Mechanical Engineering 2024 Paper I 50 marks Calculate

Q7

(a) An orthogonal cutting of metal is performed using cutting speed of 120 m/min, tool rake angle 5° and width of cut 5 mm. Cutting produces chip thickness 0·4 mm and generates main cutting and thrust forces of 600 N and 300 N, respectively. Considering uncut chip thickness as 0·2 mm, calculate the chip thickness ratio, friction force and percentage of total cutting energy used to overcome friction at chip-tool interface. (20 marks) (b) A company produces both interior and exterior paints from raw materials M1 and M2. The following table provides the basic data of the problem: | | Raw Material per ton of Paints (tons) | Maximum Daily | | | Exterior Paint | Interior Paint | Availability (tons) | | Raw Material M1 | 6 | 4 | 24 | | Raw Material M2 | 1 | 2 | 6 | | Profit per ton (₹) | 4,00,000 | 3,20,000 | | A market survey indicates that the daily demand for interior paint cannot exceed that for exterior paint by more than 1 ton. Also the maximum daily demand for interior paint is 2 tons. This company wants to determine the optimum product mix of interior and exterior paints that maximizes the total daily profit. Comment on the optimal solution if the objective function is maximization of Z = 480000x₁ + 320000x₂. (20 marks) (c) What are the elements of a quality costing system? How does an organization benefit from a quality costing system? (10 marks)

हिंदी में प्रश्न पढ़ें

(a) धातु की एक लंबिक कतन क्रिया की जाती है, जिसमें कतन गति 120 m/min, औजार नति कोण 5° तथा कतन की चौड़ाई 5 mm है। कतन 0·4 mm मोटा छीलन (चिप) पैदा करता है तथा मुख्य कतन बल और प्रणोद बल क्रमशः 600 N तथा 300 N उत्पन्न करता है। बिना कटे चिप की मोटाई 0·2 mm मानते हुए चिप मोटाई अनुपात, घर्षण बल तथा चिप-औजार के बीच घर्षण को दूर करने में लगी हुई ऊर्जा, जो कुल कतन वाली ऊर्जा के प्रतिशत में हो, ज्ञात कीजिए। (20 अंक) (b) एक कंपनी कच्चे माल M1 तथा M2 से आंतरिक और बाहरी दोनों पेंट का उत्पादन करती है। निम्न तालिका समस्या का मूल आंकड़ा प्रदान करती है : | | प्रति टन पेंट में कच्चा माल (टन) | | प्रतिदिन अधिकतम उपलब्धता (टन) | |---|---|---|---| | | बाहरी पेंट | आंतरिक पेंट | | | कच्चा माल M1 | 6 | 4 | 24 | | कच्चा माल M2 | 1 | 2 | 6 | | प्रति टन लाभ (₹) | 4,00,000 | 3,20,000 | | एक बाजार सर्वेक्षण से पता चलता है कि आंतरिक पेंट की दैनिक मांग, बाहरी पेंट की दैनिक मांग से 1 टन से अधिक नहीं हो सकती है। साथ ही आंतरिक पेंट की अधिकतम दैनिक मांग 2 टन है। कंपनी आंतरिक और बाहरी पेंट का अनुकूलतम उत्पाद मिश्रण निर्धारित करना चाहती है, जो कुल दैनिक लाभ को अधिकतम करे। अनुकूलतम समाधान पर टिप्पणी कीजिए, यदि उद्देश्य फलन, z = 480000x₁ + 320000x₂ का अधिकतमीकरण है। (20 अंक) (c) गुणवत्ता लागत प्रणाली के कौन-कौन से तत्व हैं? गुणवत्ता लागत प्रणाली से एक संस्था किस प्रकार लाभान्वित होती है? (10 अंक)

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Directive word: Calculate

This question asks you to calculate. The directive word signals the depth of analysis expected, the structure of your answer, and the weight of evidence you must bring.

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How this answer will be evaluated

Approach

Calculate requires systematic numerical solutions for parts (a) and (b) with conceptual elaboration for part (c). Allocate approximately 40% time to part (a) for metal cutting calculations, 40% to part (b) for linear programming formulation and solution, and 20% to part (c) for quality costing elements. Begin each numerical part with clear identification of given data, apply correct formulae with derivations, and conclude with practical interpretations relevant to Indian manufacturing contexts.

Key points expected

  • Part (a): Chip thickness ratio r = t/tc = 0.2/0.4 = 0.5; shear angle φ = arctan(r·cosα/(1-r·sinα)) = 28.18°; friction force F = Fc·sinα + Ft·cosα = 600sin5° + 300cos5° = 351.4 N; friction energy percentage = (F·Vc)/(Fc·V) × 100 where Vc = V·sinφ/cos(φ-α)
  • Part (a): Normal force N = Fc·cosα - Ft·sinα = 571.6 N; coefficient of friction μ = F/N = 0.615; friction angle β = arctan(μ) = 31.6°; percentage energy for friction = (F·sinβ)/(Fc·cos(β-α)) × 100 or equivalent energy ratio approach yielding ~29-30%
  • Part (b): Correct LP formulation: Maximize Z = 400000x₁ + 320000x₂ subject to 6x₁ + 4x₂ ≤ 24, x₁ + 2x₂ ≤ 6, x₂ - x₁ ≤ 1, x₂ ≤ 2, x₁, x₂ ≥ 0; corner points (0,0), (4,0), (3,1.5), (2,2), (0,2); optimal at (3, 1.5) with Z = ₹16,80,000
  • Part (b): For modified objective Z = 480000x₁ + 320000x₂, re-evaluate corner points; new optimal at (4, 0) with Z = ₹19,20,000; comment on specialization in exterior paint and sensitivity to profit coefficient changes
  • Part (c): Four elements of quality costing: prevention costs (training, process design), appraisal costs (inspection, testing), internal failure costs (scrap, rework), external failure costs (warranty, returns, reputation loss)
  • Part (c): Benefits include: quantifying 'hidden' quality costs, prioritizing prevention over appraisal/failure, supporting ISO 9001 and TQM initiatives, enabling cost-benefit analysis of quality improvements, enhancing competitiveness of Indian MSMEs in global supply chains

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10For (a): Correctly applies Merchant's circle, distinguishes between rake angle and shear angle relationships, uses proper velocity relationships; for (b): Accurately formulates all constraints including market survey conditions, identifies correct feasible region; for (c): Precisely defines all four quality cost categories with correct examples from Indian industry (e.g., BIS certification, automotive supplier quality).For (a): Uses correct basic formulae but confuses friction force direction or energy calculation approach; for (b): Correct objective and most constraints but misses one market condition; for (c): Lists three cost categories correctly but merges or misclassifies one element.For (a): Fundamental confusion between chip thickness ratio and cutting ratio, or ignores rake angle in friction calculations; for (b): Treats as transportation problem or incorrect inequality directions; for (c): Only two generic cost elements identified with no organizational benefit linkage.
Numerical accuracy20%10Part (a): r = 0.5, φ = 28.18°, F = 351.4 N, N = 571.6 N, friction energy % = 29.4% (or equivalent precise calculation); Part (b): All corner points correct, optimal (3, 1.5), Z = ₹16,80,000; modified Z optimal (4,0), Z = ₹19,20,000; all calculations show 3+ significant figures with proper units.Part (a): Correct r and φ but friction force off by 10-15% due to angle handling; Part (b): Correct graphical approach but one corner point calculation error leading to slightly wrong optimum; Part (c): Not applicable—conceptual only; overall minor arithmetic slips with mostly correct methodology.Part (a): Order-of-magnitude errors in friction force or energy percentage (e.g., >50% error); Part (b): Fundamental errors in constraint handling or simplex/graphical method application; wrong optimal point; Part (c): No numerical content expected but if quantitative examples given, they are incorrect.
Diagram quality15%7.5For (a): Clear Merchant's circle diagram with all forces (Fc, Ft, F, N) labelled with correct angles (α, β, φ); velocity diagram showing V, Vc, Vs; for (b): Accurate graphical LP solution with properly scaled axes, all constraint lines, shaded feasible region, corner points labelled, iso-profit line shown; for (c): Quality cost pyramid or P-A-F model diagram if relevant.For (a): Merchant's circle drawn but angles mislabelled or one force missing; for (b): Feasible region correct but scale poor or one constraint line inaccurate; for (c): No diagram expected but if attempted, basic structure only.For (a): No force diagram or completely wrong force relationships shown; for (b): No graphical representation or infeasible region incorrectly shaded; axes unlabelled, no scale.
Step-by-step derivation25%12.5For (a): Explicit derivation of r = t/tc, tanφ = r·cosα/(1-r·sinα), F = Fc·sinα + Ft·cosα, N = Fc·cosα - Ft·sinα, energy ratio = F·Vc/(Fc·V) with Vc/V = sinφ/cos(φ-α); for (b): Complete LP standard form, systematic corner point evaluation, sensitivity analysis comment; for (c): Logical flow from cost elements to organizational benefits with cause-effect reasoning.For (a): Correct formulae stated but skips derivation of shear angle or velocity relationships; for (b): Correct method but jumps to answer without showing all corner evaluations; for (c): Descriptive list without clear logical progression.For (a): Final answers only with no formulae or derivation; for (b): No method shown—answers guessed; for (c): Disconnected bullet points with no explanatory linkage.
Practical interpretation20%10For (a): Interprets friction energy % in context of tool wear, suggests optimal cutting conditions for Indian HSS/carbide tools, relates to specific cutting energy; for (b): Discusses resource allocation implications, sensitivity to profit changes, relevance for MSME paint manufacturers; for (c): Concrete Indian examples (Tata Steel quality costing, Maruti Suzuki supplier quality programs), quantified ROI of prevention investment.For (a): States friction is undesirable but no specific tool life or productivity link; for (b): Generic comment on LP usefulness without context; for (c): Generic benefits listed without Indian industry examples.For (a) and (b): No interpretation—pure calculation; for (c): Vague statements like 'quality is important' with no organizational benefit linkage.

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