24 Sep UPSC Civil Services (Main) Examination 2026 — Mechanical Engineering Optional Paper I: Question Paper | Plutus IAS
The questions below are from Mechanical Engineering Optional Paper I of UPSC Civil Services (Main) Examination 2026 (held 2026-08-30) — the actual paper, which is public government content. This page carries the questions for reference and revision; model answers for this paper are not published here.
Official paper (PDF): upsc.gov.in.
- (a)  A uniform beam has a mass of 50 kg per metre length. Compute the reactions at the support O. The force loads shown lie in a vertical plane. 10  (b)  The fixed-ended bar ABCD consists of three prismatic segments, as shown in the figure. The end segments have a cross-sectional area $A_1 = 840 \text{ mm}^2$ and length $L_1 = 200 \text{ mm}$. The middle segment has a cross-sectional area $A_2 = 1260 \text{ mm}^2$ and length $L_2 = 250 \text{ mm}$. Loads $P_B$ and $P_C$ are equal to $25.5 \text{ kN}$ and $17.0 \text{ kN}$, respectively. (i) Determine the reactions $R_A$ and $R_D$ at the fixed supports. (ii) Determine the compressive axial force $F_{BC}$ in the middle segment of the bar.  (c) Why is grain size important in polycrystalline materials? How is it estimated? (i) Determine the ASTM grain size number of a metal specimen if 75 grains/square inch are measured at a magnification of 100×. (ii) For the same specimen, how many grains/inch² can be estimated at a magnification of 60×? (d) In a gear drive, the power component is 0.94 times the normal thrust with the addendum for stub teeth being 85% of the module. Determine the minimum number of teeth to avoid undercutting and length of path of contact in terms of module, m, if : (i) The gear ratio is unity. (ii) The gear ratio is 2.5. (e)  A turbine shaft of diameter $d$ is under a thrust load $P$ and torque $T$ (as shown in the figure). Calculate the largest permissible value of the load $P$ if the allowable normal stress is not to exceed $\sigma_{\text{all}}$. Given : $d = 60 \text{ mm}$, $T = 2 \text{ kN.m}$, $\sigma_{\text{all}} = 220 \text{ MPa}$  (i) Explain the conceptual basis of PTS. What fundamental assumption makes it possible to preassign time values to basic motions independently of the task context ?
- What is heat treatment ? What is its purpose ? Explain different heat treatment processes with the help of common microstructural phases of steel on an Iron-Carbon equilibrium diagram. (b) A steel tube has a mean diameter of 100 mm and a thickness of 3 mm. Calculate the torque which can be transmitted by the tube with a factor of safety of 2.25, if the criterion of failure is : – (i) Maximum shear stress – (ii) Maximum strain energy – (iii) Maximum shear strain energy The elastic limit of the steel in tension is 225 $\text{MN/m}^2$ and Poisson's ratio $\nu$ is 0.3. (c) $$T (N-m) = 8000 + 1000 \sin 2\theta – 2000 \cos 2\theta$$ The effective turning moment of a two-wheeler engine crankshaft for every $180^\circ$ of crank rotation is represented by : $$T (N-m) = 8000 + 1000 \sin 2\theta – 2000 \cos 2\theta$$ where $\theta$ is the crank inclination with respect to the inner dead centre. A flywheel of 500 kg mass with radius of gyration of 750 mm is provided on the crank. Assuming external resistance as constant at an engine speed of 300 rpm, determine : – (i) The maximum, minimum and mean torques. Also represent them on a $T – \theta$ diagram. – (ii) The power developed. – (iii) Total percentage fluctuation in speed. – (iv) If, owing to size constraints, radius of gyration of the flywheel is reduced to 600 mm, will the percentage (%) fluctuation of speed increase ? If yes, then by how much ? A manufacturing firm is redesigning its assembly line and is debating whether to use traditional time study (stopwatch) or predetermined time standards (PTS). Answer the following : (ii) Describe two specific situations in which PTS would be preferred over time study for capacity planning purposes, and two situations where it would be inappropriate.
- The acceleration of a particle along a straight line is defined by $a = (2t – 9) \text{ m/s}^2$, where $t$ is in seconds. At $t = 0$, $s = 1 \text{ m}$ and $v = 10 \text{ m/s}$. When $t = 9 \text{ s}$, determine : – (I) The particle's position, – (II) The total distance travelled, and – (III) The velocity of the particle.  A pressurized steel tank is constructed with a helical weld that makes an angle $\alpha = 55^\circ$ with the longitudinal axis (as shown in the figure). The tank has radius $r = 0.6$ m, wall thickness $t = 18$ mm, and internal pressure $P = 2.8$ MPa. Also, the steel has modulus of elasticity $E = 200$ GPa and Poisson's ratio $\nu = 0.30$. Determine the following quantities for the cylindrical part of the tank : – (I) The circumferential and longitudinal stresses. – (II) The maximum in-plane and out-of-plane shear stresses. – (III) The circumferential and longitudinal strains.  (b)  The surface of an airplane wing is subjected to plane stress with normal stresses $\sigma_x$ and $\sigma_y$ and shear stress $\tau_{xy}$, as shown in the figure. At a counterclockwise angle $\theta = 32^\circ$ from the x-axis, the normal stress is 37 MPa tension, and at an angle $\theta = 48^\circ$, it is 12 MPa compression. If the stress $\sigma_x$ equals 110 MPa tension, what are the stresses $\sigma_y$ and $\tau_{xy}$?   Determine the damped natural frequency of the system shown in the figure.  (c) A glass-epoxy composite consists of full length fibers of E-glass with modulus 72.5 GPa embedded in epoxy matrix with modulus 3.5 GPa. The volume percentage of glass fibers is 35%. Compute : – (i) The modulus of elasticity of the composite in longitudinal and transverse directions. – (ii) The strain sustained by fibers and matrix, if a stress of 50 MPa is imposed on a cross-sectional area of 250 mm$^{2}$.
-  A simple beam is loaded as shown in the figure. Using the double-integration method, determine the following : – (i) The equation of the elastic curve – (ii) The slope at the end A – (iii) The deflection at the midspan  (b) A shaft carries three eccentric masses of 1 kg each at radial distances of 20, 30 and 20 mm with the central plane 50 mm apart. Their angular positions are 120° apart. If the shaft is balanced by adding two masses at a radius of 700 mm and at distances of 100 mm each from the central plane of the middle eccentric, (i) Determine the magnitudes and angular positions of the masses. (ii) If one of the masses is shifted to a radius of 50 mm due to space constraints, determine the change in its magnitude. Will there be any change in the mass and location of the other masses? (a) (b)  Determine the magnitude P of the horizontal force required to initiate motion of the block of mass m₀ for the cases : (A) P is applied to the right, and (B) P is applied to the left. Compute a general solution in each case, and then evaluate your expression for the values θ = 30°, m = m₀ = 3 kg, μₛ = 0·60, and μₖ = 0·50. 
- A strip of width 250 mm is reduced from 30 mm to 20 mm thickness in a rolling mill of roll radius 400 mm. Considering coefficient of friction 0.25 and average flow stress 300 MPa, find the following : – (i) Contact length – (ii) Rolling force – (iii) Check if the biting condition is satisfied (b) A biomedical implant requires excellent surface integrity with no thermal damage. From the perspective of non-conventional machining, discuss the following : – (i) Suitability of USM, LBM, EDM for biomedical implant machining – (ii) Surface defects in each process – (iii) Best process selection with justification – (iv) Post-processing requirements of the selected process (c) | — | — | — | — | | J₁ | M₁ → 6, M₂ → 4, M₃ → 8, M₄ → 3 | 21 | 28 | | J₂ | M₂ → 5, M₁ → 7, M₄ → 2, M₃ → 6 | 20 | 25 | | J₃ | M₃ → 4, M₂ → 8, M₁ → 3, M₄ → 5 | 20 | 30 | | J₄ | M₁ → 9, M₃ → 3, M₂ → 5, M₄ → 4 | 21 | 35 | | J₅ | M₂ → 6, M₃ → 5, M₄ → 7, M₁ → 2 | 20 | 32 | A job shop has four machines (M₁ – M₄) and five jobs (J₁ – J₅), all available at t = 0. Each job has a unique routing. The processing time (hours) and due dates are given below. | Job | Routing | Total Work | Due Date | | — | — | — | — | | J₁ | M₁ → 6, M₂ → 4, M₃ → 8, M₄ → 3 | 21 | 28 | | J₂ | M₂ → 5, M₁ → 7, M₄ → 2, M₃ → 6 | 20 | 25 | | J₃ | M₃ → 4, M₂ → 8, M₁ → 3, M₄ → 5 | 20 | 30 | | J₄ | M₁ → 9, M₃ → 3, M₂ → 5, M₄ → 4 | 21 | 35 | | J₅ | M₂ → 6, M₃ → 5, M₄ → 7, M₁ → 2 | 20 | 32 | (i) Define the critical ratio (CR). Compute the CR for each job at t = 0 and rank them by urgency (lowest CR = highest priority). State which job is the most critical and explain in two sentences why the CR is more informative than the due date or processing time alone. (ii) Apply the shortest processing time (SPT) dispatching rule — at each machine, when free, select the waiting job with the smallest operation time for that machine. Construct a complete Gantt chart. Compute for each job : Completion time Cj, Flow time Fj, Lateness Lj, and Tardiness Tj. Summarise : makespan, mean flow time, total tardiness, maximum tardiness, number of tardy jobs.
- In an orthogonal cutting operation, the following data are obtained : Cutting force $F_c = 1500 \text{ N}$ Thrust force $F_t = 900 \text{ N}$ Rake angle $\alpha = 10^\circ$ Chip thickness ratio $r = 0.4$ Determine the following : (I) Shear angle ($\phi$) (II) Friction angle ($\beta$) (III) Coefficient of friction ($\mu$) (IV) Shear force ($F_s$) using Merchant's circle relations (b) A hole is specified as $25 \pm 0.01 \text{ mm}$. Using Taylor's principle, determine the following : (I) GO gauge size (II) NO-GO gauge size Assume : (A) Gauge tolerance = 10% of work tolerance (B) Wear allowance = 10% (only GO gauge) (b) In an organization, what do you understand by the roles of flexibility, lean practices and agility ? Explain how flexibility, lean practices and agility are complementary approaches. (c)
Questions reproduced from the official paper for study purposes; verify on the exam-conducting body’s official website.
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