Your FREE FE Mechanical (Fundamentals of Engineering) Practice Test 2026 – 240+ Q&A

Correct answer: One

The correct count is one. The number of arbitrary constants in the general solution of an ordinary differential equation equals the order of the equation, so a first-order ODE produces exactly one arbitrary constant, which is fixed only when an initial or boundary condition is supplied. A second-order ODE would carry two such constants.

Correct answer: $C_{1}\cos(3x) + C_{2}\sin(3x)$

The answer is $C_{1}\cos(3x) + C_{2}\sin(3x)$. The characteristic equation $r^{2} + 9 = 0$ yields the purely imaginary roots $r = \pm 3i$, and complex conjugate roots of the form $\pm \omega i$ give oscillatory solutions $\cos(\omega x)$ and $\sin(\omega x)$ with $\omega = 3$. Real distinct roots would instead produce exponential terms.

Correct answer: 7

The dot product equals 7. The scalar (dot) product is the sum of the products of corresponding components: $(2)(4) + (3)(-1) + (1)(2) = 8 - 3 + 2 = 7$. Because the result is a single scalar rather than a vector, only the three component products are summed.

Correct answer: Their dot product

The defining condition is that their dot product is zero. Because $A \cdot B = |A||B|\cos\theta$, and $\cos(90^{\circ})$ is zero, perpendicular (orthogonal) vectors have a vanishing dot product. A zero cross product would instead indicate parallel vectors, since the cross-product magnitude depends on $\sin\theta$.

Correct answer: The zero vector

The result is the zero vector. The magnitude of a cross product is $|A||B|\sin\theta$, and for parallel vectors $\theta$ is zero (or $180^{\circ}$), making $\sin\theta$ zero and the entire product the zero vector. A perpendicular unit-vector result would require the inputs to be orthogonal, not parallel.

Correct answer: 12

The trapezoidal estimate is 12. The single-interval trapezoidal rule is $\frac{h}{2}(f_{0} + f_{1})$, where $h$ is the interval width; here $h = 2$, so $\frac{2}{2}(4 + 8) = (1)(12) = 12$. This is simply the area of a trapezoid whose parallel sides are the two function values.

Correct answer: A second-degree (parabolic) polynomial

Simpson's one-third rule fits a second-degree parabolic polynomial through three consecutive points spanning two subintervals. Because it captures curvature, it is generally more accurate than the trapezoidal rule, which fits straight-line segments and therefore ignores curvature between points.

Correct answer: Even

The number of subintervals must be even. Simpson's one-third rule processes the points in groups of two subintervals (three points each), so an even count is required to pair them completely. An odd number would leave one subinterval unpaired, which the basic rule cannot handle.

Correct answer: 13

The magnitude is 13. The magnitude of a three-dimensional vector is $\sqrt{\text{the sum of the squares of its components}}$: $\sqrt{3^{2} + 4^{2} + 12^{2}} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$. Simply adding the components would incorrectly ignore the geometric (Euclidean) nature of magnitude.

Correct answer: 10

The determinant is 10. For a $2 \times 2$ matrix $[[a, b], [c, d]]$, the determinant equals ad minus bc, so here it is $(3)(4) - (2)(1)$, which is $12 - 2$, giving 10. A nonzero determinant also confirms the matrix is invertible.

Correct answer: $5i$

The result is $5i$. A complex number in polar form $r(\cos\theta + i\sin\theta)$ converts to rectangular form $r\cos\theta + i r\sin\theta$; with r = 5 and $\theta = 90^{\circ}$, $\cos 90 = 0$ and $\sin 90 = 1$, so the value is 0 + $5i$, which is $5i$. The number therefore lies on the positive imaginary axis.

Correct answer: $2xy$

The partial derivative with respect to x is $2xy$. When differentiating partially with respect to x, the variable y is treated as a constant, so the derivative of $x^{2} y$ is $2xy$ and the derivative of the $3y$ term is zero because it contains no x. The result therefore retains the y factor as a constant multiplier.

Correct answer: $10.30\ \text{mm}$

The mean is $10.30\ \text{mm}$. Summing the five values $(10.2 + 10.4 + 10.0 + 10.6 + 10.3 = 51.5\ \text{mm})$ and dividing by the count of 5 gives $51.5/5 = 10.30\ \text{mm}$. The sample mean is simply the total of all observations divided by how many observations there are.

Correct answer: The spread or dispersion of the values about the mean

Standard deviation measures the spread or dispersion of the values about the mean. It is $\sqrt{\text{the variance}}$, which is the average of the squared deviations of each value from the mean, so a larger standard deviation means data points lie farther from the average. The most frequent value is the mode, the middle ordered value is the median, and the largest-minus-smallest gap is the range.

Correct answer: About 1.63

The sample standard deviation is about 1.63. The mean is $(8 + 10 + 12 + 10)/4 = 10$, and the squared deviations are 4, 0, 4, and 0, summing to 8. Dividing by $n-1$ (which is 3) gives a sample variance of $8/3 = 2.667$, and its square root is about 1.63. Dividing by $n$ instead of $n-1$ would incorrectly give about 1.41.

Correct answer: About 68%

About 68% of values fall within one standard deviation of the mean for a normal distribution. The empirical (68-95-99.7) rule states roughly 68% lie within one standard deviation, about 95% within two, and about 99.7% within three. The 50% figure corresponds to values below the mean, not within one standard deviation.

Correct answer: 2.0

The z-score is 2.0. The standardized value equals the observation minus the mean, divided by the standard deviation: $(51.0 - 50.0)/0.5 = 1.0/0.5 = 2.0$. This means the part lies two standard deviations above the mean of the normal distribution.

Correct answer: 0.20

The probability is 0.20. Classical probability is the number of favorable outcomes divided by the total number of equally likely outcomes: 4 defective out of 20 total parts gives $4/20 = 0.20$. The value 0.25 would result from dividing defective by good $(4/16)$ rather than by the total.

Correct answer: $400

The expected value is $400. Expected value is the sum of each outcome multiplied by its probability: $(0.6 \times 1{,}000) + (0.4 \times -500) = 600 - 200 = 400$ dollars. Each possible result is weighted by how likely it is to occur.

Correct answer: The method produces intervals that capture the true mean 95% of the time over repeated sampling

The correct interpretation is that the method produces intervals capturing the true mean 95% of the time over repeated sampling. A confidence interval is a statement about the long-run reliability of the estimation procedure, not about a single future part or the spread of individual values. The interval estimates the population mean, not the location of one observation.

Correct answer: The interval becomes narrower

The interval becomes narrower as the sample size increases. The margin of error is proportional to the standard deviation divided by $\sqrt{\text{the sample size}}$, so a larger sample shrinks the standard error and tightens the interval around the estimate. Raising the confidence level, not the sample size, is what widens an interval.

Correct answer: 96% of the variation in y is explained by the linear relationship with x

An R-squared of 0.96 means 96% of the variation in y is explained by the linear relationship with x. The coefficient of determination measures goodness of fit, the proportion of total variance in the response that the regression model accounts for. It does not measure slope accuracy, require points to lie exactly on the line, or establish causation between the variables.

Correct answer: Hold paramount the safety, health, and welfare of the public

The foremost obligation is to hold paramount the safety, health, and welfare of the public. The NCEES Model Rules and the canons of engineering ethics place public protection above duties to clients and employers, meaning an engineer must prioritize public well-being even when it conflicts with cost or schedule. Profit and client preference are subordinate to this duty.

Correct answer: Perform services only in areas of their competence

Professional ethics require the engineer to perform services only in areas of their competence. The rules of professional conduct state that engineers shall undertake assignments only when qualified by education or experience in the specific technical field, so accepting work outside one's competence violates that standard. The engineer may take the work only if qualified collaborators handle the unfamiliar portions.

Correct answer: Notify the client and appropriate authority and refuse to proceed unsafely

The required action is to notify the client and the appropriate authority and refuse to proceed unsafely. When professional judgment is overruled in a way that endangers public safety, the canons direct the engineer to inform the client or employer and notify the proper authority, because protecting the public takes precedence over the client relationship. Simply documenting and proceeding does not discharge the duty.

Correct answer: Avoid conflicts of interest and disclose any that arise

Acting as a faithful agent means avoiding conflicts of interest and disclosing any that arise. The ethics rules require engineers to serve each employer or client honestly, which includes promptly informing them of any business interest or relationship that could influence professional judgment. Blind agreement or favoring the highest payer would itself create or conceal a conflict.

Correct answer: Decline to seal work not prepared by or under the engineer's responsible charge

Professional practice requires the engineer to decline to seal work not prepared by or under the engineer's responsible charge. Affixing a professional seal certifies that the work was performed under the licensee's direct supervisory control, so sealing unreviewed outside work is a misrepresentation prohibited by licensure law. Responsible charge cannot be established by a quick glance or another's assurance.

Correct answer: Objective and truthful

Engineers are required to issue public statements that are objective and truthful. The canons direct engineers to be truthful in professional reports, statements, and testimony and to include all relevant information, which prevents misleading the public or decision makers. Bias toward an employer or persuasion at the expense of accuracy would violate this duty.

Correct answer: To protect the public by ensuring practitioners meet competency standards

The primary purpose of licensure is to protect the public by ensuring practitioners meet competency standards. State licensure laws establish minimum requirements of education, examination, and experience so that only qualified individuals offer engineering services to the public, which is fundamentally a public-protection mechanism rather than an economic or restrictive one.

Correct answer: Prohibited because it constitutes bribery and undermines fair practice

Offering a substantial gift to influence a contract award is prohibited because it constitutes bribery and undermines fair practice. The rules of professional conduct bar engineers from offering or giving anything of value to secure work or to influence official decisions, since such acts corrupt the integrity of the selection process. The harm is the same whether the gift is given openly or privately.

Correct answer: $1{,}210

The future worth is $1{,}210. The single-payment compound amount factor is $F = P \times (1 + i)^{n}$, so $F = 1{,}000 \times (1.10)^{2} = 1{,}000 \times 1.21$, which equals $1{,}210. Simple interest would have given only $1{,}200, but compounding earns interest on the first year's interest.

Correct answer: The equivalent value of that future amount in today's dollars

Present worth represents the equivalent value of that future amount in today's dollars. It is found by discounting a future cash flow back to time zero using $P = F \div (1 + i)^{n}$, which accounts for the time value of money. It is therefore not a simple undiscounted total but a present-time equivalent.

Correct answer: $5{,}000

The annual straight-line depreciation is $5{,}000. Straight-line depreciation equals the cost minus the salvage value, all divided by the useful life, so it is $(50{,}000 - 5{,}000) \div 9$, which is $45{,}000 \div 9$, giving $5{,}000 per year. This charge is the same for every year of the asset's life.

Correct answer: Greater than or equal to 1.0

A project is generally justified when the benefit-cost ratio is greater than or equal to 1.0. The benefit-cost ratio divides the equivalent worth of benefits by the equivalent worth of costs, so a value of 1.0 or above means benefits at least equal costs. A ratio below 1.0 indicates the costs outweigh the benefits.

Correct answer: The interest rate that makes the present worth of cash flows equal to zero

The rate of return is the interest rate that makes the present worth of all cash flows equal to zero. At this break-even discount rate, the present worth of benefits exactly offsets the present worth of costs, which is also where net present worth is zero. It is an internal characteristic of the cash flow series, not a rate imposed by a lender.

Correct answer: The alternative with the lower equivalent uniform annual cost

The preferred alternative is the one with the lower equivalent uniform annual cost. Annual cost analysis converts all cash flows to an equivalent uniform yearly amount, and the option that minimizes that annual cost is the more economical choice. A higher annual cost simply makes an alternative less attractive.

Correct answer: It reduces the real purchasing power of that future sum

Inflation reduces the real purchasing power of a fixed future sum. As the general price level rises, each future dollar buys fewer goods and services, so a fixed nominal amount received later is worth less in real terms than the same amount today. This is why engineering economics distinguishes real (constant) dollars from actual (then-current) dollars.

Correct answer: Sunk cost

The term is sunk cost. A sunk cost is money already spent that cannot be recovered, so it is irrelevant to choosing among future courses of action and should be excluded from the analysis. Opportunity cost, by contrast, reflects the value of the best forgone alternative and does influence decisions.

Correct answer: $5 \ \Omega$

The resistance is $5 \ \Omega$. Ohm's law states $V = I \times R$, so $R = V \div I$, which is $10 \text{ V} \div 2 \text{ A}$, giving $5 \ \Omega$. This relationship holds for any linear (ohmic) resistor.

Correct answer: $10 \ \Omega$

The equivalent series resistance is $10 \ \Omega$. Resistors in series add directly, so $R_{eq} = 4 \ \Omega + 6 \ \Omega$, which equals $10 \ \Omega$. Series resistance is always larger than the largest individual resistor.

Correct answer: $6 \ \Omega$

The equivalent parallel resistance is $6 \ \Omega$. For two equal resistors in parallel, the equivalent resistance is half of one resistor, so $12 \ \Omega \div 2$ equals $6 \ \Omega$. Parallel resistance is always smaller than the smallest individual resistor.

Correct answer: $360 \text{ W}$

The power consumed is $360 \text{ W}$. Electrical power for a resistive load is $P = V \times I$, so $P = 120 \text{ V} \times 3 \text{ A}$, which equals $360 \text{ W}$. This is the rate at which the device converts electrical energy.

Correct answer: The sum of currents entering a node equals the sum of currents leaving it

Kirchhoff's current law states that the sum of currents entering a node equals the sum of currents leaving it. This is a statement of charge conservation, meaning charge cannot accumulate at a junction. The law about voltages summing to zero around a loop is Kirchhoff's voltage law, a separate principle.

Correct answer: The amount of charge stored per unit voltage

Capacitance measures the amount of charge stored per unit voltage. It is defined by $C = Q \div V$, so a larger capacitance stores more charge for the same applied voltage. Capacitance is measured in farads and is a property of the geometry and dielectric, not of energy dissipation.

Correct answer: $120 \text{ V}$

The RMS value is approximately $120 \text{ V}$. For a sinusoid, the RMS value equals the peak value divided by $\sqrt{2}$, so $170 \div 1.414$ gives roughly $120 \text{ V}$. The RMS value represents the equivalent DC value that delivers the same power.

Correct answer: Inversely with the square of the distance

The force varies inversely with the square of the distance. Coulomb's law gives the force as proportional to the product of the two charges divided by the square of the separation, so doubling the distance reduces the force to one-fourth. This inverse-square dependence is characteristic of point-charge electrostatic forces.

Correct answer: Current lags the voltage by $90^{\circ}$

In a purely inductive circuit, the current lags the voltage by $90^{\circ}$. An inductor opposes changes in current, so the current cannot rise instantly with the voltage and reaches its peak a quarter cycle later. In a capacitor the opposite occurs, with current leading voltage by $90^{\circ}$.

Correct answer: The amount of magnetic flux passing through a unit area

Magnetic flux density describes the amount of magnetic flux passing through a unit area. It is measured in teslas and represents how concentrated the magnetic field is, with the total flux being the flux density integrated over the area. It is distinct from charge, resistance, or induced voltage.

Correct answer: The rate of change of magnetic flux through the loop

Faraday's law states that the induced EMF is proportional to the rate of change of magnetic flux through the loop. A steady, unchanging flux produces no EMF; only a time-varying flux induces a voltage. This is why generators rely on moving conductors or changing fields to produce electricity.

Correct answer: $\frac{1}{2} C V^{2}$

The energy stored is $\frac{1}{2} C V^{2}$. The energy in a capacitor's electric field is given by the expression $\frac{1}{2} C V^{2}$, which reflects that energy accumulates as the capacitor charges and the voltage builds. The simple product $C V$ gives the stored charge, not the energy.

Correct answer: Three

Three independent equilibrium equations are available in two dimensions. They are the sum of forces in the x direction equals zero, the sum of forces in the y direction equals zero, and the sum of moments about any point equals zero. These three equations allow solving for up to three unknown reactions in a planar problem.

Correct answer: $86.6 \text{ N}$

The horizontal component is $86.6 \text{ N}$. The x component of a force equals the magnitude times the cosine of the angle from the horizontal, so it is $100 \times \cos 30^{\circ}$, which is $100 \times 0.866$, giving $86.6 \text{ N}$. The vertical component would use the sine instead.

Correct answer: $15 \text{ N-m}$

The moment is $15 \text{ N-m}$. Moment equals force times the perpendicular distance, so it is $50 \text{ N} \times 0.3 \text{ m}$, which equals $15 \text{ N-m}$. Because the force is already perpendicular to the wrench, the full lever arm of $0.3 \text{ m}$ is used.

Correct answer: Tension

Such a member is in tension. When the internal force pulls the joints toward the member's center, stretching it, the member is in tension; when the force pushes the joints apart, it is in compression. Trusses carry only axial tension or compression, not bending or torsion, in ideal analysis.

Correct answer: Two equal, opposite, parallel forces that produce a pure moment

A couple consists of two equal, opposite, parallel forces that produce a pure moment. Because the two forces are equal and opposite, their net force is zero, so a couple causes rotation without any net translation. The moment of a couple is the same about every point in the plane.

Correct answer: Sum of forces in x and y equal zero (no moment equation needed)

The method of joints applies the conditions that the sum of forces in x and y equal zero, with no moment equation needed. Because all member forces at a pin joint pass through a single point, they produce no moment about that joint, leaving only two useful force equations per joint. This is why each joint can solve for at most two unknown member forces.

Correct answer: Three

This produces three reaction components in total. A pin support provides two reaction components (horizontal and vertical), and a roller provides one (perpendicular to its surface), giving three reactions. This matches the three available planar equilibrium equations, making the beam statically determinate.

Correct answer: $h/2$

The centroid is located at $h/2$ from the base. For a rectangle, which is symmetric about its horizontal midline, the centroid lies at the geometric center, so its vertical position is half the height. The $h/3$ location applies to a triangle, not a rectangle.

Correct answer: $80\,\text{N}$

The maximum static friction force is $80\,\text{N}$. Maximum static friction equals the coefficient of static friction times the normal force, and on a horizontal surface the normal force equals the weight of $200\,\text{N}$, so the friction force is $0.4 \times 200$, which is $80\,\text{N}$. Any applied horizontal force above this value initiates sliding.

Correct answer: Along the line connecting the two points of application

A two-force member carries force only along the line connecting its two points of application. For the member to be in equilibrium with forces at just two points, those forces must be equal, opposite, and collinear, which forces them to act along the line joining the two points. This principle simplifies truss and frame analysis.

Correct answer: It can directly find the force in a specific interior member

The main advantage of the method of sections is that it can directly find the force in a specific interior member. By cutting through the truss and applying equilibrium to one portion, the analyst can solve for a desired member force without working joint by joint across the entire structure. It still relies on the standard equilibrium equations.

Correct answer: $40\,\text{N}$

The equivalent concentrated load is $40\,\text{N}$. The resultant of a uniformly distributed load equals the load intensity times the length over which it acts, so it is 10 newtons per meter times 4 meters, which equals $40\,\text{N}$. This resultant acts at the centroid of the loading, which for a uniform load is at the midpoint.

Correct answer: At the midpoint of the loaded length

The resultant of a uniformly distributed load acts at the midpoint of the loaded length. Because a uniform load forms a rectangular load diagram, its centroid is at the geometric center of the rectangle. A triangular (linearly varying) load would instead place the resultant at one-third of the length from the heavier end.

Correct answer: Show all external forces and reactions acting on an isolated body

A free-body diagram is used to show all external forces and reactions acting on an isolated body. By isolating the body and representing every applied load, weight, and support reaction, the diagram sets up the equilibrium equations needed for analysis. It does not by itself compute internal stresses or deformations.

Correct answer: Equal the number of independent equilibrium equations

A statically determinate structure has unknowns that equal the number of independent equilibrium equations. When unknowns exactly match the available equations, the reactions and member forces can be solved using statics alone. If unknowns exceed the equations, the structure is statically indeterminate and requires additional compatibility relations.

Correct answer: The perpendicular distance from the point to the line of action

The moment equals the force magnitude times the perpendicular distance from the point to the line of action. This perpendicular distance, called the moment arm, is measured along the shortest line from the reference point to the force's line of action. Using a distance that is not perpendicular would overstate the true moment.

Correct answer: Only the floor through friction and the wall through its normal force

Horizontal reactions come from the floor through friction and from the wall through its normal force. A frictionless wall can only push perpendicular to itself, which is the horizontal direction, while the floor's horizontal reaction comes from friction. Together these horizontal forces keep the ladder from sliding.

Correct answer: They must be concurrent (intersect at a single point)

Three nonparallel equilibrium forces must be concurrent, meaning their lines of action intersect at a single point. If two of the forces meet at a point, the third must pass through that same point, or it would create an unbalanced moment about it. This three-force principle is a useful shortcut for solving equilibrium of rigid bodies.

Correct answer: Two straight segments meeting at the load point

A cable with a single concentrated load forms two straight segments meeting at the load point. An ideal cable carries only tension and cannot resist bending, so between discrete loads it remains straight, kinking only where the concentrated load is applied. A continuously distributed load such as self-weight would instead produce a curved shape.

Correct answer: A frame contains at least one multi-force member, while a truss has only two-force members

A frame contains at least one multi-force member, while a truss has only two-force members. Truss members are pin-connected and loaded only at joints, so they carry purely axial force, whereas frame members can carry loads along their length and therefore experience shear and bending as multi-force members. This distinction governs which analysis method applies.

Correct answer: $12\,\text{m/s}$

The final velocity is $12\,\text{m/s}$. With constant acceleration from rest, final velocity equals acceleration times time, so it is 3 meters per second squared times 4 seconds, which equals $12\,\text{m/s}$. Because the initial velocity is zero, no additional term is added.

Correct answer: Mass times acceleration

Newton's second law relates the net force to mass times acceleration. The law states that the sum of forces on a particle equals its mass multiplied by its acceleration, so acceleration is produced in the direction of the net force. Mass times velocity is momentum, a different quantity.

Correct answer: $10\,\text{kg-m/s}$

The linear momentum is $10\,\text{kg-m/s}$. Linear momentum equals mass times velocity, so it is 2 kilograms times 5 meters per second, which equals $10\,\text{kg-m/s}$. Momentum is a vector pointing in the direction of motion.

Correct answer: $18\,\text{J}$

The kinetic energy is $18\,\text{J}$. Kinetic energy equals one-half times mass times velocity squared, so it is $0.5 \times 4 \times 3^{2}$, which is $0.5 \times 4 \times 9$, giving $18\,\text{J}$. The velocity is squared, so it contributes more strongly than mass.

Correct answer: $v^{2}/r$

The centripetal acceleration is $v^{2}/r$. This acceleration is directed toward the center of the circle and keeps the object on its curved path, growing with the square of the speed and shrinking with larger radius. Even at constant speed, this acceleration is nonzero because the direction of velocity is always changing.

Correct answer: Both momentum and kinetic energy

In a perfectly elastic collision, both momentum and kinetic energy are conserved. Momentum is conserved in all collisions absent external impulses, but only elastic collisions also conserve kinetic energy because no energy is lost to deformation or heat. In an inelastic collision, kinetic energy is not conserved.

Correct answer: Relative velocity of separation to relative velocity of approach

The coefficient of restitution is the ratio of relative velocity of separation to relative velocity of approach. A value of one indicates a perfectly elastic collision, while a value of zero indicates a perfectly plastic collision in which the bodies stick together. It characterizes how much relative speed is retained after impact.

Correct answer: Zero

The horizontal acceleration is zero. With air resistance neglected, the only force on a projectile is gravity, which acts vertically, so there is no horizontal force and the horizontal velocity remains constant. The vertical motion, by contrast, accelerates downward at the gravitational rate.

Correct answer: Tangential velocity equals angular velocity times $r$

Tangential velocity equals angular velocity times $r$. A point farther from the axis moves faster for the same rotation rate, so multiplying the angular velocity by the radius gives the linear speed of that point. This relationship connects rotational and translational descriptions of motion.

Correct answer: It is proportional to $\sqrt{k/m}$

The natural frequency is proportional to $\sqrt{k/m}$. A stiffer spring raises the natural frequency while a larger mass lowers it, and the square-root relationship means quadrupling stiffness only doubles the frequency. This is the fundamental result for free vibration of a single-degree-of-freedom system.

Correct answer: The time required to complete one full cycle of motion

The period represents the time required to complete one full cycle of motion. It is the reciprocal of the frequency, so a higher frequency corresponds to a shorter period. The maximum displacement is the amplitude, a separate property of the oscillation.

Correct answer: Approaches the natural frequency

Resonance occurs when the forcing frequency approaches the natural frequency. Near this condition the system's response amplitude grows dramatically because energy is added in phase with the motion each cycle. Damping limits the peak amplitude, but lightly damped systems can experience very large vibrations at resonance.

Correct answer: The displacement in the direction of the force

The work done equals the force times the displacement in the direction of the force. Only the component of displacement parallel to the force contributes to work, so a force perpendicular to motion does no work. This definition connects directly to the work-energy theorem.

Correct answer: Its kinetic energy

The work-energy theorem states that the net work done on a particle equals the change in its kinetic energy. Positive net work speeds the particle up while negative net work slows it down, directly linking force and displacement to changes in speed. It is a scalar alternative to applying Newton's second law over distance.

Correct answer: Its linear momentum

Impulse equals the change in linear momentum. Impulse is the integral of force over time, and the impulse-momentum principle states that this equals the body's final momentum minus its initial momentum. This is why a longer contact time reduces the force needed for a given momentum change.

Correct answer: Mass moment of inertia times angular acceleration

The net torque equals the mass moment of inertia times angular acceleration. This rotational form of Newton's second law shows that a body with larger moment of inertia requires more torque to achieve the same angular acceleration. It parallels the translational law in which force equals mass times acceleration.

Correct answer: Its resistance to angular acceleration

The mass moment of inertia measures a body's resistance to angular acceleration. It depends on both the mass and how that mass is distributed relative to the rotation axis, so mass located farther from the axis increases the moment of inertia. It plays the same role in rotation that mass plays in translation.

Correct answer: It oscillates with gradually decreasing amplitude

An underdamped system oscillates with gradually decreasing amplitude. The damping is light enough to allow oscillation, but each successive cycle loses energy, so the amplitude decays exponentially toward equilibrium. A critically damped or overdamped system would return without oscillating.

Correct answer: About $19.6\,\text{m}$

It falls about $19.6\,\text{m}$. The distance fallen from rest is one-half times the gravitational acceleration times time squared, so it is $0.5 \times 9.8 \times 2^{2}$, which is $0.5 \times 9.8 \times 4$, giving about $19.6\,\text{m}$. The time appears squared, so the distance grows rapidly.

Correct answer: Kinetic and potential energy

For a conservative system, the sum of kinetic and potential energy remains constant. Energy may convert between kinetic and potential forms as the body moves, but the total mechanical energy is preserved when only conservative forces such as gravity and ideal springs act. Friction or other nonconservative forces would dissipate mechanical energy.

Correct answer: Tangential and normal components

The acceleration along a curved path has tangential and normal components. The tangential component changes the speed of the particle, while the normal (centripetal) component changes the direction of the velocity and points toward the center of curvature. Both can act simultaneously when a particle speeds up or slows down on a curve.

Correct answer: Zero

The net torque is zero. Constant angular velocity means zero angular acceleration, and since net torque equals moment of inertia times angular acceleration, the net torque must also be zero. The flywheel continues spinning steadily because no unbalanced torque acts to change its rotation rate.

Correct answer: $200\ \text{MPa}$

The normal stress is $200\ \text{MPa}$. Axial normal stress is force divided by cross-sectional area, so $(50{,}000\ \text{N})/(250\ \text{mm}^{2}) = 200\ \text{N/mm}^{2} = 200\ \text{MPa}$. Because $1\ \text{N/mm}^{2}$ equals exactly $1\ \text{MPa}$, the load and area in consistent units give the stress directly.

Correct answer: $1.40\ \text{mm}$

The elongation is $1.40\ \text{mm}$. Using $\delta = \frac{\sigma}{E} L$, the axial strain is $140\ \text{MPa}/200{,}000\ \text{MPa} = 0.0007$, and multiplying by the $2000\ \text{mm}$ length gives $0.0007 \times 2000 = 1.40\ \text{mm}$. The deformation equals strain times original length because strain is elongation per unit length.

Correct answer: Normal strain

Stress is directly proportional to normal strain. Hooke's law is $\sigma = E \epsilon$, where the modulus of elasticity $E$ is the constant of proportionality linking stress and strain in the linear-elastic region. The relationship is linear, not quadratic, and it holds only below the proportional limit.

Correct answer: 0.0006

The lateral strain magnitude is 0.0006. Poisson's ratio is the ratio of lateral strain magnitude to longitudinal strain, so lateral strain equals $\nu$ times longitudinal strain: $0.30 \times 0.0020 = 0.0006$. The lateral strain is contraction (opposite sign) when the longitudinal strain is tensile.

Correct answer: $\sigma = \frac{M c}{I}$

The flexure formula is $\sigma = \frac{M c}{I}$, where $M$ is the bending moment, $c$ is the distance from the neutral axis to the point of interest, and $I$ is the area moment of inertia about the neutral axis. The expression $\frac{V Q}{I t}$ gives transverse shear stress instead, and $\frac{T r}{J}$ applies to torsion.

Correct answer: $96\ \text{MPa}$

The maximum bending stress is $96\ \text{MPa}$. The moment of inertia is $\frac{(50)(100^{3})}{12} = 4.1667 \times 10^{6}\ \text{mm}^{4}$, and the extreme fiber distance $c$ is half the depth, $50\ \text{mm}$. Then $\sigma = \frac{M c}{I} = \frac{(8 \times 10^{6}\ \text{N-mm})(50)}{4.1667 \times 10^{6}} = 96\ \text{MPa}$, occurring at the top and bottom surfaces.

Correct answer: At the neutral axis

Bending stress is zero at the neutral axis. Because $c$ is the distance measured from the neutral axis, setting $c$ to zero makes the flexure stress vanish there; stress grows linearly with distance and reaches its maximum at the extreme fibers farthest from the neutral axis. The neutral axis passes through the centroid for pure bending.

Correct answer: $47.7\ \text{MPa}$

The maximum shear stress is about $47.7\ \text{MPa}$. The polar moment of inertia is $J = \frac{\pi (20^{4})}{2} = 2.513 \times 10^{5}\ \text{mm}^{4}$, and the torsion formula $\tau = \frac{T r}{J}$ gives $\frac{(600{,}000\ \text{N-mm})(20)}{2.513 \times 10^{5}} = 47.7\ \text{MPa}$. The maximum occurs at the outer radius because shear stress varies linearly from zero at the center.

Correct answer: Linearly, from zero at the center to a maximum at the surface

Shear stress varies linearly, from zero at the center to a maximum at the outer surface. In $\tau = \frac{T r}{J}$ the variable $r$ is the radial distance from the axis, so stress is directly proportional to that distance. This is why hollow shafts use material efficiently, placing it where the stress is highest.

Correct answer: $\phi = \frac{T L}{G J}$

The angle of twist is $\phi = \frac{T L}{G J}$. The twist increases with torque and length and decreases with the torsional rigidity $G J$ in the denominator. This parallels axial deformation $\frac{P L}{A E}$, with $G J$ playing the stiffness role that $A E$ plays for axial loading.

Correct answer: $\frac{P L}{4}$

The maximum bending moment is $\frac{P L}{4}$. Each support reaction is $\frac{P}{2}$, and the moment at midspan is the reaction times half the span, $\frac{P}{2} \cdot \frac{L}{2} = \frac{P L}{4}$. This peak occurs directly under the load, where the shear diagram crosses zero.

Correct answer: The shear force

The slope of the moment diagram equals the shear force. The fundamental relationship dM/dx = V means the rate of change of moment with position is the shear at that location. Consequently, the bending moment reaches a local maximum or minimum exactly where the shear diagram passes through zero.

Correct answer: At the fixed support, with magnitude $P \cdot L$

The maximum moment is $P \cdot L$ at the fixed support. The internal bending moment in a tip-loaded cantilever grows linearly from zero at the free end to P times the full length L at the wall, so the wall is the critical section. The free end carries the load but has zero moment there.

Correct answer: Linearly with a constant slope

The shear varies linearly with a constant slope under a uniform distributed load. Because dV/dx equals the negative of the distributed load intensity, a constant load gives a constant slope, producing a straight sloped line in the shear diagram. The moment diagram over that same segment becomes a parabola.

Correct answer: $30 \text{ MPa}$

The radius is $30 \text{ MPa}$. With no shear stress, Mohr's circle radius equals the magnitude of (sigma_x - sigma_y)/2 = (80 - 20)/2 = $30 \text{ MPa}$. The radius represents the maximum in-plane shear stress, and the two given normal stresses are the principal stresses because shear is already zero.

Correct answer: The average of $\sigma_x$ and $\sigma_y$

The center lies at the average normal stress, $(\sigma_x + \sigma_y)/2$. Mohr's circle is centered on the horizontal (normal-stress) axis at this average, and its radius accounts for the shear and the difference of normal stresses. The principal stresses are then the center value plus and minus the radius.

Correct answer: Principal stresses and maximum shear stress

Mohr's circle graphically gives the principal stresses and the maximum shear stress. The extreme points where the circle crosses the normal-stress axis are the principal stresses, while the top and bottom of the circle give the maximum in-plane shear stress. Material constants like the modulus of elasticity are not obtained from the circle.

Correct answer: Plus and minus the radius $\sqrt{((\sigma_x - \sigma_y)/2)^{2} + \tau_{xy}^{2}}$

The principal stresses are the average normal stress plus and minus $\sqrt{((\sigma_x - \sigma_y)/2)^{2} + \tau_{xy}^{2}}$. This square-root term is the Mohr's-circle radius, combining the half-difference of normal stresses with the shear stress. The principal directions are oriented so that the shear stress vanishes on those planes.

Correct answer: The shear stress is zero

Principal planes are those on which the shear stress is zero. The stress-transformation equations are solved for the angle that makes the transformed shear stress vanish, and on those planes the normal stresses reach their extreme (principal) values. The planes of maximum shear are oriented 45 degrees away from the principal planes.

Correct answer: $P_{cr} = \pi^{2} E I / L_e^{2}$

Euler's critical load is $P_{cr} = \pi^{2} E I / L_e^{2}$. The buckling capacity rises with the flexural rigidity $E I$ and falls with the square of the effective length, so longer slender columns buckle at much lower loads. The least moment of inertia is used because the column buckles about its weakest axis.

Correct answer: It decreases to one-fourth

The critical load drops to one-fourth. Because Euler's formula has the length squared in the denominator, doubling the length multiplies the denominator by 4 and divides the buckling load by 4. This strong sensitivity to length is why slenderness is the dominant factor in column stability.

Correct answer: 2.0

The effective length factor is 2.0 for a fixed-free column. The effective length Le equals K times the actual length, and the fixed-free (flagpole) condition is the least stable common case, doubling the effective length and sharply reducing the buckling load. A pinned-pinned column, by contrast, uses K equal to 1.0.

Correct answer: Adding the axial stress and the bending stress algebraically

The contributions are added algebraically, $\sigma = P/A \pm M c / I$. Because both produce normal stresses acting in the same direction, superposition allows them to be summed, with the bending term added on the side where it is tensile and subtracted on the opposite side. The largest combined stress governs the design check.

Correct answer: The ultimate tensile strength

The highest point on the curve is the ultimate tensile strength. It is the maximum engineering stress the material sustains, after which necking begins and the engineering stress falls until fracture. The yield strength occurs earlier at the onset of permanent deformation, and fracture happens later at a lower engineering stress.

Correct answer: The 0.2 percent offset method

The yield strength is defined by the 0.2 percent offset method. A line parallel to the initial elastic slope is drawn from a strain of 0.002, and its intersection with the curve marks the offset yield strength. This convention is needed because many metals transition gradually from elastic to plastic behavior without a distinct yield point.

Correct answer: The modulus of elasticity

The slope of the linear elastic portion is the modulus of elasticity. Within this region stress is proportional to strain, and the ratio of stress to strain equals Young's modulus, a measure of stiffness. The ultimate tensile strength is a stress value at the curve's peak, not a slope.

Correct answer: 24 percent

The percent elongation is 24 percent. Percent elongation is the change in gauge length divided by the original gauge length, times 100: (62 - 50)/50 * 100 = 12/50 * 100 = 24 percent. This value quantifies ductility, the amount of plastic deformation a material undergoes before fracture.

Correct answer: Toughness

The area under the full stress-strain curve up to fracture measures toughness. Toughness is the total energy per unit volume a material can absorb before it breaks, combining both strength and ductility. Stiffness relates only to the elastic slope, while hardness is measured by indentation rather than from the tensile curve.

Correct answer: Austenite

Austenite is the phase that transforms in the eutectoid reaction. At the eutectoid point, solid austenite (face-centered cubic gamma iron) decomposes upon slow cooling into the two-phase lamellar structure of ferrite plus cementite known as pearlite. The reaction involves one solid transforming into two solids, with no liquid present.

Correct answer: Pearlite

This lamellar ferrite-and-cementite structure is called pearlite. It forms by the eutectoid decomposition of austenite during slow, near-equilibrium cooling, producing the characteristic alternating layers. Martensite, by contrast, is a non-equilibrium phase produced by rapid quenching and does not appear on the equilibrium iron-carbon diagram.

Correct answer: 2.1 percent carbon

The steel-to-cast-iron boundary lies near 2.1 percent carbon. Iron-carbon alloys below roughly 2.1 percent carbon are classified as steels, while those above are cast irons, which is the maximum solubility of carbon in austenite. The value 0.76 percent marks the eutectoid composition, and 6.7 percent corresponds to pure cementite.

Correct answer: Quenching (hardening)

Quenching, the hardening step, produces martensite. Rapid cooling of austenite suppresses the diffusion needed to form ferrite and cementite, trapping carbon and creating the hard, brittle martensitic structure. Annealing and normalizing both use slower cooling that yields softer equilibrium structures rather than martensite.

Correct answer: To reduce brittleness and relieve internal stresses

Tempering reduces brittleness and relieves internal stresses. Reheating quenched martensite allows some carbon to precipitate, trading a modest loss in hardness for a substantial gain in toughness and reduced residual stress. It does not raise hardness to a maximum; quenching alone produces the hardest, most brittle condition.

Correct answer: Full annealing

Full annealing is the appropriate choice. Heating into the austenite region followed by very slow furnace cooling yields a soft, ductile, coarse pearlitic structure that relieves stresses from cold work and improves machinability. Quenching would do the opposite, creating hard, brittle martensite.

Correct answer: It has iron as its principal constituent

A ferrous metal is one whose principal constituent is iron. Steels and cast irons are the major ferrous alloys, built on an iron base with carbon and other alloying elements. Aluminum-based, copper-based, and similar non-iron alloys are classified instead as nonferrous metals.

Correct answer: Stainless steel

Stainless steel is the corrosion-resistant ferrous alloy. Adding at least roughly 10.5 percent chromium forms a thin, adherent, self-healing chromium-oxide passive layer that protects the underlying metal. Plain carbon steel and ordinary cast irons lack this chromium content and therefore rust readily.

Correct answer: Increases hardness and strength but decreases ductility

Higher carbon content increases hardness and strength while decreasing ductility. Carbon forms more hard cementite within the steel, raising strength and hardness but making the material more brittle and less able to deform plastically. This trade-off is why low-carbon steels are used where formability matters and high-carbon steels where wear resistance is needed.

Correct answer: Endurance (fatigue) limit

This threshold stress is the endurance limit, also called the fatigue limit. For many ferrous metals, stress amplitudes below the endurance limit allow effectively infinite cyclic life, which appears as a horizontal asymptote on the S-N (stress versus number of cycles) curve. Static properties such as yield and ultimate strength describe single-load behavior, not repeated cyclic loading.

Correct answer: Progressive crack initiation and growth under repeated stresses well below the ultimate strength

Fatigue is the progressive initiation and growth of a crack under repeated cyclic stresses well below the ultimate strength. A small crack typically nucleates at a stress concentration and advances a little each cycle until the remaining section can no longer carry the load, causing sudden final fracture. It is distinct from a single-cycle overload failure.