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PEC EPE Mechanical practice questions

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Question 1Engineering Mathematics and Fundamentals

A 3 × 3 real matrix A has eigenvalues 2, −3 and 5. What is the determinant of A?

  1. 4
  2. −30 Correct answer.
  3. 30
  4. 10

Explanation

For any square matrix the determinant equals the product of its eigenvalues, so det(A) = (2)(−3)(5) = −30. The sum of the eigenvalues, 4, gives the trace, not the determinant, which is a common source of error in matrix problems.

Question 2Engineering Mathematics and Fundamentals

Evaluate the limit as x approaches 0 of sin(5x) divided by 2x.

  1. 5/2 Correct answer.
  2. 0
  3. 5
  4. 2/5

Explanation

Write the expression as (5/2)·sin(5x)/(5x); since sin(u)/u tends to 1 as u tends to 0, the limit is 5/2. Applying L'Hôpital once gives 5cos(5x)/2, which also tends to 5/2.

Question 3Engineering Mathematics and Fundamentals

Two coplanar forces of 3 kN and 4 kN act at the same point on a machine bracket, at 90° to each other. What is the magnitude of their resultant?

  1. 5 kN Correct answer.
  2. 1 kN
  3. 25 kN
  4. 7 kN

Explanation

For two forces acting at right angles, the resultant magnitude is the square root of the sum of the squares: √(3² + 4²) = 5 kN. The resultant acts at about 53° to the 3 kN force, following the familiar 3-4-5 triangle.

Question 4Engineering Mathematics and Fundamentals

A solid steel rod of 20 mm diameter carries an axial tensile load of 50 kN. What is the average normal stress in the rod?

  1. 318 MPa
  2. 159 MPa Correct answer.
  3. 39.8 MPa
  4. 79.6 MPa

Explanation

Average normal stress is load divided by cross-sectional area, and A = πd²/4 = π(20)²/4 ≈ 314 mm². Therefore σ = 50 000 N / 314.2 mm² ≈ 159 MPa, well below the yield strength of typical structural steel.

Question 5Engineering Mathematics and Fundamentals

In a piston-cylinder device, 250 kJ of heat is transferred to the gas while the gas expands and performs 80 kJ of boundary work on the piston. Neglecting changes in kinetic and potential energy, what is the change in internal energy of the gas?

  1. 330 kJ
  2. 170 kJ Correct answer.
  3. −170 kJ
  4. 80 kJ

Explanation

For a closed system the first law is ΔU = Q − W, with heat added to the system and work done by the system both taken as positive. Hence ΔU = 250 − 80 = 170 kJ, so the internal energy rises by 170 kJ.

Question 6Engineering Mathematics and Fundamentals

Water flows steadily through a horizontal pipe that contracts from 150 mm diameter to 75 mm diameter. If the velocity in the larger section is 2.0 m/s and water is treated as incompressible, what is the velocity in the smaller section?

  1. 1.0 m/s
  2. 8.0 m/s Correct answer.
  3. 4.0 m/s
  4. 0.5 m/s

Explanation

Continuity for steady incompressible flow in a full pipe gives A₁V₁ = A₂V₂, and area varies with the square of diameter. Thus V₂ = V₁(d₁/d₂)² = 2.0 × (150/75)² = 8.0 m/s, so halving the diameter quadruples the velocity.

Question 7Engineering Mathematics and Fundamentals

What is the general solution of the homogeneous differential equation y'' + 4y' + 13y = 0, where C₁ and C₂ are arbitrary constants?

  1. y = C₁e^(−2t) + C₂e^(3t)
  2. y = e^(−2t)(C₁cos√13 t + C₂sin√13 t)
  3. y = e^(2t)(C₁cos3t + C₂sin3t)
  4. y = e^(−2t)(C₁cos3t + C₂sin3t) Correct answer.

Explanation

The characteristic equation r² + 4r + 13 = 0 has roots r = −2 ± 3i, a complex conjugate pair with a negative real part. The solution is therefore a decaying oscillation, y = e^(−2t)(C₁cos3t + C₂sin3t).

Question 8Engineering Mathematics and Fundamentals

The diameter of a machined shaft is normally distributed with mean 50.0 mm and standard deviation 0.5 mm. Using the standard normal table value P(Z < 2.00) = 0.9772, what is the probability that a shaft diameter exceeds 51.0 mm?

  1. 0.9772
  2. 0.0228 Correct answer.
  3. 0.0456
  4. 0.1587

Explanation

Standardising gives z = (51.0 − 50.0)/0.5 = 2.00, so the required probability is P(Z > 2.00) = 1 − 0.9772 = 0.0228. About 2.3% of shafts would exceed the 51.0 mm limit.

Question 9Engineering Mathematics and Fundamentals

A 5 kg block starts from rest and slides 4 m down a plane inclined at 30° to the horizontal. The coefficient of kinetic friction between the block and the plane is 0.20. Taking g = 9.81 m/s², what is the speed of the block at the bottom?

  1. 4.9 m/s
  2. 6.3 m/s
  3. 8.9 m/s
  4. 5.1 m/s Correct answer.

Explanation

The work-energy principle with friction gives v = √(2gL(sin30° − μcos30°)) = √(2 × 9.81 × 4 × (0.500 − 0.173)) ≈ 5.1 m/s. The friction term reduces the speed from the frictionless value of about 6.3 m/s.

Question 10Engineering Mathematics and Fundamentals

For the Laplace transform F(s) = (3s + 5)/(s² + 4s + 13), what is the corresponding time-domain function f(t) for t ≥ 0?

  1. e^(2t)[3cos3t − (1/3)sin3t]
  2. e^(−2t)[3cos3t − (1/3)sin3t] Correct answer.
  3. e^(−2t)[3cos3t + 3sin3t]
  4. e^(−2t)[(1/3)cos3t − 3sin3t]

Explanation

Completing the square gives s² + 4s + 13 = (s + 2)² + 9, and writing 3s + 5 = 3(s + 2) − 1 splits the transform into a cosine part and a sine part. Inverting term by term yields f(t) = e^(−2t)[3cos3t − (1/3)sin3t], where the shift s + 2 produces the exponential decay.

Question 11Engineering Mathematics and Fundamentals

The complex number z = 1 + i√3 is expressed in polar form. What is its principal argument in radians?

  1. π/2
  2. π/4
  3. π/3 Correct answer.
  4. π/6

Explanation

For a complex number z = a + ib, the principal argument satisfies tan θ = b/a with quadrant adjustment. Here a = 1 and b = √3, so tan θ = √3 and the point is in the first quadrant. The principal value is therefore θ = π/3.

Question 12Engineering Mathematics and Fundamentals

What is the rank of the matrix [[1, 2], [2, 4]]?

  1. 3
  2. 0
  3. 1 Correct answer.
  4. 2

Explanation

The second row is exactly twice the first row, so the rows are linearly dependent. Only one independent row remains, which means the rank is 1. The determinant is zero, confirming the matrix is not full rank.

Question 13Engineering Mathematics and Fundamentals

For vectors a = 2i − j + 3k and b = i + 4j − 2k, what is the scalar product a · b?

  1. 8
  2. -8 Correct answer.
  3. 6
  4. 0

Explanation

The scalar product is formed by multiplying corresponding components and adding: (2)(1) + (−1)(4) + (3)(−2). This gives 2 − 4 − 6 = −8. The result is negative because the vectors point partly in opposite directions.

Question 14Engineering Mathematics and Fundamentals

What is the solution of the separable differential equation dy/dx = 2xy with initial condition y(0)=1?

  1. y = 1 + x^2
  2. y = e^{x^2} Correct answer.
  3. y = e^{2x}
  4. y = cosh(2x)

Explanation

Separate variables to get dy/y = 2x dx, then integrate: ln|y| = x² + C. Applying y(0)=1 gives C=0, so y = e^{x²}. This function satisfies both the differential equation and the initial condition.

Question 15Engineering Mathematics and Fundamentals

A production lot has a 10% defective rate. If five parts are selected independently, what is the probability that exactly one is defective?

  1. 0.40951
  2. 0.32805 Correct answer.
  3. 0.59049
  4. 0.06561

Explanation

This is a binomial probability with n=5, p=0.1, and k=1. The probability is C(5,1)(0.1)^1(0.9)^4 = 5 × 0.1 × 0.6561 = 0.32805. The combination factor accounts for the five possible positions of the single defective part.

Question 16Engineering Mathematics and Fundamentals

Using the trapezoidal rule with h = 0.5, what is the estimate of ∫_0^1 e^x dx?

  1. 1.6487
  2. 1.8591
  3. 2.7183
  4. 1.7539 Correct answer.

Explanation

With h=0.5, the nodes are x=0, 0.5, and 1. The trapezoidal rule gives T = (h/2)[f(0)+2f(0.5)+f(1)] = 0.25[1+2(1.6487)+2.7183]. This evaluates to 1.7539.

Question 17Engineering Mathematics and Fundamentals

For the scalar field φ(x,y,z)=x^2 y + y z, what is the gradient vector at the point (1,2,3)?

  1. (4, 4, 3)
  2. (4, 4, 2) Correct answer.
  3. (4, 2, 2)
  4. (2, 4, 2)

Explanation

The gradient components are ∂φ/∂x = 2xy, ∂φ/∂y = x² + z, and ∂φ/∂z = y. At (1,2,3) these become 4, 4, and 2. Hence the gradient vector is (4,4,2).

Question 18Engineering Mathematics and Fundamentals

The matrix A = [[4, 1], [2, 3]] has which pair of eigenvalues?

  1. −2 and 9
  2. 2 and 5 Correct answer.
  3. 3 and 4
  4. 1 and 6

Explanation

The characteristic equation is det(A−λI) = (4−λ)(3−λ) − 2 = λ² − 7λ + 10 = 0. Factoring gives (λ−5)(λ−2)=0, so the eigenvalues are 2 and 5. Their sum equals the trace 7 and their product equals the determinant 10.

Question 19Engineering Mathematics and Fundamentals

A force field F(x,y) = y i + x j acts on a particle moving along the parabola y = x^2 from (0,0) to (1,1). What is the work done by F?

  1. 1/2
  2. 1 Correct answer.
  3. 4/3
  4. 2

Explanation

Parameterize the path by r(t)=(t,t²), so dr=(1,2t)dt and F=(t²,t). Then F·dr = t² + 2t² = 3t², and the work is ∫₀¹ 3t² dt = [t³]₀¹ = 1. The field is conservative with potential xy, which also gives 1 between these points.

Question 20Engineering Mathematics and Fundamentals

For the function f(x)=1 on the interval 0 < x < π, what is the first sine-series Fourier coefficient b₁?

  1. 2/π
  2. 1/π
  3. π/2
  4. 4/π Correct answer.

Explanation

For a sine series on 0<x<π, b_n = (2/π)∫₀^π f(x) sin(nx) dx. With f(x)=1 and n=1, b₁ = (2/π)∫₀^π sin x dx = (2/π)(2) = 4/π. The coefficient is positive and larger than 1 because the average of sin x over the interval is 2/π.

Question 21Engineering Mathematics and Fundamentals

For the function f(x, y) = x³y², what is the value of the mixed partial derivative ∂²f/∂x∂y at the point (2, 1)?

  1. 12
  2. 6
  3. 48
  4. 24 Correct answer.

Explanation

Differentiate first with respect to x to obtain 3x²y², then differentiate that result with respect to y to obtain 6x²y. Substituting x = 2 and y = 1 gives 6(4)(1) = 24.

Question 22Engineering Mathematics and Fundamentals

For the planar vectors a = i + 2j and b = 3i + 4j, what is the vector product a × b?

  1. −2k Correct answer.
  2. 10k
  3. 2k
  4. −10k

Explanation

For planar vectors, a × b = (a_x b_y − a_y b_x)k. Here that is (1 × 4 − 2 × 3)k = (4 − 6)k = −2k.

Question 23Engineering Mathematics and Fundamentals

Which expression gives the Newton–Raphson iteration for solving f(x) = 0?

  1. x_{n+1} = x_n − f(x_n)/f′(x_n) Correct answer.
  2. x_{n+1} = x_n + f(x_n)/f′(x_n)
  3. x_{n+1} = f(x_n)/f′(x_n) − x_n
  4. x_{n+1} = x_n − f′(x_n)/f(x_n)

Explanation

The Newton–Raphson update is derived from the tangent line to f at x_n and is x_{n+1} = x_n − f(x_n)/f′(x_n). The correction term must be subtracted from the current estimate.

Question 24Engineering Mathematics and Fundamentals

For x > 0, what is the integrating factor for the first-order linear differential equation y′ + (2/x)y = x²?

  1. x² Correct answer.
  2. e^{2x}
  3. 2 ln x
  4. x

Explanation

For y′ + P(x)y = Q(x), the integrating factor is e^{∫P dx}. Here P = 2/x, so ∫P dx = 2 ln x and the factor is e^{2 ln x} = x².

Question 25Engineering Mathematics and Fundamentals

What is the Laplace transform of t e^{−2t} for t ≥ 0?

  1. 2/(s + 2)²
  2. 1/(s − 2)²
  3. s/(s + 2)²
  4. 1/(s + 2)² Correct answer.

Explanation

The transform of t is 1/s², and multiplication by e^{−2t} shifts the transform by replacing s with s + 2. Hence L{t e^{−2t}} = 1/(s + 2)².

Question 26Engineering Mathematics and Fundamentals

The time to failure of a bearing is modeled by an exponential distribution with a mean of 2000 hours. What is the probability that it fails before 1000 hours?

  1. 0.5000
  2. 0.3935 Correct answer.
  3. 0.6065
  4. 0.1353

Explanation

For an exponential distribution with mean μ, P(T < t) = 1 − e^{−t/μ}. With μ = 2000 h and t = 1000 h, P = 1 − e^{−0.5} ≈ 0.3935.

Question 27Engineering Mathematics and Fundamentals

A rectangular section is 60 mm wide and 120 mm deep. What is its second moment of area about an axis along its base?

  1. 8.64 × 10⁶ mm⁴
  2. 17.28 × 10⁶ mm⁴
  3. 69.12 × 10⁶ mm⁴
  4. 34.56 × 10⁶ mm⁴ Correct answer.

Explanation

For a rectangle about its base, I = bh³/3. Substituting b = 60 mm and h = 120 mm gives 60 × 120³ / 3 = 34.56 × 10⁶ mm⁴.

Question 28Engineering Mathematics and Fundamentals

For the matrix A = [[2, 1], [5, 3]], what is A⁻¹?

  1. [[3, −1], [−5, 2]] Correct answer.
  2. [[3, −1], [5, 2]]
  3. [[3, 1], [5, 2]]
  4. [[2, −1], [−5, 3]]

Explanation

For A = [[a, b], [c, d]], the inverse is (1/(ad − bc))[[d, −b], [−c, a]] when the determinant is nonzero. Here ad − bc = 6 − 5 = 1, so A⁻¹ = [[3, −1], [−5, 2]].

Question 29Engineering Mathematics and Fundamentals

Using Green’s theorem, what is the value of ∮(x dy − y dx) taken counterclockwise around the circle x² + y² = 4?

  1. 8π Correct answer.
  2. 16π
  3. 0
  4. 4π

Explanation

Write the integrand as P dx + Q dy with P = −y and Q = x. Then ∂Q/∂x − ∂P/∂y = 1 − (−1) = 2, so the integral equals 2 times the circle’s area, namely 2 × 4π = 8π.

Question 30Engineering Mathematics and Fundamentals

A solid circular shaft of diameter 40 mm transmits 15 kW at 500 rpm. What is the maximum shear stress in the shaft?

  1. 22.8 MPa Correct answer.
  2. 45.6 MPa
  3. 11.4 MPa
  4. 5.7 MPa

Explanation

Torque T = P/ω, with ω = 2π(500)/60 ≈ 52.36 rad/s, so T ≈ 286.5 N·m. For a solid shaft, τmax = 16T/(πd³); with d = 0.040 m this gives τmax ≈ 22.8 MPa.

Question 31Mechanics of Materials and Machine Design

A 20 mm diameter solid steel bar carries an axial tensile load of 50 kN. What is the nominal axial stress in the bar?

  1. 79.5 MPa
  2. 318 MPa
  3. 159 MPa Correct answer.
  4. 39.8 MPa

Explanation

Nominal axial stress equals the load divided by the cross-sectional area. For a 20 mm diameter bar the area is π(20)²/4 ≈ 314 mm², so 50,000 N ÷ 314 mm² gives about 159 MPa, well below the yield strength of ordinary structural steel.

Question 32Mechanics of Materials and Machine Design

A steel bar in uniaxial tension develops an axial strain of 800 microstrain. Taking Poisson's ratio for the steel as 0.30, what is the lateral strain in the bar?

  1. −240 µε Correct answer.
  2. −267 µε
  3. +240 µε
  4. +267 µε

Explanation

Poisson's ratio links lateral and axial strain through the relation ε_lateral = −ν · ε_axial. Multiplying 800 µε by 0.30 gives 240 µε, and the negative sign shows that the bar contracts laterally while it extends axially.

Question 33Mechanics of Materials and Machine Design

A rectangular beam 50 mm wide and 100 mm deep carries a bending moment of 2 kN·m about its horizontal centroidal axis. What is the maximum bending stress?

  1. 48 MPa
  2. 6 MPa
  3. 24 MPa Correct answer.
  4. 12 MPa

Explanation

Bending stress follows σ = Mc/I, using c = h/2 = 50 mm and I = bh³/12 = 4.17 × 10⁶ mm⁴. Substituting M = 2 × 10⁶ N·mm gives about 24 MPa at the extreme fibre, which is where the stress is highest.

Question 34Mechanics of Materials and Machine Design

A solid circular shaft transmits 20 kW at 300 rev/min. If the allowable shear stress is 40 MPa, what minimum shaft diameter is required by the torsion formula alone?

  1. 54.4 mm
  2. 34.6 mm
  3. 43.2 mm Correct answer.
  4. 21.6 mm

Explanation

Torque is power divided by angular velocity, so T = 20,000 W ÷ 31.42 rad/s ≈ 637 N·m. Rearranging the torsion formula τ = 16T/(πd³) for τ = 40 MPa gives a minimum diameter of about 43 mm.

Question 35Mechanics of Materials and Machine Design

A steel column of length 2 m is pin-ended in both planes, with E = 200 GPa and a minimum second moment of area of 2.0 × 10⁶ mm⁴. What is the Euler critical buckling load?

  1. 2014 kN
  2. 3948 kN
  3. 247 kN
  4. 987 kN Correct answer.

Explanation

For a pin-ended column the effective length equals the actual length, so Pcr = π²EI/L². With E = 200 GPa, I = 2.0 × 10⁶ mm⁴ and L = 2 m, the critical load works out to roughly 990 kN, provided the column is long and slender enough for elastic buckling.

Question 36Mechanics of Materials and Machine Design

A machined steel part has an endurance limit of 200 MPa and an ultimate tensile strength of 600 MPa. Using the Goodman relation, what completely reversed stress amplitude is permissible when the mean stress is 150 MPa?

  1. 100 MPa
  2. 200 MPa
  3. 175 MPa
  4. 150 MPa Correct answer.

Explanation

The Goodman line is written σa/Se + σm/Sut = 1, so the allowable amplitude falls as mean stress rises. Substituting Se = 200 MPa, Sut = 600 MPa and σm = 150 MPa gives σa = 200(1 − 0.25) = 150 MPa.

Question 37Mechanics of Materials and Machine Design

A plane stress element has σx = 80 MPa, σy = 20 MPa and τxy = 30 MPa, with no other stress components. What are the two principal stresses?

  1. 80 MPa and 20 MPa
  2. 122.4 MPa and 22.4 MPa
  3. 92.4 MPa and 7.6 MPa Correct answer.
  4. 100 MPa and 0 MPa

Explanation

Mohr's circle is centred at (σx + σy)/2 = 50 MPa with a radius of √(30² + 30²) ≈ 42.4 MPa. Adding and subtracting the radius from the centre gives principal stresses of about 92.4 MPa and 7.6 MPa.

Question 38Mechanics of Materials and Machine Design

A square-thread power screw has a lead angle of 3.0° and a friction angle of 8.5°. What is the efficiency of the screw when raising the load?

  1. 13%
  2. 35%
  3. 26% Correct answer.
  4. 65%

Explanation

For a square thread the raising efficiency is η = tan λ / tan(λ + φ). With λ = 3.0° and φ = 8.5°, this becomes 0.0524 / 0.2035 ≈ 0.26, so about 26 percent of the input work is converted into useful lifting work.

Question 39Mechanics of Materials and Machine Design

A thick-walled steel cylinder with internal radius 50 mm and external radius 100 mm is subjected to an internal pressure of 100 MPa with closed ends. What is the maximum hoop stress in the wall?

  1. 100 MPa
  2. 266.7 MPa
  3. 33.3 MPa
  4. 166.7 MPa Correct answer.

Explanation

Lamé's equations place the peak hoop stress at the bore, where σθ = p(ro² + ri²)/(ro² − ri²). Substituting p = 100 MPa, ri = 50 mm and ro = 100 mm gives 100 × 12,500 / 7,500 ≈ 167 MPa, tensile at the inner surface.

Question 40Mechanics of Materials and Machine Design

A shaft surface element is in plane stress with σx = 100 MPa and τxy = 50 MPa, all other components zero. What is the von Mises (distortion energy) equivalent stress?

  1. 100 MPa
  2. 132.3 MPa Correct answer.
  3. 141.4 MPa
  4. 122.5 MPa

Explanation

For plane stress with σy = 0, the von Mises stress reduces to √(σx² + 3τxy²). Substituting the values gives √(10,000 + 7,500) ≈ 132 MPa, which is the stress to compare with the yield strength in a ductile design check.

Question 41Mechanics of Materials and Machine Design

For an isotropic steel with Young's modulus E = 210 GPa and Poisson's ratio ν = 0.30, what is the shear modulus G?

  1. 161.5 GPa
  2. 80.8 GPa
  3. 105 GPa
  4. 70.0 GPa Correct answer.

Explanation

For an isotropic linear-elastic material, the shear modulus is related to Young's modulus and Poisson's ratio by G = E/[2(1+ν)]. Substituting E = 210 GPa and ν = 0.30 gives G = 210/(2×1.30) = 80.8 GPa. This relation assumes small strains and isotropic behavior.

Question 42Mechanics of Materials and Machine Design

A thin-walled cylindrical pressure vessel has an internal diameter of 1.0 m, a wall thickness of 10 mm, and an internal pressure of 2 MPa. What is the hoop stress in the wall?

  1. 200 MPa
  2. 100 MPa Correct answer.
  3. 25 MPa
  4. 50 MPa

Explanation

For a thin-walled cylindrical pressure vessel, the hoop stress is σ_h = pr/t, where r is the internal radius. Here r = 0.5 m, p = 2 MPa and t = 0.010 m, so σ_h = (2×0.5)/0.010 = 100 MPa. The longitudinal stress would be half this value, 50 MPa.

Question 43Mechanics of Materials and Machine Design

A flat-belt drive has a coefficient of friction of 0.30 and a wrap angle of π radians. At impending slip, what is the ratio of tight-side tension to slack-side tension?

  1. 2.57 Correct answer.
  2. 1.35
  3. 3.14
  4. 1.57

Explanation

At impending slip in a flat-belt drive, the tension ratio is T1/T2 = e^{μθ}, with θ in radians. Using μ = 0.30 and θ = π rad gives e^{0.30π} = e^{0.942} ≈ 2.57. A larger wrap angle or friction coefficient would increase this ratio.

Question 44Mechanics of Materials and Machine Design

A solid circular steel shaft has a diameter of 40 mm and a length of 1.2 m. It transmits a torque of 500 N·m, and the shear modulus is 80 GPa. What is the angle of twist over the shaft length?

  1. 0.015 rad
  2. 0.120 rad
  3. 0.030 rad Correct answer.
  4. 0.060 rad

Explanation

For a circular shaft in pure torsion, the angle of twist is θ = TL/(GJ), with J = πd^4/32 for a solid shaft. Here J ≈ 2.513×10^-7 m^4, so θ = (500×1.2)/(80×10^9×2.513×10^-7) ≈ 0.030 rad. The result is in radians because consistent SI units are used.

Question 45Mechanics of Materials and Machine Design

A simply supported beam has a span of 4 m and carries a central point load of 10 kN. If E = 200 GPa and I = 20×10^6 mm^4, what is the maximum deflection at midspan?

  1. 6.67 mm
  2. 3.33 mm Correct answer.
  3. 1.67 mm
  4. 13.3 mm

Explanation

For a simply supported beam with a central point load, the maximum deflection is δ = PL^3/(48EI). Converting I = 20×10^6 mm^4 to 20×10^-6 m^4 gives EI = 4.0×10^6 N·m^2. Then δ = (10,000×4^3)/(48×4.0×10^6) = 0.00333 m = 3.33 mm.

Question 46Mechanics of Materials and Machine Design

A notched steel component has a theoretical stress concentration factor Kt = 2.2 and a notch sensitivity q = 0.80. What is the fatigue notch factor Kf?

  1. 2.40
  2. 1.96 Correct answer.
  3. 1.76
  4. 2.20

Explanation

The fatigue notch factor is related to the theoretical stress concentration factor by Kf = 1 + q(Kt − 1), where q is the notch sensitivity. With q = 0.80 and Kt = 2.2, Kf = 1 + 0.80(1.2) = 1.96. The factor Kf is used with nominal stress in fatigue calculations, while Kt applies to the local elastic peak stress.

Question 47Mechanics of Materials and Machine Design

A preloaded bolted joint has an initial preload of 20 kN and is subjected to an external tensile load of 10 kN. If the joint stiffness ratio C = 0.25, what is the total bolt load?

  1. 30.0 kN
  2. 20.0 kN
  3. 25.0 kN
  4. 22.5 kN Correct answer.

Explanation

In a preloaded bolted joint, the bolt load under external tensile load Fe is Fb = Fi + C Fe, where C is the joint stiffness ratio (bolt stiffness divided by total stiffness). Here Fi = 20 kN, Fe = 10 kN and C = 0.25, so Fb = 20 + 0.25×10 = 22.5 kN. Only the bolt's share C of the external load adds to its preload.

Question 48Mechanics of Materials and Machine Design

A ball bearing has a basic dynamic load rating C = 30 kN and operates with an equivalent dynamic load P = 5 kN. What is its basic rating life in million revolutions?

  1. 6 million rev
  2. 1,296 million rev
  3. 216 million rev Correct answer.
  4. 36 million rev

Explanation

For a ball bearing, the basic rating life in million revolutions is L10 = (C/P)^3, where C is the dynamic load rating and P is the equivalent dynamic load. With C = 30 kN and P = 5 kN, C/P = 6, so L10 = 6^3 = 216 million revolutions. Roller bearings use an exponent of 10/3 instead of 3.

Question 49Mechanics of Materials and Machine Design

A helical compression spring has a wire diameter of 4 mm, a mean coil diameter of 32 mm, and an axial load of 200 N. Using the Wahl factor, what is the maximum shear stress in the wire?

  1. 255 MPa
  2. 302 MPa Correct answer.
  3. 380 MPa
  4. 340 MPa

Explanation

The Wahl factor corrects the nominal torsional shear stress for direct shear and coil curvature: Kw = (4C−1)/(4C−4) + 0.615/C, with spring index C = D/d. For C = 8, Kw ≈ 1.184. The maximum shear stress is τ = Kw 8FD/(πd^3) = 1.184×8×200×0.032/(π×0.004^3) ≈ 302 MPa.

Question 50Mechanics of Materials and Machine Design

A single-plate clutch has one pair of friction surfaces with outer radius 120 mm and inner radius 80 mm. The axial clamping force is 4 kN and the coefficient of friction is 0.30. Using the uniform-wear theory, what is the torque capacity?

  1. 160 N·m
  2. 144 N·m
  3. 96 N·m
  4. 120 N·m Correct answer.

Explanation

Under the uniform-wear assumption for a clutch, the friction radius is the mean radius rm = (ro + ri)/2, and the torque capacity for one pair of surfaces is T = μ F rm. Here rm = (0.120 + 0.080)/2 = 0.100 m, so T = 0.30×4000×0.100 = 120 N·m. Uniform-pressure theory would give a slightly different torque for the same dimensions.