 units, from setting 20,000 + 8Q equal to 35,000 + 5Q and solving for Q
- C. 15,000 units, from dividing only the fixed-cost difference by the variable cost of Machine A
- D. 1,875 units, from dividing the sum of the fixed costs by the sum of the variable costs
Show answer & explanation
Answer: B
Setting the two total-cost equations equal, 20,000 + 8Q = 35,000 + 5Q, gives 3Q = 15,000, so Q = 5,000 units is the break-even volume. Using the wrong variable-cost difference or combining the fixed and variable costs by addition instead of solving the break-even equation both produce a volume that does not actually equalize the two machines' total costs.
42. An engineer invests $10,000 today in a fund that grows at an annual compound interest rate of 8%. Approximately how much will the investment be worth after 10 years, using F = P(1+i)^n?
- A. $25,937, from using a 10% growth rate instead of the given 8% rate
- B. $21,589, from evaluating 10,000 × (1.08)^10
- C. $10,800, from applying only a single year of growth instead of 10 years
- D. $18,000, from computing simple rather than compound interest
Show answer & explanation
Answer: B
Compound growth requires raising the growth factor (1 + i) to the power of the number of periods, so 10,000 × (1.08)^10 ≈ $21,589. Computing simple interest instead of compounding, applying growth for only one period, or substituting a different interest rate all produce future values that do not correspond to 10 years of compounding at the stated 8% rate.
43. Two resistors, 4 Ω each, are connected in parallel across a 12 V source. What is the total current supplied by the source?
- A. 0.75 A, from multiplying the two resistances together instead of applying the parallel-combination formula
- B. 1.5 A, from treating the resistors as connected in series rather than in parallel
- C. 6 A, from combining the resistors into an equivalent parallel resistance of 2 Ω
- D. 3 A, from using a single 4 Ω resistor as if it were the equivalent resistance
Show answer & explanation
Answer: C
For two equal resistors in parallel, the equivalent resistance is half of one resistor's value, so 4 Ω combined in parallel gives 2 Ω, and the current is I = V/R = 12/2 = 6 A. Adding the resistances as if they were in series, or using only one resistor's value as the equivalent, both produce an equivalent resistance that does not reflect how current actually splits between two parallel paths.
44. A pin-jointed truss joint has two members meeting at right angles: one horizontal carrying 300 N of pull on the joint and one vertical carrying 400 N of pull on the joint, with no other loads applied. For the joint to remain in equilibrium, what must be the magnitude of the single additional force needed to balance it?
- A. 100 N, from taking the difference between the two member forces
- B. 500 N, from combining the 300 N and 400 N forces as perpendicular vector components
- C. 350 N, from averaging the two member forces
- D. 700 N, from adding the two member forces directly instead of combining them as vectors
Show answer & explanation
Answer: B
Because the two member forces act at right angles, their combined effect on the joint is the vector resultant, found from the Pythagorean relationship sqrt(300² + 400²) = 500 N, and the balancing force must equal this resultant in magnitude. Simply adding, subtracting, or averaging the two force magnitudes ignores that they act along perpendicular directions rather than along the same line.
45. A 50 kg crate rests on a plane inclined at 20° to the horizontal, with a coefficient of static friction of 0.3 between the crate and the plane. Based on comparing the tangent of the incline angle to the coefficient of static friction, will the crate remain in equilibrium without additional restraint?
- A. No, because tan(20°) ≈ 0.36 exceeds the coefficient of static friction of 0.3, so the crate will slide
- B. Yes, because the normal force is always sufficient to prevent sliding on any incline
- C. No, but only because the crate's mass is too large for friction to act
- D. Yes, because the coefficient of friction exceeds the sine of the incline angle
Show answer & explanation
Answer: A
A block on an incline stays in place without additional restraint only if the coefficient of static friction is at least as large as the tangent of the incline angle; here tan(20°) ≈ 0.36 is greater than the given coefficient of 0.3, so friction alone cannot hold the crate and it will slide. Comparing friction to the sine of the angle instead of the tangent, or assuming friction is always sufficient regardless of geometry, both misstate the actual equilibrium condition, and mass itself does not change whether the ratio of forces permits sliding.
46. Three forces act concurrently at a point: F1 = 100 N at 0°, F2 = 100 N at 90°, and F3 acts at 225° (measured from the positive x-axis). For the point to be in static equilibrium, what must be the magnitude of F3?
- A. 70.7 N, from taking half of the resultant of F1 and F2
- B. 100 N, from assuming F3 needs to balance only one of the two other forces
- C. 141.4 N, from matching the magnitude of the resultant of F1 and F2
- D. 200 N, from simply summing the magnitudes of F1 and F2
Show answer & explanation
Answer: C
Since F1 and F2 act at right angles, their resultant has magnitude sqrt(100² + 100²) ≈ 141.4 N, and for equilibrium F3 must be equal in magnitude and opposite in direction to that resultant, which is consistent with the stated 225° direction. Balancing only one force, simply adding the two magnitudes, or halving the resultant all fail to account for the actual vector sum that F3 must cancel.
47. A car accelerates uniformly from rest to 25 m/s in 8 seconds. What distance does the car travel during this time?
- A. 200 m, from using the final velocity directly instead of the average velocity
- B. 12.5 m, from reporting the average velocity value alone as if it were the distance
- C. 50 m, from an arithmetic error while combining average velocity and time
- D. 100 m, from multiplying the average velocity by the elapsed time
Show answer & explanation
Answer: D
For uniform acceleration from rest, the average velocity is half the final velocity, or 12.5 m/s, and distance equals average velocity times time, so 12.5 × 8 = 100 m. Using the final velocity instead of the average overstates the distance, since the car was traveling slower than 25 m/s for most of the interval.
48. A 1000 kg car traveling at 20 m/s brakes to a stop over a distance of 50 m. What average braking force is required, using the work-energy principle?
- A. 4000 N, from equating the car's kinetic energy to the work done by the braking force over the stopping distance
- B. 400 N, from a decimal-placement error in the final division
- C. 8000 N, from omitting the one-half factor in the kinetic energy expression
- D. 10000 N, from using an incorrect stopping distance of 20 m instead of the given 50 m
Show answer & explanation
Answer: A
The car's kinetic energy is (1/2)(1000)(20²) = 200,000 J, and by the work-energy principle this energy equals the braking force times the stopping distance, so F = 200,000/50 = 4000 N. Omitting the one-half factor in the kinetic-energy expression doubles the apparent energy, and substituting the wrong distance value changes the denominator used to back out the force.
49. A solid disk flywheel with a mass moment of inertia of 5 kg·m² is subjected to a net torque of 15 N·m. What is the resulting angular acceleration of the flywheel?
- A. 0.33 rad/s², from inverting the ratio of inertia to torque
- B. 75 rad/s², from multiplying torque and inertia instead of dividing
- C. 3 rad/s², from dividing the net torque by the mass moment of inertia
- D. 10 rad/s², from an arithmetic slip in the division
Show answer & explanation
Answer: C
The rotational analog of Newton's second law states that angular acceleration equals net torque divided by mass moment of inertia, so α = 15/5 = 3 rad/s². Multiplying the two quantities instead of dividing, or inverting the ratio, both misapply the relationship and produce a value that does not describe how quickly this flywheel actually speeds up under the given torque.
50. A steel rod with a cross-sectional area of 500 mm² is subjected to an axial tensile load of 100 kN. What is the normal stress in the rod?
- A. 50 MPa, from using an incorrect area of 2000 mm² in the calculation
- B. 200 MPa, from dividing the axial load by the given cross-sectional area
- C. 20 MPa, from a decimal-placement error in the division
- D. 500 MPa, from using an incorrect area of 200 mm² in the calculation
Show answer & explanation
Answer: B
Normal stress under axial loading is force divided by cross-sectional area, so σ = 100,000 N / 500 mm² = 200 N/mm² = 200 MPa. Substituting an area that is too large or too small directly shifts the computed stress away from the value that this rod's actual dimensions and load would produce.
51. A solid circular shaft with a diameter of 40 mm transmits a torque of 500 N·m. What is the maximum shear stress at the outer surface of the shaft, using τ = 16T/(πd³)?
- A. 1.59 MPa, from using d² instead of d³ in the denominator
- B. 19.9 MPa, from using a factor of 8 in the numerator instead of the correct factor of 16
- C. 79.6 MPa, from using a factor of 32 in the numerator instead of the correct factor of 16
- D. 39.8 MPa, from evaluating the formula with all quantities in consistent units
Show answer & explanation
Answer: D
Converting the diameter to meters and evaluating τ = 16(500)/(π(0.04)³) gives approximately 39.8 MPa (equivalently 16T/(πd³) with d in mm and T in N·mm gives the same result). Using the wrong numerical factor in the numerator, as would apply to a different loading case, or substituting d² for d³ in the denominator, both produce a stress value that does not match the actual cubic dependence of torsional stress on shaft diameter.
52. A steel bar is rigidly fixed at both ends so it cannot expand or contract. If the bar's temperature increases by 40°C, and steel has a coefficient of thermal expansion of about 12×10⁻⁶/°C and a modulus of elasticity of 200 GPa, what is the magnitude of the thermal stress induced in the bar?
- A. 96 MPa, from evaluating σ = EαΔT with the given values
- B. 192 MPa, from using a temperature change of 80°C instead of the given 40°C
- C. 48 MPa, from using a temperature change of 20°C instead of the given 40°C
- D. 9.6 MPa, from a decimal-placement error of a factor of 10
Show answer & explanation
Answer: A
For a bar fully restrained against thermal expansion, the induced stress is σ = EαΔT = (200×10⁹)(12×10⁻⁶)(40) = 96×10⁶ Pa = 96 MPa. Using half or double the actual temperature change directly scales the result proportionally, since the relationship is linear in ΔT, which is why those substitutions land exactly at half or double the correct stress.
53. At a point in a loaded machine component, the stress state consists of a normal stress of 80 MPa (tension) and a shear stress of 30 MPa on the same plane, with no stress on the perpendicular plane. What is the maximum principal stress at this point, using σ1 = σx/2 + sqrt((σx/2)² + τxy²)?
- A. 110 MPa, from simply summing the given normal and shear stresses
- B. 90 MPa, from evaluating 40 + sqrt(40² + 30²)
- C. 10 MPa, from computing the minimum rather than the maximum principal stress
- D. 50 MPa, from omitting the σx/2 term and using only the square-root term
Show answer & explanation
Answer: B
With σx/2 = 40 MPa and τxy = 30 MPa, the square-root term is sqrt(40² + 30²) = sqrt(2500) = 50 MPa, so σ1 = 40 + 50 = 90 MPa. Simply adding the given normal and shear stresses ignores the actual principal-stress formula, dropping the σx/2 term leaves only the radius of the Mohr's-circle construction rather than the full principal stress, and subtracting instead of adding the two terms yields the minimum principal stress (−10 MPa in magnitude 10 MPa) rather than the maximum.
54. A machine shaft is subjected to a cyclic stress amplitude below the material's endurance limit. Based on typical fatigue behavior described by an S-N curve, what is the expected outcome for the shaft's fatigue life?
- A. The shaft will fail immediately, because any cyclic stress causes fatigue cracking
- B. The shaft will fail after exactly 10³ cycles regardless of the stress level
- C. The shaft's fatigue life is independent of stress amplitude once the stress is below the ultimate tensile strength
- D. The shaft can theoretically withstand an essentially infinite number of cycles without fatigue failure
Show answer & explanation
Answer: D
Many ferrous materials exhibit an endurance limit on their S-N curve, below which the material can sustain cyclic loading for an essentially unlimited number of cycles without fatigue failure. Assuming failure at a fixed cycle count regardless of stress level, or assuming any cyclic loading causes immediate cracking, both mischaracterize the S-N relationship, which shows fatigue life decreasing as stress amplitude increases above the endurance limit rather than being constant or immediate.
55. A tank contains water (density 1000 kg/m³) to a depth of 4 m. Using g = 9.81 m/s², what is the gauge pressure at the bottom of the tank?
- A. 39.2 kPa, from evaluating P = ρgh with the given values
- B. 392 kPa, from a decimal-placement error of a factor of 10
- C. 4000 kPa, from a unit-handling error that omits gravitational acceleration and misapplies units
- D. 9.81 kPa, from forgetting to multiply by the depth
Show answer & explanation
Answer: A
Gauge pressure at a depth in a static fluid is P = ρgh = 1000 × 9.81 × 4 = 39,240 Pa ≈ 39.2 kPa. Leaving out the depth term entirely, or mishandling the units so that gravitational acceleration is effectively dropped, both produce a pressure value disconnected from the actual product of density, gravity, and depth.
56. A sphere of frontal area 0.02 m² moves through air (density 1.2 kg/m³) at a velocity of 20 m/s, with a drag coefficient of 0.5. What is the drag force on the sphere, using F_D = 0.5·Cd·ρ·A·V²?
- A. 4.8 N, from omitting the leading factor of 0.5 in the drag equation
- B. 2.0 N, from omitting the air density term from the calculation
- C. 2.4 N, from evaluating the drag equation with all given values correctly substituted
- D. 0.12 N, from using velocity instead of velocity squared in the calculation
Show answer & explanation
Answer: C
Substituting directly into the drag equation gives F_D = 0.5(0.5)(1.2)(0.02)(20²) = 0.5(0.5)(1.2)(0.02)(400) = 2.4 N. Dropping the leading one-half factor doubles the result, using velocity rather than velocity squared drastically understates the speed-dependent term, and omitting the density term removes a factor that scales the whole expression, each producing a force that does not match what these specific conditions actually generate.
57. A solid block with a volume of 0.05 m³ and a density of 600 kg/m³ is fully submerged in water (density 1000 kg/m³). Using g = 9.81 m/s², what is the net vertical force on the block due to buoyancy and gravity combined?
- A. 196.2 N upward, from subtracting the block's weight from the buoyant force computed with the water's density
- B. 294.3 N downward, from accounting only for the block's weight and ignoring buoyancy
- C. 490.5 N upward, from accounting only for the buoyant force and ignoring the block's weight
- D. 98.1 N upward, from using an incorrect density difference of 200 kg/m³ instead of the actual 400 kg/m³
Show answer & explanation
Answer: A
The buoyant force equals the weight of displaced water, 1000 × 0.05 × 9.81 = 490.5 N upward, while the block's own weight is 600 × 0.05 × 9.81 = 294.3 N downward, so the net force is 490.5 − 294.3 = 196.2 N upward. Considering only the weight or only the buoyant force ignores that both act simultaneously on a submerged object, and using an incorrect density difference changes the effective net force without reflecting the actual 400 kg/m³ gap between water and the block.
58. A closed system receives 500 kJ of heat while performing 200 kJ of work on its surroundings. What is the change in internal energy of the system, using ΔU = Q − W?
- A. 300 kJ, from evaluating ΔU = Q − W with the given values
- B. 700 kJ, from adding the heat and work values instead of subtracting
- C. −300 kJ, from a sign error that computes W − Q instead of Q − W
- D. 100 kJ, from an arithmetic slip that doubles the work term before subtracting
Show answer & explanation
Answer: A
The first law of thermodynamics for a closed system gives ΔU = Q − W = 500 − 200 = 300 kJ, meaning the system's internal energy increases by that amount. Adding the two quantities instead of subtracting overstates the internal energy change, and reversing the sign of the subtraction produces a negative value that would incorrectly suggest the system's energy decreased despite receiving more heat than the work it performed.
59. An engineer proposes a heat engine that receives 1000 kJ of heat from a hot reservoir and converts all of it into work with no heat rejected to a cold reservoir. What does the second law of thermodynamics say about this proposed engine?
- A. It is possible as long as the engine operates reversibly
- B. It is possible only if the engine is operating exactly at its Carnot efficiency
- C. It is impossible only because friction always exists in real machines, not because of any fundamental thermodynamic limit
- D. It is impossible, since no heat engine operating in a cycle can convert heat completely into work without rejecting some heat to a lower-temperature reservoir
Show answer & explanation
Answer: D
The second law of thermodynamics establishes that no cyclic heat engine can convert all of the heat it receives into work; some heat must always be rejected to a lower-temperature reservoir, a limitation that is fundamental and not merely a consequence of friction or other real-world imperfections. Reversibility improves efficiency toward the Carnot limit but does not eliminate this requirement, and even a perfectly reversible Carnot engine still rejects some heat unless the cold reservoir is at absolute zero.
60. In a counterflow heat exchanger, hot water enters at 90°C and leaves at 50°C, while cold water enters at 20°C and leaves at 60°C. Both streams have equal mass flow rate and specific heat. What can be concluded about the heat exchanger's energy balance?
- A. The heat exchanger must be operating in parallel flow rather than counterflow, based on these temperatures
- B. The energy balance is satisfied, because both streams experience the same 40°C temperature change, consistent with their equal mass flow rates and specific heats
- C. The energy balance is violated because the cold stream's exit temperature exceeds the hot stream's exit temperature
- D. The energy balance cannot be evaluated without knowing the overall heat transfer coefficient
Show answer & explanation
Answer: B
An energy balance across a heat exchanger requires that the heat lost by the hot stream equal the heat gained by the cold stream; here the hot stream drops by 40°C and the cold stream rises by 40°C, and since both streams have equal mass flow rate and specific heat, this matching temperature change confirms the balance holds. Comparing the streams' exit temperatures to each other, rather than comparing each stream's own temperature change, misapplies the energy-balance requirement, and neither the flow arrangement nor the overall heat transfer coefficient is needed to confirm that energy is conserved between the two streams.
61. A simple proportional feedback control system is used to regulate the speed of a motor. Increasing the proportional gain of the controller generally has what effect on the system's steady-state error and stability margin?
- A. Gain has no effect on either steady-state error or stability in a feedback system
- B. Steady-state error always increases while stability always improves, regardless of the gain value
- C. Steady-state error tends to decrease, but excessively high gain can reduce the stability margin and cause oscillation or instability
- D. Increasing gain always improves both steady-state error and stability with no trade-off
Show answer & explanation
Answer: C
Raising proportional gain in a feedback control loop generally reduces steady-state error by amplifying the corrective response to any error signal, but pushing gain too high narrows the system's stability margin and can drive the response toward oscillation or outright instability, which is a well-known trade-off in control system design. Assuming gain has no effect, or assuming it improves both error and stability without any limit, both ignore this fundamental trade-off between responsiveness and stability that proportional control exhibits.

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The FE Mechanical is administered by National Council of Examiners for Engineering and Surveying (NCEES), with 110 scored questions and a Pass/Fail (NCEES does not publish the passing score; scored on correct answers only, no penalty for wrong answers) result.

This free FE Mechanical practice test has 61 original questions written to National Council of Examiners for Engineering and Surveying (NCEES)'s official content outline, last checked against it on August 9, 2026. Every question shows a worked explanation, and nothing here requires a signup.

As of 2026, the FE Mechanical exam fee is $225 (separate nonrefundable $25 international scheduling fee for foreign-entity locations).

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Official sources

Every exam fact on this page traces to a primary document published by the body that administers the exam.

Fundamentals of Engineering (FE) Mechanical CBT Exam SpecificationsNational Council of Examiners for Engineering and Surveying (NCEES)ncees.org
NCEES Examinee Guide (May 2026)National Council of Examiners for Engineering and Surveying (NCEES)ncees.org
FE Exam | NCEESNational Council of Examiners for Engineering and Surveying (NCEES)ncees.org
Exam Scoring | NCEESNational Council of Examiners for Engineering and Surveying (NCEES)ncees.org

Last verified against the official exam content outline: August 9, 2026

Frequently asked questions

How many questions are in an FE Mechanical practice test?

A good practice test mirrors the real NCEES FE Mechanical exam, which has 110 questions. Working through a full-length set at that length is the best way to build the stamina the actual appointment demands.

What score do I need on FE Mechanical practice questions to feel exam-ready?

NCEES does not publish a passing score for the FE Mechanical exam, and scoring counts only correct answers with no penalty for wrong ones. Since there's no official cutoff to target in practice, aim to consistently answer a strong majority correct across every blueprint topic rather than chasing a specific number.

Is this FE Mechanical practice test free and does it require signup?

Yes — you can work through the FE Mechanical practice questions here without creating an account or paying anything.

How should I use FE Mechanical practice questions to study effectively?

Work practice questions topic by topic first, then switch to full-length timed sets as your exam date nears. Give extra attention to heavily weighted areas like Mechanical Design and Analysis, which carries 10-15 questions on the real exam, since gains there move your overall score the most.

How long should a timed FE Mechanical practice session run?

The real exam appointment runs 6 hours, so a realistic full-length practice run should be timed to that same window. Shorter, topic-focused sessions are fine for daily review, but at least one full 6-hour simulation before test day helps you gauge pacing and fatigue.

Which topics should get the most practice questions?

Focus extra reps on the highest-weighted blueprint areas: Fluid Mechanics, Thermodynamics, and Dynamics, Kinematics, and Vibrations each carry 10-15 questions, the largest shares on the exam. Lighter areas like Ethics and Professional Practice and Engineering Economics carry only 4-6 questions each, so they warrant proportionally less review time.