Mechanical Engineering 2024 Paper I 50 marks Compulsory Solve

Q1

(a) A rod of length 2 m and weight 450 N rests on ground at one end. The other end is supported by a cord through which a force of 200 N is applied. What is the minimum angle α for which equilibrium is possible? The coefficient of static friction between the rod and the floor is 0·4 : (10 marks) (b) A shaft of span 1·5 m and diameter 20 mm is simply supported at the ends. It carries a 125 kg concentrated mass at midspan. If E = 200 GPa, calculate its fundamental frequency. (10 marks) (c) Two thick washers are placed on the two ends of a copper tube and a steel bolt of pitch 1·8 mm is made to pass through these washers as shown in the figure below. The nut is tightened first with hand so that there are no stresses in the tube. Now, the nut is further tightened with the spanner through one-fourth of a turn. Calculate the axial stresses developed in the tube and the bolt. Young's modulus values for steel and copper are 200 GPa and 105 GPa, respectively : (10 marks) (d) Explain critical cooling rate using continuous cooling transformation (CCT) diagram and write its importance in hardening heat treatment of alloy steel. (10 marks) (e) A single-cylinder reciprocating engine has speed 250 rpm, stroke 300 mm, mass of the reciprocating parts 60 kg and mass of the revolving parts 50 kg at 150 mm radius. If two-thirds of the reciprocating parts and all the revolving parts are to be balanced, find (i) the balance mass required at a radius of 400 mm and (ii) the residual unbalanced force when the crank has rotated 60° from top dead centre. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक 2 m लम्बी तथा 450 N वजन की छड़ धरती पर अपने एक छोर पर स्थित है। इसका दूसरा छोर एक रस्सी से आलम्बित है, जिस पर 200 N का बल लग रहा है। सन्तुलन के लिए न्यूनतम कोण α क्या होगा? छड़ तथा फर्श के बीच का स्थैतिक घर्षण गुणांक 0·4 है : (10 अंक) (b) एक शाफ्ट की विस्तृति 1·5 m तथा व्यास 20 mm है, जिसे इसके दोनों किनारों पर साधारण आलम्ब सहारा दे रहे हैं। शाफ्ट के बीच में 125 kg का सकेन्द्रित द्रव्यमान स्थित है। यदि E = 200 GPa है, तो इसकी मूल आवृत्ति ज्ञात कीजिए। (10 अंक) (c) तांबे की ट्यूब के दोनों सिरों पर दो मोटे वॉशर रखे गए हैं और 1·8 mm पिच के एक इस्पात से बने बोल्ट को इन वॉशरों से गुजारा गया है जैसा नीचे दिए गए चित्र में दर्शाया गया है। नट को पहले हाथ से कस दिया जाता है ताकि ट्यूब में कोई प्रतिबल न हो। अब, नट को स्पैनर की सहायता से एक-चौथाई चक्कर देकर फिर से कस दिया जाता है। ट्यूब और बोल्ट में विकसित अक्षीय प्रतिबलों की गणना कीजिए। इस्पात तथा तांबे के यंग मापांक क्रमशः: 200 GPa तथा 105 GPa हैं : (10 अंक) (d) सतत शीतलन रूपांतरण (CCT) आरेख की सहायता से कांतिक शीतलन दर की व्याख्या कीजिए तथा इसका महत्व ऐलॉय इस्पात के कठोरीकरण ताप उपचार में बताइए। (10 अंक) (e) एक एकल सिलिंडर प्रत्यागामी इंजन की गति 250 rpm, स्ट्रोक 300 mm, प्रत्यागामी भागों का द्रव्यमान 60 kg तथा 150 mm त्रिज्या पर घूर्णी भागों का द्रव्यमान 50 kg है। यदि दो-तिहाई प्रत्यागामी भागों और सभी घूर्णी भागों को संतुलित किया जाना है, तो (i) 400 mm की त्रिज्या पर आवश्यक संतुलन द्रव्यमान और (ii) क्रैंक के उच्च निष्चल्य स्थिति (टॉप डेड सेंटर) से 60° घूर्णन पर अवशिष्ट असंतुलित बल ज्ञात कीजिए। (10 अंक)

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Approach

Solve each sub-part systematically, allocating approximately 12 minutes per 10-mark section. For (a), draw FBD and apply equilibrium with friction cone analysis; for (b), use Rayleigh's method or standard formula for simply supported beam with central mass; for (c), analyze bolt-tube assembly as statically indeterminate problem with compatibility condition; for (d), explain CCT diagram features and link to industrial hardening practice; for (e), apply balancing of reciprocating masses with analytical method. Present derivations first, then numerical substitution, with clear sub-headings for each part.

Key points expected

  • Part (a): Correct FBD with normal reaction N, friction force μN, tension T=200N; moment equilibrium about ground contact gives tan(α) ≥ (W - Tsinα)/(Tcosα - μW); iterative or quadratic solution yields α_min ≈ 51.3°
  • Part (b): Fundamental frequency f = (1/2π)√(k_eq/m_eq) where k = 48EI/L³ for simply supported beam with central load; I = πd⁴/64; calculate f ≈ 22.6 Hz
  • Part (c): Compatibility condition δ_bolt + δ_tube = pitch/4 = 0.45mm; force equilibrium P_bolt = P_tube = P; solve for P using 1/k_bolt + 1/k_tube = 0.45/P; stresses σ_steel ≈ 127 MPa, σ_copper ≈ 60 MPa
  • Part (d): Critical cooling rate defined as minimum cooling rate to avoid pearlite nose in CCT diagram; importance in preventing soft pearlite/ferrite formation, ensuring martensite for hardness in alloy steels like AISI 4340 used in automotive gears
  • Part (e): Balancing mass m_b = [(2/3)m_rec × r + m_rev × r_rev]/r_b = [(40×0.15) + (50×0.15)]/0.4 = 33.75 kg; residual unbalance force at 60° using exact and approximate methods with ω = 26.18 rad/s

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies: (a) impending slip condition with friction angle; (b) lumped mass SDOF vibration model; (c) statically indeterminate assembly with stiffness compatibility; (d) CCT diagram features (pearlite nose, martensite start/finish); (e) partial balancing of reciprocating engines with primary and secondary effects distinguished.Correct basic concepts for most parts but confuses (a) friction direction or (c) thermal expansion vs mechanical deformation; (d) mixes CCT with TTT diagram features; (e) uses full reciprocating mass instead of 2/3.Fundamental errors: treats (a) as frictionless, (b) as fixed-fixed beam, (c) as simple bolt stress without compatibility, (d) describes only iron-carbon diagram, (e) omits angular velocity in force calculation.
Numerical accuracy20%10All five parts: (a) α_min = 51.3° ± 0.5°; (b) f = 22.5-23.0 Hz; (c) σ_steel ≈ 127 MPa, σ_copper ≈ 60 MPa; (d) typical critical cooling rates cited (e.g., 8-10°C/s for 0.4%C steel); (e) m_b = 33.75 kg, residual force ≈ 1.8 kN at 60°. All intermediate steps shown with correct unit conversions (GPa to Pa, mm to m, rpm to rad/s).Correct final answers for 3-4 parts with minor arithmetic slips in one; or correct method but calculator errors in (b) frequency or (e) trigonometric evaluation at 60°.Multiple incorrect answers; order-of-magnitude errors (e.g., frequency in kHz, stresses in GPa); consistent failure to convert units (mm⁴ to m⁴ in moment of inertia).
Diagram quality20%10Clear FBD for (a) showing all forces at correct angles; (b) beam deflection shape with central mass; (c) assembly sketch with deformation exaggerated; (d) hand-drawn CCT diagram with time-temperature axes, transformation curves, and critical cooling rate line labelled; (e) crank mechanism with rotating vectors for balancing mass position.Diagrams present but missing labels or incorrect force directions in (a); CCT diagram drawn but axes unlabelled or curves generic without specific alloy reference.No diagrams for (a), (c), (d), (e); or diagrams contradict written solution (e.g., tension shown as compression).
Step-by-step derivation20%10Each part shows complete derivation: (a) ΣFx=0, ΣFy=0, ΣM=0 leading to quadratic in tan(α); (b) energy method or direct stiffness derivation; (c) stiffness expressions k=EA/L derived, compatibility equation set up; (d) logical progression from TTT to CCT, then CCR definition; (e) primary unbalanced force expression derived, then balancing condition applied.Derivations present but skips key steps (e.g., jumps from equilibrium to final formula in (a) or (b)); or uses memorized formula without showing origin.Final answers only with no working; or incorrect derivations leading to fortuitously correct answers by wrong method.
Practical interpretation20%10Links (a) to ladder safety angles; (b) to critical speed avoidance in turbine shafts; (c) to bolted joint preload in pressure vessels; (d) to selection of quenching medium (oil vs polymer vs water) for Indian automotive components; (e) to engine mounting design and vibration isolation in locomotive engines (e.g., WDG-4 diesel-electric).Brief mention of practical relevance for 2-3 parts without specific Indian industry context; generic statements like 'important in design'.No interpretation; treats all parts as purely academic exercises; or provides irrelevant applications (e.g., discussing beam theory for shaft vibration problem).

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