Mechanical Engineering 2023 Paper I 50 marks Compulsory Solve

Q1

(a) A heavy carriage wheel of weight W, and radius r is to be dragged over an obstacle of height h by a horizontal force P applied to the centre of the wheel. Show that P is slightly greater than $\frac{W \cdot \sqrt{2rh - h^2}}{r - h}$. (10 marks) (b) A disc mounted on a shaft is having three masses of 6 kg, 5 kg and 4 kg, which are attached at a radial distance of 70 mm, 80 mm and 50 mm at the angular positions of 45°, 135° and 240° respectively. The angular positions are measured counter-clockwise from the reference line along the x-axis. Calculate the amount of countermass at the radial distance of 85 mm required for the static balance. (10 marks) (c) From a balloon ascending with a velocity of 25 m/s above the surface of a lake, a stone is let fall and the sound of the splash is heard 5 seconds later. Find the height of the balloon when the stone was dropped, assuming that the velocity of sound is 340 m/s. (10 marks) (d) What is the importance of the atomic packing factor ? Compute the atomic packing factor for the FCC crystal structure. (10 marks) (e) A 1·5 m long shaft having diameter 2·0 cm is held at the ends by long bearings. The weight of a disc at the centre of the shaft is 20 kg. If the modulus of elasticity of the material of shaft is 2 × 10⁶ kg/cm², then calculate the critical speed of the shaft in cycles per minute. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक भारी वाहन के चक्के को जिसका भार W तथा त्रिज्या r है, h ऊँचाई के अवरोध पर चक्के के केन्द्र की ओर निर्देशित क्षैतिज बल P से खींचा जाना है । सिद्ध कीजिए कि P, $\frac{W \cdot \sqrt{2rh - h^2}}{r - h}$ से थोड़ा अधिक है । (10 अंक) (b) एक शाफ्ट पर एक चक्रिका लगी है जिस पर 70 mm, 80 mm तथा 50 mm त्रिज्या दूरी पर 6 kg, 5 kg तथा 4 kg के तीन द्रव्यमान क्रमशः 45°, 135° तथा 240° कोणीय स्थितियों पर हैं । कोणीय स्थितियाँ x-अक्ष की दिशा में निर्देश रेखा से बामावर्त दिशा में मापी गई हैं । स्थैतिक संतुलन के लिए आवश्यक 85 mm त्रिज्या दूरी पर प्रति-द्रव्यमान की गणना कीजिए । (10 अंक) (c) एक गुब्बारे से जो कि 25 m/s के वेग से एक तालाब की सतह से ऊपर जा रहा है, एक पत्थर गिरने दिया जाता है और छपाके की आवाज 5 सेकंड के बाद सुनाई देती है । ध्वनि का वेग 340 m/s मानते हुए गुब्बारे की ऊँचाई उस समय ज्ञात कीजिए जब पत्थर को गिराया जाता है । (10 अंक) (d) परमाणवीय पैकिंग गुणक का क्या महत्व है ? FCC क्रिस्टल संरचना का परमाणवीय पैकिंग गुणक ज्ञात कीजिए । (10 अंक) (e) एक 2·0 cm व्यास के 1·5 m लम्बे शाफ्ट के सिरों को लम्बी बेयरिंग्स में रखा जाता है । शाफ्ट के केन्द्र पर स्थित एक चक्रिका का भार 20 kg है । यदि शाफ्ट के पदार्थ का प्रत्यास्थता मापांक 2 × 10⁶ kg/cm² हो, तो शाफ्ट की क्रांतिक गति चक्र प्रति मिनट में परिकलित कीजिए । (10 अंक)

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Approach

Solve each sub-part systematically, allocating approximately 20% time to each. For (a), draw the FBD and apply moment equilibrium about the contact point. For (b), resolve masses into x-y components and find resultant unbalance. For (c), set up simultaneous equations for stone fall time and sound travel time. For (d), define APF and derive the FCC packing calculation. For (e), apply Rayleigh's method or Dunkerley's equation for critical speed. Present derivations first, then numerical substitutions, with clear unit conversions throughout.

Key points expected

  • Part (a): Moment equilibrium about contact point B gives P(r-h) = W√(2rh-h²); shows P > W√(2rh-h²)/(r-h) due to friction/rolling resistance
  • Part (b): Σmr cosθ and Σmr sinθ computed; resultant unbalance found; countermass m_c = √(ΣFx² + ΣFy²)/(85 mm) with proper angle
  • Part (c): Quadratic equation in t₁ (fall time): ½gt₁² = h₀ + 25t₁; total time t₁ + t₂ = 5 where t₂ = (h₀ + 25t₁)/340; solves to h₀ ≈ 108 m
  • Part (d): APF = (volume of atoms in unit cell)/(volume of unit cell); FCC: 4 atoms, APF = π√2/6 ≈ 0.74; relates to ductility and slip systems
  • Part (e): δ = WL³/(48EI) for simply supported beam; ω_n = √(g/δ); converts to rpm; critical speed ≈ 845-850 rpm
  • All parts: Consistent unit handling (mm→m, kg→N, cm⁴→m⁴ where needed); g = 9.81 m/s² or 981 cm/s² appropriately
  • Diagrams: Free body diagram for (a) showing geometry; force polygon or component diagram for (b); shaft-deflection sketch for (e)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies moment equilibrium for (a), static balancing principles for (b), relative motion with sound propagation for (c), crystallography fundamentals for (d), and beam vibration theory for (e); recognizes that (a) requires P to be slightly greater due to practical considerations.Uses correct basic formulas but misapplies equilibrium conditions in one part (e.g., wrong moment arm in (a) or incorrect boundary conditions in (e)); minor conceptual gaps in (d) explanation.Fundamental errors: treats (a) as force equilibrium ignoring moments, uses dynamic balancing for (b), ignores balloon velocity in (c), confuses FCC with BCC in (d), or uses cantilever formula for (e).
Numerical accuracy20%10All five numerical answers correct within reasonable rounding: (b) countermass ~4.8 kg at appropriate angle, (c) h₀ ≈ 108 m, (e) critical speed ~845 rpm; handles unit conversions flawlessly (cm⁴ to m⁴, mm to m).Three to four parts correct with minor arithmetic slips in one; consistent use of g=10 m/s² instead of 9.81 with acknowledgment; small errors in decimal places or angle calculation in (b).Multiple calculation errors; wrong order of magnitude in (c) or (e); mixes CGS and SI units without conversion; uses diameter instead of radius in (a); incorrect I = πd⁴/64 formula application.
Diagram quality20%10Clear FBD for (a) showing wheel geometry, contact points, force directions and angle θ where cosθ = (r-h)/r; component force diagram for (b); shaft deflection sketch for (e) with disc at centre; all diagrams labelled with variables and geometric relationships.Diagrams present but incomplete labelling; missing angle indication in (a) or unclear coordinate system in (b); (e) shows simply supported ends but no deflection curve.No diagrams despite (a) and (e) requiring them; or diagrams drawn without any labels; wrong geometry (e.g., shows force P at wrong location in (a)).
Step-by-step derivation20%10Shows complete derivation chain: (a) geometry → moment equation → P expression; (b) tabular component resolution → resultant calculation; (c) simultaneous equations setup with clear time variables; (d) unit cell volume derivation; (e) δ = WL³/(48EI) derivation from beam theory.Skips some intermediate steps but logical flow maintained; jumps from formula to answer in one part; uses standard results without derivation in (e).Final answers stated without working; no equation setup shown for (c); uses memorized formula directly for (a) and (e) with no explanation of terms.
Practical interpretation20%10Interprets (a) as wheel/road obstacle problem relevant to vehicle design; (b) as rotating machinery balancing critical for turbines/pumps; (c) as projectile motion application; (d) links APF to FCC ductility and metal forming; (e) discusses critical speed avoidance in rotating shafts for Indian power systems (50 Hz).Brief practical mention for 2-3 parts; generic statements about importance without specific application context.No interpretation provided; treats all parts as pure mathematics; no connection to engineering applications or real-world relevance.

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