Physics 2023 Paper I 50 marks Derive

Q3

(a) (i) What are the requisite conditions for observation of interference pattern on a screen ? (5 marks) (ii) Derive the expression for fringe width and intensity at a point on the screen in a double slit experiment. (10 marks) (b) (i) Prove that the separation of two colliding particles is same, when observed in centre of mass and laboratory systems. (10 marks) (ii) Determine the kinetic energy of a thin disc of mass 0·5 kg and radius 0·2 m rotating with 100 rotations per second around the axis passing through its centre and perpendicular to its plane. (5 marks) (c) Write equation for damped harmonic oscillations and obtain expression for logarithmic decrement. In a damped harmonic motion, the first amplitude is 10 cm, which reduces to 2 cm after 50 oscillations, each of period 4 seconds. Determine the logarithmic decrement. Also, calculate the number of oscillations in which the amplitude decreases to 25%. (20 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) व्यतिकरण पैटर्न को पर्दे पर प्रेक्षण के लिए आवश्यक शर्तों को लिखिए । (5 अंक) (ii) द्वि-झिरी प्रयोग में पर्दे के किसी बिंदु पर फ्रिंज की चौड़ाई तथा तीव्रता के लिए व्यंजक व्युत्पन्न कीजिए । (10 अंक) (b) (i) सिद्ध कीजिए कि द्रव्यमान केन्द्र और प्रयोगशाला निकायों में प्रेक्षित दो टकराने वाले (संघनी) कणों के बीच की दूरी समान होती है । (10 अंक) (ii) एक 0·5 kg द्रव्यमान और 0·2 m अर्ध्व्यास की पतली चक्रिका उसके तल के लम्बवत् और उसके केन्द्र से होकर गुजरते अक्ष के परितः 100 घूर्णन प्रति सेकण्ड की दर से घूर्णन कर रही है । चक्रिका की गतिज ऊर्जा की गणना कीजिए । (5 अंक) (c) अवमंदित सरल आवर्त दोलनों के लिए समीकरण लिखिए और लघुगणकीय अपक्षय का व्यंजक व्युत्पन्न कीजिए । एक अवमंदित सरल आवर्त गति में, प्रथम आयाम 10 cm है, जो कि 50 दोलनों के बाद घटकर 2 cm हो जाता है, जिसमें प्रत्येक दोलन का आवर्तकाल 4 सेकण्ड है । लघुगणकीय अपक्षय की गणना कीजिए । उन दोलनों की संख्या की गणना भी कीजिए जिसमें कि आयाम घटकर 25% रह जाता है । (20 अंक)

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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.

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How this answer will be evaluated

Approach

This multi-part question demands rigorous derivation and proof-based responses across interference, collision dynamics, and damped oscillations. Allocate approximately 30% time to part (a) covering interference conditions and fringe width derivation, 30% to part (b) on collision frame invariance and rotational kinetic energy, and 40% to part (c) given its higher weightage on damped oscillations and logarithmic decrement calculations. Structure each sub-part with clear statement of principles → mathematical derivation → numerical application where applicable.

Key points expected

  • (a)(i) Conditions: coherent sources, monochromatic light, narrow slits, comparable amplitudes, and constant phase difference
  • (a)(ii) Derivation of fringe width β = λD/d and intensity distribution I = 4I₀cos²(δ/2) with proper phase difference relation
  • (b)(i) Proof that relative position vector r = r₂ - r₁ is frame-invariant using Galilean transformation: r' = r in CM and lab frames
  • (b)(ii) Calculation of rotational KE = ½Iω² = ½(½MR²)(2πν)² with correct moment of inertia for disc
  • (c) Damped oscillator equation: d²x/dt² + 2βdx/dt + ω₀²x = 0; derivation of logarithmic decrement δ = ln(xₙ/xₙ₊₁) = βT
  • (c) Numerical: δ = (1/50)ln(10/2) = 0.0322, and n = ln(4)/δ ≈ 43 oscillations for 25% amplitude reduction

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Precisely states all interference conditions including temporal and spatial coherence; correctly identifies frame-independence as consequence of Galilean invariance; accurately distinguishes underdamped/critically damped/overdamped regimes for (c)States basic conditions but misses coherence types or confuses reference frames; writes damped equation but with sign errors or missing damping termConfuses interference with diffraction conditions; treats CM and lab frames as producing different separations; writes incorrect differential equation form
Derivation rigour25%12.5Complete step-by-step derivations: path difference → phase difference → fringe width; explicit Galilean transformation proof for (b)(i); full solution of damped harmonic oscillator with characteristic equation and logarithmic decrement derivationDerivations with gaps or skipped steps; assumes results without proof; correct final formulas but incomplete intermediate stepsStates formulas without derivation; mathematically incorrect steps; confuses derivation with statement of results
Diagram / FBD15%7.5Clear double-slit geometry with path difference construction; collision diagram showing CM and lab frame velocities; damped oscillator displacement-time envelope curve showing exponential decayBasic diagrams without proper labeling; missing key geometric constructions; diagrams present but not effectively used in derivationsNo diagrams where essential; incorrect ray diagrams; confusing or irrelevant sketches
Numerical accuracy20%10Correct rotational KE = 986.96 J or ~987 J; accurate logarithmic decrement δ = 0.0322; precise calculation of n ≈ 43 oscillations with proper rounding and unit consistencyCorrect method with minor calculation errors; wrong powers of 10; inconsistent units (rpm vs rad/s)Major calculation errors; incorrect formula substitution; missing numerical parts entirely
Physical interpretation20%10Explains why coherence is essential for sustained interference; interprets frame invariance as fundamental to Newtonian mechanics; relates logarithmic decrement to quality factor and energy dissipation rate; connects to practical applications like LIGO damping or mechanical Q-factorBrief physical comments without depth; mentions applications superficially; misses connection between mathematical results and physical meaningPurely mathematical treatment with no physical insight; incorrect interpretation of results; confuses physical quantities

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