Mechanical Engineering 2025 Paper I 50 marks Calculate

Q2

(a) (i) A cylinder C weighing 1000 N is supported by two cylinders A and B weighing 500 N each as shown in the figure. Calculate the contact forces at P, Q and R. (10 marks) (ii) The state of plane stress at a critical point in a machine member is as follows: σₓ = 80 MPa tensile, σᵧ = 40 MPa compressive and complementary shear is 25 MPa. The yield strength of the material is 250 MPa. Determine the factor of safety with respect to the yield strength using: (10 marks) (I) maximum shear criterion, and (II) maximum distortion energy criterion. (b) In a Proell governor with open arms, the mass of each ball is 7 kg and the mass of the sleeve (central load) is 140 kg. The length of each arm is 180 mm and the length of extension of lower arms to which the balls are attached is 90 mm. The distance of pivots of the arms from axis of rotation is 25 mm and radius of rotation of the balls is 190 mm when the arms are inclined 40° to the axis of rotation. Determine the equilibrium speed for the given configuration. (20 marks) Further, if the frictional effect is included by incorporating a frictional force of 40 N at the sleeve, determine the coefficient of insensitiveness and range of speed when the governor is insensitive. (c) Classify the nanomaterials depending on the number of dimensions in the nano range. Support your answer with the geometry of classified nanomaterials. (10 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) चित्र में दर्शाए अनुसार 1000 N वजन का एक बेलन C प्रत्येक 500 N वजन रखने वाले दो बेलनों A तथा B द्वारा समर्थित है। P, Q तथा R पर संपर्क बलों की गणना कीजिए। (10 अंक) (ii) एक मशीन अवयव में एक क्रांतिक बिंदु पर द्विविम प्रतिबल की अवस्था इस प्रकार है: σₓ = 80 MPa तनन, σᵧ = 40 MPa संपीडन तथा पूरक अपरूपण 25 MPa है। सामग्री का प्रारंभ सामर्थ्य 250 MPa है। निम्नलिखित का प्रयोग करते हुए प्रारंभ सामर्थ्य के संबंध में सुरक्षा गुणक निर्धारित कीजिए: (10 अंक) (I) अधिकतम अपरूपण मानदंड, और (II) अधिकतम विकृति ऊर्जा मानदंड। (b) खुली भुजाओं वाले एक प्रोएल अधिनियंत्रक (गवर्नर) में प्रत्येक गेंद का द्रव्यमान 7 kg तथा स्लीव (केंद्रीय भार) का द्रव्यमान 140 kg है। प्रत्येक भुजा की लंबाई 180 mm तथा निचली भुजाओं के विस्तार की लंबाई 90 mm है जिनके साथ गेंदें जुड़ी हैं। भुजाओं के कीलक की घूर्णन अक्ष से दूरी 25 mm तथा भुजाओं के घूर्णन अक्ष के प्रति 40° झुकाव पर गेंदों के घूर्णन की त्रिज्या 190 mm है। दिए गए विन्यास हेतु संतुलन गति निर्धारित कीजिए। (20 अंक) आगे, यदि घर्षण प्रभाव को स्लीव पर 40 N के घर्षण बल के साथ शामिल करके जोड़ा जाए, तो असंवेदिता गुणांक तथा अधिनियंत्रक के असंवेदनशील होने की गति सीमा का निर्धारण कीजिए। (c) नैनो श्रेणी में आयामों की संख्या के आधार पर नैनो सामग्रियों (नैनोमटीरियल्स) को वर्गीकृत कीजिए। वर्गीकृत नैनो सामग्रियों की ज्यामिति के साथ अपने उत्तर को समर्थित कीजिए। (10 अंक)

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How this answer will be evaluated

Approach

Calculate the contact forces in (a)(i) using free-body diagrams and equilibrium equations for three-cylinder system; compute factor of safety in (a)(ii) applying both Tresca and von Mises yield criteria with principal stress transformation. For (b), derive Proell governor equilibrium speed using instantaneous centre method, then incorporate friction to find coefficient of insensitiveness and speed range. For (c), classify nanomaterials by dimensional constraints with schematic geometries. Allocate approximately 20% time to (a)(i), 20% to (a)(ii), 40% to (b), and 20% to (c) based on marks distribution.

Key points expected

  • (a)(i) Contact forces: P = 577.35 N, Q = 577.35 N, R = 1500 N using angle geometry (30° between cylinder centres) and ΣF=0, ΣM=0
  • (a)(ii)(I) Maximum shear criterion: τ_max = (σ₁-σ₂)/2, compare with σ_y/2; FOS = 250/(2×57.02) ≈ 2.19
  • (a)(ii)(II) Distortion energy: σ_eq = √(σ₁²+σ₂²-σ₁σ₂), compare with σ_y; FOS = 250/96.82 ≈ 2.58
  • (b) Proell governor: equilibrium speed N = 60ω/2π where ω² = (Mg + mg/2)(1+tan²α)tanα / [mr(1+k tan²α)]; k = extension/arm length ratio
  • (b) Coefficient of insensitiveness = (ω₂² - ω₁²)/(2ω²) = F_friction/(Mg + mg/2); speed range ΔN = N_max - N_min
  • (c) 0D: quantum dots (spherical, all dimensions nano); 1D: nanotubes/nanowires (one dimension bulk); 2D: graphene/quantum wells (two dimensions bulk); 3D: bulk nanostructured materials (zero dimensions nano)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies equilibrium conditions for three-cylinder contact problem; properly uses Mohr's circle or stress transformation for principal stresses; applies instantaneous centre method for Proell governor; distinguishes Tresca vs von Mises criteria; accurately classifies nanomaterials by dimensional confinement with correct geometric examples.Uses correct equilibrium equations but minor error in angle identification; applies yield criteria formulas correctly but may confuse principal stress signs; governor derivation mostly correct but omits k factor; nanomaterial classification correct but geometry descriptions vague.Confuses normal and tangential force components; applies yield criteria to given stresses without finding principal stresses; treats Proell as Porter governor; misclassifies nanomaterials or gives wrong dimensional examples.
Numerical accuracy20%10Contact forces P=Q=577.35 N, R=1500 N exact; principal stresses σ₁=90.97 MPa, σ₂=-50.97 MPa; FOS_Tresca=2.19, FOS_vonMises=2.58; governor speed N≈138-142 rpm; coefficient of insensitiveness ≈0.028-0.032; speed range ≈4-5 rpm; all calculations to 2 decimal places with proper units.Final answers correct but intermediate rounding errors; contact forces within 5%; FOS values within 0.2; governor speed within 10 rpm; coefficient of insensitiveness order of magnitude correct.Contact forces wrong by >20% due to angle error; principal stresses calculated incorrectly; FOS based on wrong stress values; governor speed calculation missing g or using wrong radius; no unit consistency.
Diagram quality15%7.5Clear FBD for three cylinders with all forces labelled at correct angles; Mohr's circle drawn with centre, radius, principal planes marked; Proell governor geometry with instantaneous centre I, all lengths (180mm, 90mm, 25mm, 190mm), angle 40° labelled; nanomaterial schematics showing 0D sphere, 1D cylinder, 2D sheet, 3D bulk with nano features.FBDs drawn but angles not marked; governor diagram shows configuration but missing key dimensions; nanomaterial classification listed without sketches or with generic shapes.No diagrams despite figure references; free-hand sketches without labels; wrong geometry (e.g., closed arms for Proell); nanomaterials described only textually.
Step-by-step derivation25%12.5Shows complete equilibrium equations for each cylinder with angle geometry derivation; principal stress formula derived or Mohr's circle construction shown; governor speed derived from moment equilibrium about instantaneous centre with clear force polygon; friction effect derived showing ω_max and ω_min expressions; all algebraic steps visible.Key equations stated but some algebraic steps skipped; principal stresses found by formula without derivation; governor equilibrium stated without I-centre explanation; final formulas quoted with partial substitution.Final answers only with no working; jumps to formulas without setup; governor treated as simple conical pendulum; no derivation of coefficient of insensitiveness.
Practical interpretation20%10Comments on why von Mises gives higher FOS than Tresca for ductile materials; explains significance of coefficient of insensitiveness for governor stability (hunting prevention); relates nanomaterial classification to applications (quantum dots in displays, graphene in electronics, nanowires in sensors); notes Indian context (IISc/IIT research in nanomaterials, BHEL governor applications).States which criterion is conservative; mentions that friction causes speed range; gives one application example for nanomaterials.No interpretation of results; treats all calculations as abstract exercises; no connection to engineering practice or Indian industry context.

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