Mechanical Engineering 2024 Paper I 50 marks Compulsory Calculate

Q5

(a) A cutting tool is used to machine alloy steel at cutting speed of 40 m/min. Considering the values of constants c and n for cutting tool material as 300 and 0·5, respectively as per tool-life equation, calculate the tool life in minutes. If the cutting speed is increased by 80%, then calculate the percentage change in tool life. (10 marks) (b) The stepper motor of a point-to-point controlled NC machine has specification sensitivity of 3°/pulse. The pitch of the lead screw is 2·4 mm. Determine the expected positioning accuracy. (10 marks) (c) A factory working in two shifts, each of 8 hours, produces 28000 tube lights using a set of workstations. Using this information, compute the actual cycle time of the plant operation. There are 6 tasks required to manufacture the tube light. The sum of all task times is equal to 10 seconds. How many workstations are required to maintain this level of production assuming that combining of tasks into those workstations is a feasible alternative? (10 marks) (d) A car manufacturing unit uses large quantities of a component made of steel. The demand is continuous and inventory planning could be done independent of the production plan. The annual demand for the component is 2500 boxes. The company procures the item from a supplier at the rate of ₹ 1,250 per box. The company estimates the cost of carrying inventory to be 20 percent/unit/annum and the cost of ordering as ₹ 1,200 per order. The company works for 250 days in a year. How should the company design an inventory control system for this item? What is the total cost of the plan? (10 marks) (e) A manufacturer of toys for children in the age group of 2 to 4 years commissioned a market research firm to understand the factors that influenced the demand for the product. After some detailed studies, the research firm concluded that the demand was a simple linear function of number of newlywed couples in the city. Based on this assumption, build a model for forecasting the demand for the product using the data in the table given below, which is collected from a residential area in a city: | New Marriages (X) | Demand for Toys (Y) | |---|---| | 200 | 165 | | 235 | 184 | | 210 | 180 | | 195 | 145 | | 225 | 190 | | 240 | 169 | | 217 | 180 | | 225 | 170 | (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक कतन औजार का उपयोग ऐलॉय इस्पात को 40 m/min की कतन गति से मशीन के लिए किया जाता है। औजार-आयु समीकरण में कतन औजार के पदार्थ के स्थिरांक c तथा n को क्रमशः 300 तथा 0·5 मानते हुए औजार-आयु को मिनट में ज्ञात कीजिए। यदि कतन गति को 80% से बढ़ाया जाए, तो औजार की आयु में प्रतिशत बदलाव निकालिए। (10 अंक) (b) बिंदुशः (पॉइंट-टु-पॉइंट) नियंत्रित NC मशीन के स्टेपर मोटर की विभेदक संवेदनशीलता 3°/पल्स है। अग्रण पेंच का पिच 2·4 mm है। अनुमानित स्थिति-निर्धारण सटीकता ज्ञात कीजिए। (10 अंक) (c) प्रत्येक 8 घंटे की दो शिफ्टों में काम करने वाली एक फैक्ट्री, कार्यस्थलों (वर्कस्टेशनों) के एक समूह का उपयोग करके 28000 ट्यूब लाइट का उत्पादन करती है। इस जानकारी का उपयोग करके संयंत्र संचालन के वास्तविक चक्र समय की गणना कीजिए। ट्यूब लाइट के निर्माण के लिए 6 कार्यों की आवश्यकता होती है। सभी कार्य-समय का योग 10 सेकेंड के बराबर है। उत्पादन के इस स्तर को बनाए रखने के लिए कितने कार्यस्थलों की आवश्यकता है, यदि उन कार्यस्थलों में कार्यों का संयोजन एक व्यावहारिक विकल्प है? (10 अंक) (d) एक कार निर्माण इकाई बड़ी मात्रा में इस्पात से बने घटक का उपयोग करती है। माँग निरंतर है और इन्वेंट्री (सामग्री सूची) योजना उत्पादन योजना से स्वतंत्र की जा सकती है। घटक की वार्षिक माँग 2500 बॉक्स है। कंपनी आपूर्तिकर्ता से ₹ 1,250 प्रति बॉक्स की दर से यह वस्तु खरीदती है। कंपनी का अनुमान है कि इन्वेंट्री ले जाने की लागत 20 प्रतिशत प्रति इकाई प्रति वर्ष और ऑर्डर की लागत ₹ 1,200 प्रति ऑर्डर होगी। कंपनी साल में 250 दिन काम करती है। कंपनी को इस वस्तु की इन्वेंट्री नियंत्रण प्रणाली कैसे डिजाइन करनी चाहिए? योजना की कुल लागत क्या है? (10 अंक) (e) 2 से 4 वर्ष की आयु के बच्चों के लिए खिलौनों के एक निर्माता ने उत्पाद की माँग को प्रभावित करने वाले कारकों को समझने के लिए एक बाजार अनुसंधान फर्म की स्थापना की। कुछ विस्तृत अध्ययनों के बाद, अनुसंधान फर्म ने निष्कर्ष निकाला कि माँग शहर में नवविवाहित जोड़ों की संख्या का एक सरल रैखिक फलन है। इस धारणा के आधार पर नीचे दी गई तालिका के आँकड़ों, जो किसी शहर के आवासीय क्षेत्र से एकत्र किये गये हैं, का उपयोग करके उत्पाद की माँग का पूर्वानुमान लगाने के लिए एक मॉडल बनाइए: | नवविवाहित (X) | खिलौनों की माँग (Y) | |---|---| | 200 | 165 | | 235 | 184 | | 210 | 180 | | 195 | 145 | | 225 | 190 | | 240 | 169 | | 217 | 180 | | 225 | 170 | (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 numerical solutions for all five sub-parts with systematic working. For (a) apply Taylor's tool life equation VT^n=C; for (b) compute positioning accuracy from stepper motor resolution and lead screw pitch; for (c) determine cycle time and balance assembly line workstations; for (d) apply EOQ model for inventory control; for (e) perform linear regression analysis for demand forecasting. Present each part with formula, substitution, and final answer clearly labelled.

Key points expected

  • (a) Tool life T = (C/V)^(1/n) = (300/40)^2 = 56.25 min; at 1.8× speed, new T = 17.36 min; percentage decrease = 69.1%
  • (b) Pulses per revolution = 360°/3° = 120; linear displacement per pulse = 2.4mm/120 = 0.02 mm; positioning accuracy = ±0.02 mm
  • (c) Available time = 2×8×3600 = 57600 sec; actual cycle time = 57600/28000 = 2.057 sec; theoretical stations = 10/2.057 ≈ 4.86; minimum stations needed = 5 (rounded up)
  • (d) EOQ = √(2×2500×1200)/(0.20×1250) = 154.9 ≈ 155 boxes; total cost = purchase + ordering + carrying = ₹31,25,000 + ₹19,355 + ₹19,375 = ₹31,63,730
  • (e) Regression: ΣX=1747, ΣY=1383, ΣXY=303,355, ΣX²=383,849, n=8; b=0.743, a=-2.86; Y = -2.86 + 0.743X; correlation coefficient r ≈ 0.67
  • All five parts show correct formula application, unit consistency, and final answers rounded appropriately with % or ₹ symbols where relevant

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies and applies: (a) Taylor's tool life equation with proper exponent handling; (b) stepper motor resolution-to-linear conversion; (c) cycle time vs takt time distinction and line balancing; (d) EOQ model assumptions and cost components; (e) least-squares regression with correlation analysis. No conceptual confusion between similar formulas.Uses correct formulas for most parts but may confuse (c) cycle time with takt time, or (e) population vs sample formulas; minor errors in EOQ cost component identification.Applies wrong formulas (e.g., uses simple average for regression, confuses tool life with cutting time, treats inventory problem as ABC analysis); fundamental misunderstanding of manufacturing concepts.
Numerical accuracy20%10All calculations precise: (a) T=56.25 min, 69.1% decrease; (b) 0.02 mm exact; (c) cycle time 2.057 sec, 5 stations; (d) EOQ=155, total cost ₹31,63,730; (e) regression coefficients b≈0.74, a≈-2.9, r≈0.67. No rounding errors beyond acceptable limits; carries sufficient decimal places.Final answers approximately correct but with minor arithmetic slips (e.g., EOQ off by 5-10 units, percentage change miscalculated, regression slope sign error); most working shown.Major calculation errors (order of magnitude wrong, incorrect percentage formula, EOQ squared instead of square-rooted, regression using n instead of n-1 incorrectly); answers without working.
Diagram quality15%7.5For (d): EOQ cost curves showing ordering, carrying, and total cost vs quantity with EOQ point marked; for (e): scatter plot with regression line, data points, and equation labelled; clear axes with units. Optional but beneficial: tool wear curve for (a), assembly line layout sketch for (c).Basic EOQ diagram or scatter plot present but poorly labelled, missing axes titles, or not to scale; no diagrams for other parts where they could aid clarity.No diagrams despite visual representation being valuable; or completely wrong diagrams (e.g., shear stress diagram for tool life, bar chart for regression).
Step-by-step derivation25%12.5Every sub-part shows complete derivation: (a) logs shown for exponent; (b) unit conversion chain clear; (c) time calculations with 57600 sec explicit; (d) EOQ formula derived from d(TC)/dQ=0 or at least full substitution; (e) normal equations or shortcut formulas with all summations tabulated. Cross-checks shown where possible.Most steps shown but skips 'obvious' algebraic manipulations; EOQ formula quoted without derivation; regression uses calculator directly without showing summation table; some substitution steps implied.Final answers only with no working; or fragmented steps with logical gaps; no verification of reasonableness (e.g., EOQ larger than annual demand unchecked).
Practical interpretation20%10Interprets results contextually: (a) comments on tool material suitability for high-speed machining; (b) assesses if ±0.02mm meets typical CNC tolerances; (c) discusses line efficiency and idle time; (d) recommends reorder point, safety stock, and supplier lead time considerations; (e) evaluates r=0.67 as moderate correlation, suggests other demand factors. Links to Indian manufacturing context (e.g., MSME inventory practices).Brief contextual note for 2-3 parts (e.g., states EOQ is 'economical' without elaboration; mentions 'good fit' for regression without interpreting r); generic statements.Purely mathematical treatment; no interpretation of whether results are practical, optimal, or realistic; ignores that 69% tool life reduction may require carbide/ceramic tools, or that r=0.67 suggests model limitations.

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