(a) A cylindrical workpiece of 60 mm diameter is turned down to 50 mm over a length of 120 mm on a CNC lathe with cutting speed…

(a) Machining time for straight turning (10 marks)

Data: D = 60 mm, L = 120 mm, V = 120 m/min, f = 0.3 mm/rev. The pass reduces the diameter to 50 mm (depth of cut 5 mm), which is done in one pass.

Spindle speed: N = 1000 V/(πD) = 1000 × 120/(π × 60) = 636.6 rpm.

Feed rate: f_m = f N = 0.3 × 636.6 = 191 mm/min.

Machining time: t = L/(f N) = 120/191 = 0.628 min ≈ 37.7 s.

This ignores the approach and over-travel of the tool. The speed is set on the initial workpiece diameter (60 mm).

(b) Break-even and cross-over analysis of processes X and Y (10 marks)

Data: X has F = ₹9,60,000 and v = ₹55/unit, Y has F = ₹15,00,000 and v = ₹38/unit, and the price is ₹110/unit. The contribution per unit is 110 − 55 = ₹55 for X and 110 − 38 = ₹72 for Y.

(i) Break-even volumes.

• X: Q = F/(p − v) = 9,60,000/55 = 17,455 units (17,454.5).
• Y: Q = 15,00,000/72 = 20,834 units (20,833.3).

(ii) Cross-over point. The total costs are equal when 9,60,000 + 55Q = 15,00,000 + 38Q, so 17Q = 5,40,000 and Q = 31,765 units (31,764.7). Below this volume X has the lower total cost, and above it Y is cheaper because of its lower variable cost.

(iii) Profit at 35,000 units.

• X: 35,000 × 55 − 9,60,000 = 19,25,000 − 9,60,000 = ₹9,65,000.
• Y: 35,000 × 72 − 15,00,000 = 25,20,000 − 15,00,000 = ₹10,20,000.

Since 35,000 exceeds the cross-over volume of 31,765, process Y is recommended, with ₹55,000 more profit per year.

Margin of safety under Y. MoS = (35,000 − 20,833)/35,000 = 14,167/35,000 = 40.5%. (Under X it would be 50.1%, so Y is the riskier choice if demand falls below about 31,765 units.)

(c) Power density of a laser beam (10 marks)

Data: E = 5 J, t = 0.01 s, A = 0.5 mm² = 0.5 × 10⁻⁶ m².

Power: P = E/t = 5/0.01 = 500 W.

Power density = P/A = 500/(0.5 × 10⁻⁶) = 1 × 10⁹ W/m², that is 1000 W/mm² or 1 × 10⁵ W/cm² (1 GW/m²).

Comment. In laser beam machining the material is removed by melting and vaporisation of a small spot. What matters is the power density, not the energy alone: it must be high enough that the absorbed energy heats the spot faster than conduction carries it away into the bulk. Focusing the beam to a smaller area, or shortening the pulse at the same energy, raises the power density and confines the heat-affected zone.

(d) Control charts and a p-chart (10 marks)

(i) X̄ and R charts. Data: X̄̄ = 48.6 mm, R̄ = 4.2 mm, n = 5, A₂ = 0.577, D₃ = 0, D₄ = 2.114.

X̄ chart:

• UCL = X̄̄ + A₂R̄ = 48.6 + 0.577 × 4.2 = 48.6 + 2.423 = 51.02 mm.
• CL = 48.6 mm.
• LCL = X̄̄ − A₂R̄ = 48.6 − 2.423 = 46.18 mm.

R chart:

• UCL = D₄R̄ = 2.114 × 4.2 = 8.88 mm.
• CL = 4.2 mm.
• LCL = D₃R̄ = 0.

(ii) p-chart. Total inspected = 20 × 200 = 4000 units, with 184 defective:

p̄ = 184/4000 = 0.046 (4.6%).

The standard deviation for n = 200 is σ_p = √(p̄(1 − p̄)/n) = √(0.046 × 0.954/200) = 0.01481.

• UCL_p = p̄ + 3σ_p = 0.046 + 0.0444 = 0.0904 (9.04%).
• LCL_p = p̄ − 3σ_p = 0.046 − 0.0444 = 0.0016 (0.16%).

A lot with a fraction defective above 0.0904 signals an out-of-control condition.

(e) Surface finish in machining, and profile finishing techniques (10 marks)

Factors affecting surface finish.

• Cutting conditions: feed is the main one, since the theoretical peak-to-valley roughness is R_max = f²/(8r) (with r the tool nose radius), so a smaller feed gives a finer surface. Cutting speed matters through the built-up edge, which forms at low speed and roughens the surface, and depth of cut has a smaller effect.
• Tool: a larger nose radius, sharp edge, suitable rake and clearance angles, and low wear all improve finish. Tool material and coating affect adhesion.
• Workpiece: material, hardness and microstructure (ductile materials tear, and free-machining grades finish better).
• Machine and set-up: rigidity, vibration and chatter, spindle run-out, slide accuracy and work holding.
• Cutting fluid, which lubricates, cools and reduces the built-up edge, and the temperature and chip flow.

Profile finishing techniques in modern manufacturing.

• Conventional: fine grinding, honing, lapping, superfinishing, buffing and polishing, and burnishing.
• Advanced: abrasive flow machining, magnetic abrasive finishing, magnetorheological finishing, chemical mechanical polishing, electropolishing and chemical polishing, ultrasonic and laser polishing, and vibratory or barrel (mass) finishing.