Question Statement:
A copper ball and a steel ball having diameters d1 and d2, respectively, are initially at a uniform temperature of 200⁰C. Both the balls are exposed to the atmosphere at 30⁰C. If both the balls attain a temperature of 120⁰C after equal exposure duration, then the ratio of d1 to d2 is ___________________ (rounded off to 3 decimal places). Assume Biot Number to be less than 0.1.
Step-by-Step Solution & Explanation
With Biot number < 0.1, lumped-capacitance cooling applies: (T−T∞)/(T_i−T∞) = exp(−hAt/(ρVc_p)). For a sphere, V/A = D/6, so the exponent becomes −6ht/(ρc_pD). Since both balls reach the SAME dimensionless temperature ratio θ=(120−30)/(200−30)=90/170 after the SAME time t, and (for this idealized comparison) the same convective coefficient h is assumed for both:
6ht/(ρ₁c_{p1}D₁) = 6ht/(ρ₂c_{p2}D₂) ⇒ D₁/D₂ = (ρ₂c_{p2})/(ρ₁c_{p1})
D₁/D₂ (copper/steel) = (7800×460)/(8950×383) = 3,588,000/3,427,850 = 1.047.
This question was officially marked “Marks to All” (MTA) by the exam authority — i.e., every candidate was awarded full marks regardless of their answer, most likely because the problem as posed does not actually require (and did not intend to require) the thermal conductivity values given in the original table, or because of an ambiguity in whether identical h should be assumed for both balls. We report our independently derived value (d₁/d₂ ≈ 1.047) using the standard lumped-capacitance method, alongside the official MTA status.
Appendix A — Consolidated Answer Table
Appendix B — Section-wise Weightage
Preparation observations
• Soil and Water Conservation Engineering carries the single largest weightage (17%) among the seven core AG sections in this paper, spanning hydrology, fluid mechanics, soil mechanics, surveying and erosion-control design — candidates should prioritise numerical fluency across all five of these sub-areas rather than any one alone.
• General Aptitude (15%) and Farm Machinery (14%) are the next-largest blocks; GA marks are comparatively low-effort to secure with consistent verbal/quantitative/analytical practice, offering strong return on preparation time.
• Farm Power (13%) and Dairy & Food Engineering (12%) both lean heavily on multi-step numerical problems (engine thermodynamics, tractor mechanics, heat and mass transfer) — practicing unit-conversion discipline (bar/kPa, rpm/rad·s⁻¹, mm/cm) is essential, as several official answers in this key hinge on exactly this.
• Irrigation & Drainage Engineering (10%) and Agricultural Process Engineering (8%) are smaller but consistently featured blocks; design-formula recall (sprinkler spacing, Thiem equation, cake filtration) is more decisive here than raw computation speed.
• Engineering Mathematics (11%) questions in this paper were comparatively direct application (limits, matrices, RK4, binomial probability) rather than proof-heavy — a well-drilled formula sheet covers most of this section's marks efficiently.
• A number of NAT questions in this paper (flagged in the explanations above) show a gap between our independently re-derived values and the officially listed answer ranges; these are documented transparently in-line rather than silently resolved, and are worth independent re-verification by the candidate against the original source/solutions.