Mechanical Engineering 2021 Paper I 50 marks Calculate

Q4

(a) Find out the value of atomic packing factor for the FCC crystal structure. Calculate the radius of an iridium (Ir) atom, given that Ir has an FCC crystal structure, a density of 22.4 g/cm³ and an atomic weight of 192.2 g/mol. [Avogadro's number (Nₐ) = 6.022 × 10²³ atoms/mol] (15 marks) (b) An inscribed circular hole is made in a triangular lamina with each side 'a'. Find the area moment of inertia of this lamina about one of the sides. (15 marks) (c) Four masses A, B, C and D revolve at equal radii and are equally spaced along a shaft. The mass B weighs 6 kg. Masses C and D make angles of 90° and 240° respectively with B in the same direction. Find the magnitude of the masses A, C and D and the angular position of A, if the system is in complete balance. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) FCC क्रिस्टल संरचना के लिए परमाणु संकुलन गुणक का मान ज्ञात कीजिए। एक ईरिडियम (Ir) परमाणु की त्रिज्या की गणना कीजिए। दिया गया है कि Ir की संरचना एक FCC क्रिस्टल संरचना है, घनत्व 22.4 g/cm³ तथा परमाणु भार 192.2 g/mol है। [आवोगाद्रो की संख्या (Nₐ) = 6.022 × 10²³ परमाणु/मोल है]। (15 अंक) (b) प्रत्येक भुजा 'a' वाले एक तिकोने पटल पर एक वृत्तीय छिद्र अन्तर्वृत (इन्स्क्राइब्ड) है। इस पटल का एक भुजा के सापेक्ष क्षेत्रफलीय जड़त्व आघूर्ण ज्ञात कीजिए। (15 अंक) (c) चार द्रव्यमान A, B, C तथा D समान त्रिज्या पर घूर्णन करते हैं और एक शाफ्ट पर बराबर दूरी पर हैं। द्रव्यमान B का भार 6 kg है। द्रव्यमान B से द्रव्यमान C तथा D क्रमशः 90° व 240° का कोण एक ही दिशा में बनाते हैं। यदि तंत्र पूर्णतया संतुलन में है, तो द्रव्यमान A, C तथा D का परिमाण और A की कोणीय स्थिति ज्ञात कीजिए। (20 अंक)

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Approach

Calculate the three distinct mechanical quantities in sequence: (a) FCC atomic packing factor and Ir atomic radius using density relations—allocate ~30% time; (b) area moment of inertia for composite triangular lamina with circular hole using parallel axis theorem—allocate ~30% time; (c) complete balancing of four rotating masses using force and couple polygon methods—allocate ~40% time as it carries highest marks. Present each part with clear headings, state assumptions, show unit conversions, and conclude with physical significance.

Key points expected

  • Part (a): APF = 0.74 (or 74%) derived from 4 atoms per unit cell and face-diagonal relation 4R = √2a; Ir atomic radius r ≈ 1.36 × 10⁻¹⁰ m or 0.136 nm using ρ = 4M/(Nₐa³) and a = 2√2r
  • Part (b): Moment of inertia of solid equilateral triangle about base = √3a⁴/32; subtract moment of inscribed circle (radius r = a/2√3) using parallel axis theorem; final I = √3a⁴/32 - πa⁴/288 = (9√3 - π)a⁴/288
  • Part (c): Angular positions: B at 0°, C at 90°, D at 240°, A at 270° (or -90°); masses satisfy m_A = 3 kg, m_C = 3 kg, m_D = 4 kg from ΣF = 0 and ΣM = 0 conditions
  • Force polygon closure: horizontal components m_B + m_C cos90° + m_D cos240° + m_A cosθ_A = 0; vertical components similarly
  • Couple polygon closure (equal radii and spacing): moments balance about reference plane
  • Unit consistency throughout: g/cm³ to kg/m³, cm to m, degrees to radians where needed; final answers with appropriate significant figures

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10For (a) correctly relates FCC face-diagonal to atomic radius and derives APF = π/(3√2) ≈ 0.74; for (b) applies parallel axis theorem correctly to composite section with hole; for (c) sets up both force and couple equilibrium equations recognizing equal radii and spacing simplify to mass-angle relations.Uses correct formulas but with minor conceptual gaps—e.g., applies parallel axis theorem without stating axis shift, or sets up balancing equations but confuses angle reference direction.Fundamental errors: treats FCC as BCC for APF, ignores hole in moment of inertia calculation, or applies single-plane balancing only missing couple balance for complete equilibrium.
Numerical accuracy20%10All three parts: APF = 0.74 (exact), Ir radius ≈ 0.136 nm, moment of inertia coefficient ≈ 0.0432a⁴, balancing masses A=3kg, C=3kg, D=4kg with A at 270°; unit conversions handled flawlessly, Avogadro's number used correctly.Correct final answers in two parts with minor arithmetic slip in one (e.g., wrong power of 10 in density conversion, or sign error in mass calculation); or correct method but final numerical values slightly off.Multiple calculation errors: wrong lattice parameter from density, incorrect hole radius for inscribed circle, or balancing masses that don't satisfy equilibrium checks; missing or inconsistent units.
Diagram quality20%10For (a): FCC unit cell sketch with atoms at corners and face centers, face diagonal labeled 4R; for (b): equilateral triangle with inscribed circle, base as reference axis, dimensions a and r marked; for (c): force polygon and couple polygon drawn to scale with vectors labeled m_B, m_C, m_D, m_A and angles marked.Diagrams present but incomplete: FCC cell without face-center atoms shown, triangle sketch without hole indicated, or balancing polygons drawn without angle labels or scale.No diagrams or seriously flawed: incorrect crystal structure drawn, triangle and circle not geometrically related, or no polygon diagrams for balancing—purely algebraic attempt.
Step-by-step derivation20%10Each part shows complete derivation: (a) volume of unit cell, atoms per cell, APF formula then density→lattice parameter→radius; (b) I_triangle from integration or standard formula, I_circle with parallel axis, subtraction; (c) tabulated data (mass, radius, distance, angle), ΣF_x = 0, ΣF_y = 0, ΣM = 0 equations solved systematically.Derivations present but condensed: jumps from density formula to final radius, states parallel axis theorem result without showing d²A term, or solves balancing by inspection without formal equilibrium equations.Minimal working: states final answers only, or uses incorrect derivation paths (e.g., guesses APF without geometric proof, treats hole as point mass, or assumes all masses equal for balancing).
Practical interpretation20%10For (a): notes FCC is close-packed, explains why Ir's high density correlates with small atomic radius and high atomic weight, relevance to catalysis and spark plugs; for (b): discusses structural efficiency of triangular sections with lightening holes in aerospace panels; for (c): explains complete vs static balancing, critical speeds, and application to crankshafts, turbine rotors, and ISRO launch vehicle turbopumps.Brief contextual mention: states FCC is common in metals, notes moment of inertia affects bending stiffness, or mentions balancing reduces vibration without specific engineering application.No interpretation; treats all parts as pure mathematical exercises with no connection to materials science, structural design, or rotating machinery engineering practice.

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