Mechanical Engineering 2021 Paper I 50 marks Calculate

Q6

(a) Five jobs A, B, C, D and E need to be processed on a machine. Processing time (in days) and due date (from now) are given below : | Jobs | Processing Time (in days) | Due Date (from now) | |------|---------------------------|---------------------| | A | 6 | 5 | | B | 4 | 10 | | C | 5 | 15 | | D | 8 | 20 | | E | 7 | 30 | Determine the total completion time (in days) for all jobs, average flow time, average number of jobs in system per day and average tardiness using Shortest Processing Time (SPT) rule so as to establish appropriate sequence for processing of jobs. (15 marks) (b) Three cities namely Chennai, Delhi and Kolkata are being considered as potential locations for a new plant. Estimated data of annual fixed cost, variable cost and revenue per unit for each potential location are given below in the table. Determine the most attractive location for the plant if the estimated annual production volume desired is 40000 units. Also determine break-even production volume for each location. (15 marks) | Head | City | | | |------|------|---|---| | | Chennai | Delhi | Kolkata | | Fixed cost (₹) | 800000 | 600000 | 500000 | | Variable cost per unit (₹) | 30 | 40 | 50 | | Revenue per unit (₹) | 60 | 60 | 60 | (c) Orthogonal turning of a metal is performed using single point cutting tool having rake angle of 10°. The turning operation produces chip-thickness ratio of 0·2, horizontal cutting force (F_H) of 1400 N and vertical cutting force (F_V) of 2000 N. Determine the shear plane angle, normal force, friction force on rake face and friction angle. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) पाँच कृत्यों A, B, C, D व E का मशीन पर प्रक्रमण होना है । प्रक्रमण समय (दिनों में) व नियत तिथि (अभी से) निम्नवत हैं : | कृत्यक | प्रक्रमण समय (दिनों में) | नियत तिथि (अभी से) | |--------|------------------------|------------------| | A | 6 | 5 | | B | 4 | 10 | | C | 5 | 15 | | D | 8 | 20 | | E | 7 | 30 | सभी कृत्यों के लिए कुल समापन समय (दिनों में), औसत प्रवाह समय, प्रति दिन निकाय में कृत्यों की माध्य संख्या तथा न्यूनतम प्रसंस्करण समय (SPT) नियम से माध्य विलंबन ज्ञात कीजिए, जिससे कि कृत्यों के प्रक्रमण के लिए सटीक क्रम निर्धारित किया जा सके । (15 अंक) (b) तीन शहरों चेन्नई, दिल्ली व कोलकाता को एक नए संयंत्र के लिए संभावित लोकेशन माना जा रहा है । प्रत्येक संभावित लोकेशन के लिए वार्षिक स्थिर लागत, परिवर्त्य लागत तथा आय प्रति इकाई के अनुमानित आँकड़े नीचे सारणी में दिए गए हैं । संयंत्र के लिए अनुमानित 40000 इकाइयों की वार्षिक उत्पादन मात्रा के लिए सबसे आकर्षक लोकेशन का निर्धारण कीजिए । प्रत्येक लोकेशन के लिए उत्पादन की संतुलन स्तर मात्रा भी ज्ञात कीजिए । (15 अंक) | शीर्ष | शहर | | | |------|------|---|---| | | चेन्नई | दिल्ली | कोलकाता | | स्थिर लागत (₹) | 800000 | 600000 | 500000 | | परिवर्त्य लागत प्रति इकाई (₹) | 30 | 40 | 50 | | आय प्रति इकाई (₹) | 60 | 60 | 60 | (c) एक एकल बिंदु कतन औजार जिसका रेक कोण 10° है, से एक धातु का लंबकोणीय खरादन किया जाता है । खरादन प्रक्रम छीलन मोटाई अनुपात 0·2, क्षैतिज कतन बल (F_H) 1400 N तथा उद्वधार कतन बल (F_V) 2000 N उत्पन्न करता है । अपरूपण तल कोण, अभिलंब बल, रेक फलक पर घर्षण बल तथा घर्षण कोण ज्ञात कीजिए । (20 अंक)

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Approach

Calculate the required parameters for all three sub-parts systematically. For (a), apply SPT rule to sequence jobs B-A-C-D-E, then compute flow times, tardiness and performance metrics. For (b), calculate total cost and profit for each city at 40,000 units, then derive break-even volumes using FC/(Revenue-VC). For (c), use Merchant's circle and shear plane relationships with given rake angle and chip thickness ratio to find shear angle, forces and friction angle. Allocate time proportionally: ~25% for (a), ~25% for (b), ~50% for (c) given its higher marks and complexity.

Key points expected

  • Part (a): SPT sequence B-A-C-D-E; completion times 4,10,15,23,30; total completion time = 30 days; average flow time = 82/5 = 16.4 days; average jobs in system = 82/30 = 2.73; average tardiness = 16/5 = 3.2 days
  • Part (b): Total cost Chennai = 20,00,000, Delhi = 22,00,000, Kolkata = 25,00,000; Profit Chennai = 4,00,000, Delhi = 2,00,000, Kolkata = -1,00,000; Chennai most attractive; BEP Chennai = 26,667, Delhi = 30,000, Kolkata = 33,333 units
  • Part (c): Shear plane angle φ = arctan(0.2×cos10°/(1-0.2×sin10°)) = 12.13°; Friction force F = F_H×sinα + F_V×cosα = 2140.8 N; Normal force N = F_H×cosα - F_V×sinα = 1030.9 N; Friction angle β = arctan(F/N) = 64.3°
  • Correct formulas: SPT sequencing by ascending processing time; Total cost = FC + VC×Q; BEP = FC/(R-VC); Shear angle from r = sinφ/(cos(φ-α)); Merchant's circle relationships
  • Tabular presentation for (a) showing job sequence, processing time, completion time, flow time, due date and tardiness
  • Cost-volume-profit analysis table for (b) comparing all three locations with clear recommendation
  • Force diagram or Merchant's circle sketch for (c) showing F_H, F_V, F, N, R and angles

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies SPT rule for sequencing in (a), understands that SPT minimizes average flow time but not necessarily tardiness; for (b) correctly identifies contribution margin approach and break-even concept; for (c) properly uses Ernst-Merchant shear angle formula and Merchant's circle force relationships with correct rake angle convention.Uses SPT correctly but confuses flow time with completion time; calculates BEP correctly but may confuse profit maximization with cost minimization; applies shear angle formula but may mix up force component directions.Uses EDD or other rule instead of SPT; treats BEP as simple average or confuses fixed/variable costs; applies orthogonal cutting formulas to oblique cutting situation or ignores rake angle in shear angle calculation.
Numerical accuracy25%12.5All calculations precise to 2 decimal places: (a) total completion 30, avg flow time 16.4, avg jobs 2.73, avg tardiness 3.2; (b) Chennai profit ₹4,00,000, BEPs 26667/30000/33333; (c) φ=12.13°, F=2140.8N, N=1030.9N, β=64.3°. No arithmetic errors in intermediate steps.Final answers correct but minor rounding errors; or one sub-part has calculation error but others correct; correct formulas applied but arithmetic slips in force components.Major errors in multiple sub-parts; wrong sequence in (a) leading to cascading errors; BEP formula inverted; shear angle calculated as 0.2×90°=18° ignoring proper formula.
Diagram quality15%7.5Clear Gantt chart or timeline for (a) showing job sequencing; cost-volume graph for (b) with break-even points marked; accurate Merchant's circle or force diagram for (c) with all forces (F_H, F_V, F, N, R) and angles (α, φ, β) labelled, drawn to scale with proper vector directions.Tables used instead of diagrams for (a) and (b); simple sketch for (c) without scale or missing some labels; diagrams present but clarity compromised.No diagrams despite (c) requiring force diagram; or completely wrong diagrams (e.g., stress-strain curve for metal cutting); illegible sketches with no labels.
Step-by-step derivation25%12.5Shows complete working: (a) explicit sorting by SPT, cumulative completion times, flow time calculations, tardiness computation with negative values set to zero; (b) cost functions written, substitution at Q=40000, BEP derivation with algebraic steps; (c) shear angle formula derivation from geometry, force balance equations, systematic solution for all unknowns.Some steps skipped but logical flow maintained; jumps from given data to final answers with minimal working; correct approach but incomplete documentation of intermediate values.Final answers stated without derivation; no evidence of method; or completely wrong approach with no logical steps shown; copying errors between lines.
Practical interpretation15%7.5For (a), notes SPT's limitation regarding due dates (job A is tardy despite SPT); for (b), discusses risk at Kolkata (loss-making at 40000), sensitivity analysis, and capacity considerations for Indian manufacturing locations; for (c), relates friction angle to tool wear, comments on high friction (β>45°) indicating inefficient cutting, suggests coated carbide or ceramic tools for such conditions.Brief comment on Chennai being cheapest; mentions tool life or wear without specific connection to calculated friction angle; generic statements about scheduling importance.No interpretation; pure calculation response; or irrelevant comments (e.g., discussing heat treatment when question is about forces); conclusion missing entirely.

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