Physics 2021 Paper I 50 marks Derive

Q2

(a) A body of mass m at rest splits into two masses m₁ and m₂ by an explosion. After the split the bodies move with a total kinetic energy T in opposite direction. Show that their relative speed is √(2Tm/m₁m₂). (15 marks) (b) A light rod of length 100 cm is suspended from the ceiling, horizontally by means of two vertical wires of equal length tied to its ends. One of the wires is made of steel and its cross-section is 0·05 sq. cm and the other is of brass of cross-section 0·1 sq. cm. Find the position along the rod at which a weight may be hung to produce (i) Equal stresses in both the wires, (ii) Equal strain in both the wires. Young's modulus of elasticity of brass and steel are 1·0 × 10¹¹ N/m² and 2·0 × 10¹¹ N/m² respectively. (15 marks) (c) Show that the phenomenon of Fraunhofer diffraction at two vertical slits is modulation of two terms viz. double slit interference and single slit diffraction. Obtain the condition for positions of maxima and minima. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) एक द्रव्यमान m का पिंड, जो कि स्थिर अवस्था में है, एक विस्फोट के दौरान दो भागों m₁ और m₂ द्रव्यमानों के दो पिंडों में विभाजित हो जाता है । विभाजन के बाद दोनों पिंड कुल गतिज ऊर्जा T के साथ विपरीत दिशाओं में गति करते हैं । दिखाइए कि दोनों की आपेक्षिक चाल √(2Tm/m₁m₂) है । (15 अंक) (b) 100 cm लम्बाई के एक हल्के छड़ को दोनों किनारों पर समान लम्बाई के दो ऊर्ध्वाधर तारों से बाँधकर छत से क्षैतिज दिशा में लटकाया गया है । एक तार इस्पात से बना है जिसका अनुप्रस्थ-परिच्छेद 0·05 sq. cm है तथा दूसरा तार ब्रास (पीतल) का बना है जिसका अनुप्रस्थ-परिच्छेद 0·1 sq. cm है । छड़ पर उस स्थान को ज्ञात कीजिए जहाँ पर एक भार को लटकाया जा सके जिससे कि दोनों तारों में (i) बराबर प्रतिबल, (ii) बराबर विकृति उत्पन्न किया जा सके । ब्रास (पीतल) और इस्पात के यंग के प्रत्यास्थता गुणांक क्रमशः 1·0 × 10¹¹ N/m² और 2·0 × 10¹¹ N/m² हैं । (15 अंक) (c) दिखाइए कि दो उर्ध्वाधर छिद्रों से होने वाले फ्राउनहोफर विवर्तन की परिघटना दो पदों, जैसे कि द्वि-छिरी से व्यतिकरण और एकल छिरी से विवर्तन का मॉडुलन होता है । अधिकतम और न्यूनतम मान की स्थितियों की शर्त प्राप्त कीजिए । (20 अंक)

Evaluate your answer to this question → See all 2021 Physics questions

Directive word: Derive

This question asks you to derive. 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

This question demands rigorous derivation and proof across three distinct physics domains. Begin with conservation laws for the explosion problem, apply static equilibrium and elasticity theory for the suspended rod, and conclude with wave optics derivation showing modulation of interference and diffraction patterns. Structure as: (a) momentum-energy derivation with clear algebraic steps, (b) force analysis with two cases for stress and strain conditions, (c) Fraunhofer setup with mathematical demonstration of intensity modulation and conditions for extrema.

Key points expected

  • Conservation of linear momentum (initial zero momentum implies equal and opposite momenta of fragments) and expression of total kinetic energy in terms of reduced mass
  • Static equilibrium conditions: sum of vertical forces equals weight, and torque balance about suspension point for both stress and strain cases
  • Correct application of Young's modulus relation: stress = Y × strain, and proper unit conversion from CGS to SI or consistent handling of cm and m²
  • Fraunhofer setup with plane wavefront, lens arrangement, and mathematical derivation showing I(θ) = I₀(sinβ/β)²cos²α where β and α relate to single slit and double slit parameters
  • Clear identification that (sinβ/β)² represents single slit diffraction envelope modulating the cos²α interference term
  • Conditions for maxima: dsinθ = nλ (interference) and bsinθ = mλ (diffraction minima), with missing orders explained

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies conservation of momentum in explosion, identifies tension-stress-strain relationships in composite rod, and accurately describes Fraunhofer far-field condition with proper distinction between interference and diffraction phenomena; no conceptual confusion between near-field and far-field diffractionMinor errors in applying equilibrium conditions or混淆 stress-strain definitions; Fraunhofer condition stated but not clearly distinguished from Fresnel; correct final formulas but shaky conceptual foundationFundamental errors like assuming kinetic energy conservation in explosion, confusing stress with strain in elasticity, or treating double slit as pure interference ignoring diffraction modulation entirely
Derivation rigour25%12.5Step-by-step algebraic rigour: momentum conservation leading to v₁/v₂ = m₂/m₁, kinetic energy expression yielding reduced mass μ = m₁m₂/m, and final relative velocity derivation; torque equations properly set up for both cases in (b); intensity derivation from Huygens-Fresnel principle with clear separation of termsCorrect final formulas but skips critical intermediate steps (e.g., jumps from momentum to relative velocity without showing algebra); torque balance present but algebraic manipulation errors; intensity formula stated without derivation from path differenceMissing essential steps, circular reasoning, or mathematically invalid operations (e.g., dividing by zero scenarios); no derivation for intensity modulation, only statement of final result
Diagram / FBD15%7.5Clear FBD for explosion showing opposite velocities, labeled free body diagram for rod with tension forces, weight vector, and distance markers x and (100-x); Fraunhofer setup with collimating lens, double slit, converging lens, and screen with angle θ marked; path difference geometry clearly indicatedDiagrams present but missing key labels or force directions; Fraunhofer setup incomplete (e.g., missing lens); rough sketches without proper scaling or annotationNo diagrams despite explicit need for geometric visualization; or seriously misleading diagrams (wrong force directions, incorrect optical setup)
Numerical accuracy20%10Precise calculation for part (b): correct unit conversion (cm to m, cm² to m²), proper handling of significant figures, accurate substitution into torque equations yielding x = 40 cm for equal stress and x = 50 cm for equal strain; intermediate steps shown with proper dimensional analysisCorrect method but arithmetic errors or unit conversion mistakes (e.g., factor of 100 error in area); final answers numerically wrong but method identifiable; inconsistent significant figuresMajor numerical errors, wrong formulas substituted, or complete omission of numerical work for part (b); answers without any working shown
Physical interpretation20%10Insightful commentary: explains why reduced mass appears in relative velocity formula, discusses practical implications of stress/strain equality in engineering design (e.g., bridge cables, ISRO payload structures), and clearly explains missing orders in diffraction pattern with real-world spectroscopy applications; connects to Indian context like use of diffraction gratings in astronomical instruments at ARIES NainitalBrief mention of physical significance without elaboration; standard textbook commentary on interference fringes; no connection to practical applications or Indian scientific contextPurely mathematical treatment with zero physical insight; no interpretation of what equal stress/strain means physically; no explanation of why modulation occurs in diffraction pattern

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 Physics 2021 Paper I