Mechanical Engineering 2021 Paper I 50 marks Calculate

Q7

A car servicing company is interested in reducing the waiting time for its customers. They select four customers randomly each day and find the waiting time for each customer while his/her car is serviced. From these observations, the sample average and range are found. This process is repeated for 25 days. The summary data for these observations are as under: $$\sum_{i=1}^{25} \bar{X}_i = 1000, \quad \sum_{i=1}^{25} R_i = 250$$ (i) Find out $\bar{X}$ and R chart control limits. (ii) Assuming that the process is in control and the distribution of waiting time is normal, find the percentage of customers who will not have to wait for more than 50 minutes. (iii) Find the 2σ control limits. (Factors for Computing Centerline and Three-Sigma Control Limits and Cumulative Standard Normal Distribution table are appended in the question paper) (b) How do lean systems function? What are the characteristics of lean systems? Also, discuss the benefits and risks of lean systems. (c) In a case of open die forging, derive the expression for determining forging force per unit length for forging a flat strip between two parallel dies. Also, state the assumptions made while deriving the above mentioned expression.

हिंदी में प्रश्न पढ़ें

एक कार सर्विसिंग कंपनी अपने ग्राहकों के लिए प्रतीक्षा काल घटाने में रुचि रखती है। वे प्रत्येक दिन चार ग्राहक यादृच्छिक रूप से चुनकर प्रत्येक ग्राहक का प्रतीक्षा काल ज्ञात करते हैं जब उसकी कार सर्विस हो रही है। इन प्रेक्षणों से प्रतिदर्श औसत व सीमा ज्ञात की जाती हैं। इस प्रक्रिया को 25 दिन तक दोहराया जाता है। इन प्रेक्षणों के सारांश आँकड़े निम्नवत हैं: $$\sum_{i=1}^{25} \bar{X}_i = 1000, \quad \sum_{i=1}^{25} R_i = 250$$ (i) $\bar{X}$ व R चार्ट नियंत्रण सीमाएँ ज्ञात कीजिए। (ii) यह मानते हुए कि प्रक्रिया नियंत्रण में है तथा प्रतीक्षा काल का वितरण सामान्य है, उन ग्राहकों का प्रतिशत ज्ञात कीजिए जिनको 50 मिनट से ज्यादा प्रतीक्षा नहीं करनी होगी। (iii) 2σ नियंत्रण सीमाएँ ज्ञात कीजिए। (माध्य रेखा व तीन-सिग्मा नियंत्रण सीमाओं की गणना के लिए गुणक तथा संचयी मानक नामल वितरण सारणी प्रश्न-पत्र के साथ संलग्न हैं) (b) लीन निकाय किस प्रकार कार्य करते हैं? लीन निकायों के क्या लक्षण हैं? लीन निकायों के लाभों व जोखिमों की भी विवेचना कीजिए। (c) खुली डाई फोर्जन के संदर्भ में, दो समांतर डाइयों के बीच एक चपटी पट्टी के फोर्जन के लिए प्रति इकाई लंबाई पर लगने वाले फोर्जन बल को ज्ञात करने के लिए व्यंजक व्युत्पन्न कीजिए। उपर्युक्त व्यंजक को व्युत्पन्न करने के दौरान मानी गई पूर्वधारणाएं भी बताइए।

Evaluate your answer to this question → See all 2021 Mechanical Engineering questions

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.

See our UPSC directive words guide for a full breakdown of how to respond to each command word.

How this answer will be evaluated

Approach

Calculate the SQC control limits for part (a) using n=4, A2=0.729, D3=0, D4=2.282; derive the forging force expression in part (c) starting from equilibrium and yield criterion; explain lean systems conceptually in part (b). Allocate approximately 40% time to (a) due to three numerical sub-parts, 30% each to (b) and (c), ensuring all five sub-parts are addressed with clear headings.

Key points expected

  • Part (a)(i): Grand mean X̿ = 1000/25 = 40 min, R̄ = 250/25 = 10 min; X̄ chart: UCL = 40 + 0.729×10 = 47.29, LCL = 40 - 0.729×10 = 32.71; R chart: UCL = 2.282×10 = 22.82, LCL = 0×10 = 0
  • Part (a)(ii): σ̂ = R̄/d2 = 10/2.059 = 4.857 min; Z = (50-40)/4.857 = 2.06; P(X≤50) = Φ(2.06) ≈ 98.03% or ~98% customers
  • Part (a)(iii): 2σ X̄ limits: 40 ± 2×(4.857/√4) = 40 ± 4.857 → 35.14 to 44.86 min; or using A2 factor approximation
  • Part (b): Lean systems function via JIT, jidoka, kaizen, pull production, waste elimination (muda); characteristics: zero inventory, small lot sizes, quick changeovers, multifunctional workers, cellular layouts; benefits: cost reduction, quality improvement, flexibility; risks: supply chain vulnerability, worker stress, disruption sensitivity (e.g., Maruti Suzuki lean implementation challenges)
  • Part (c): Forging force per unit length F/L = 2k·a·exp(2μx/h) integrated or F/L = σ̄·h·(exp(2μa/h)-1)/(μ/h) for sticking friction; assumptions: plane strain, rigid-plastic material, constant friction (Coulomb or sticking), uniform deformation, isothermal, parallel flat dies, no barreling

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies SQC chart formulas with proper factor selection (A2, D3, D4, d2 for n=4); understands difference between 3σ and 2σ limits; correctly identifies lean principles (JIT, kanban, kaizen) and their interconnection; applies slab method correctly for forging with proper stress equilibrium and yield criterion.Uses correct formulas but with minor factor errors or confuses R-chart and X̄-chart purposes; describes lean tools but misses systemic integration; attempts slab method but confuses stress boundary conditions.Applies wrong control chart constants or uses σ directly without R̄/d2 conversion; treats lean as mere cost-cutting without JIT/flow concepts; uses incorrect friction model or misses plane strain assumption in forging.
Numerical accuracy20%10All calculations precise: X̿=40, R̄=10, X̄ UCL/LCL=47.29/32.71, R UCL=22.82, σ̂=4.857, Z=2.06, percentage≈98%, 2σ limits≈35.14-44.86; intermediate steps shown; proper significant figures.Correct method but arithmetic slip (e.g., 47.3 instead of 47.29); or uses d2=2.0 approximating to σ̂=5; final percentage within ±2% of correct value.Major calculation errors (e.g., forgets √n in standard error, uses wrong d2, or Z-score calculation inverted); percentage answer nonsensical (>100% or <50%).
Diagram quality15%7.5Sketches X̄ and R charts with centerlines, UCL, LCL clearly marked; shows sample points; draws forging setup with die geometry, strip dimensions (h, a), friction arrows, and stress element; lean flow diagram showing pull system with kanban signals.Charts drawn without proper scale or missing LCL on R-chart; forging sketch shows dies and strip but misses friction direction or stress element; lean described textually without flow diagram.No diagrams despite need for visual explanation; or incorrect chart types (e.g., p-chart instead of X̄-R); forging sketch misleading or absent.
Step-by-step derivation25%12.5Forging derivation: starts with equilibrium equation dσx/dx = ±2τ/h, applies yield criterion σy - σx = 2k = σ̄, integrates with boundary condition σx=0 at x=a, obtains exponential solution; SQC part shows explicit substitution of factors; lean explanation logically structured from principles to practices to outcomes.Forging derivation jumps steps or assumes linear pressure distribution; SQC shows final formulas with values substituted but skips factor lookup demonstration; lean lists characteristics without causal flow.No derivation shown for forging—only final formula stated; SQC answers appear without formula reference; lean is bullet points without explanation of 'how' they function together.
Practical interpretation20%10Interprets control limits in service context: X̄ chart monitors process mean shift, R chart monitors variability consistency; notes 98% within 50 min implies 2% customer dissatisfaction risk; discusses when 2σ vs 3σ limits preferred (sensitivity vs false alarms); lean risks illustrated with Indian manufacturing examples (e.g., auto component suppliers during COVID disruption); forging result related to press selection and die design.States what charts monitor but limited interpretation of business impact; mentions lean risks generically without contextual example; forging force related to equipment capacity superficially.No interpretation—purely mathematical/nominal answers; or incorrect interpretation (e.g., suggests process out of control when within limits); lean benefits listed without risk acknowledgment.

Practice this exact question

Write your answer, then get a detailed evaluation from our AI trained on UPSC's answer-writing standards. Free first evaluation — no signup needed to start.

Evaluate my answer →

More from Mechanical Engineering 2021 Paper I