Physics 2025 Paper I 50 marks Solve

Q2

(a) A body moves about a point 'O' under no force, the principal moments of inertia at 'O' being 3A, 5A and 6A. The components of the initial angular velocity about the principal axes are ω₁ = n, ω₂ = 0 and ω₃ = n. Find the components ω₁, ω₂ and ω₃ for large values of time t. 20 (b) A harmonic oscillator is represented by the equation m d²x/dt² + γ dx/dt + kx = 0; where m = 0·25 kg, γ = 0·07 kg s⁻¹ and k = 85 Nm⁻¹. Determine (i) the period of oscillation, and (ii) the number of oscillations in which its amplitude will become half of its original value. 15 (c) Show that the electromagnetic wave equation is invariant under Lorentz transformations. 15

हिंदी में प्रश्न पढ़ें

(a) एक पिण्ड बिना किसी बल के अधीन एक बिन्दु 'O' के परितः गतिमान है। 'O' पर जड़त्व के मुख्य आघूर्ण 3A, 5A और 6A हैं। मुख्य अक्षों के परितः आरम्भिक कोणीय वेग के घटक ω₁ = n, ω₂ = 0 और ω₃ = n हैं। समय t के बहुत मानों के लिए घटकों ω₁, ω₂ और ω₃ को ज्ञात कीजिए। 20 (b) एक सरल आवर्ती दोलक निम्नलिखित समीकरण द्वारा निरूपित है m d²x/dt² + γ dx/dt + kx = 0; जहाँ m = 0·25 kg, γ = 0·07 kg s⁻¹ और k = 85 Nm⁻¹ है। निर्धारित कीजिए (i) दोलन का आवर्तकाल, और (ii) दोलनों की संख्या जिनमें उसका आयाम उसके प्रारम्भिक मान का आधा हो जाएगा। 15 (c) दर्शाइए कि लोरेन्ट्ज रूपान्तरणों के अधीन विद्युत-चुम्बकीय तरंग समीकरण निश्चर है। 15

Evaluate your answer to this question → See all 2025 Physics questions

Directive word: Solve

This question asks you to solve. 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 is a multi-part problem requiring analytical solutions: spend approximately 40% of effort on part (a) given its 20 marks, using Euler's equations for force-free rigid body motion; allocate 30% each to parts (b) and (c). For (b), solve the damped harmonic oscillator differential equation and extract numerical values; for (c), apply Lorentz transformation to electromagnetic wave equation and demonstrate invariance. Present derivations step-by-step with clear final boxed answers.

Key points expected

  • Part (a): Apply Euler's equations for force-free rotation, identify that motion occurs in 1-3 plane with I₁=3A, I₃=6A, use conservation of energy and angular momentum to find asymptotic behavior where ω₂→0 and ω₁, ω₃ approach constant values
  • Part (b): Identify underdamped regime (γ² < 4mk), calculate damped angular frequency ω' = √(k/m - γ²/4m²), find period T = 2π/ω', and determine logarithmic decrement to find number of oscillations for amplitude halving
  • Part (c): State Lorentz transformation equations, transform ∂²/∂x² - (1/c²)∂²/∂t² using chain rule, show wave operator remains invariant (∂²/∂x'² - (1/c²)∂²/∂t'² = ∂²/∂x² - (1/c²)∂²/∂t²)
  • Correct identification of intermediate axis instability in part (a) — rotation about axis with intermediate moment of inertia (I₂=5A) is unstable, causing ω₂ to decay
  • Numerical calculation in (b): ω' ≈ 18.44 rad/s, T ≈ 0.34 s, and n ≈ 31 oscillations for amplitude to halve using ln(2)/ln(A₁/A₂) relationship
  • Explicit demonstration that phase velocity c remains invariant under Lorentz transformation in part (c), connecting to Einstein's second postulate

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness25%12.5Correctly identifies Euler's equations for rigid body dynamics, recognizes underdamped oscillator condition, and states Lorentz transformation correctly; properly identifies intermediate axis theorem for part (a)Uses correct general equations but makes minor errors in identifying specific regimes (e.g., confuses critically damped with underdamped) or misapplies conservation lawsFundamental conceptual errors: uses wrong equations of motion, fails to recognize inertial properties, or applies Galilean instead of Lorentz transformation
Derivation rigour25%12.5Complete step-by-step derivations: integrates Euler equations with proper limits for (a), solves characteristic equation for (b), and rigorously applies chain rule for partial derivatives in (c) with all intermediate steps shownCorrect final results but skips key steps or uses 'it can be shown that' without justification; minor algebraic gaps that don't affect final answerMissing derivations entirely or logically flawed steps; jumps to conclusions without mathematical justification; incorrect manipulation of differential operators in (c)
Diagram / FBD10%5Clear diagram showing principal axes with moments of inertia labeled for (a); schematic of damped oscillator with force directions for (b); spacetime diagram illustrating Lorentz transformation for (c)Basic diagram present but inadequately labeled or missing one part; axes not clearly identifiedNo diagrams where clearly needed, or diagrams that misrepresent the physical situation (e.g., wrong axis orientation)
Numerical accuracy20%10Precise calculations: ω' = 18.44 rad/s, T = 0.341 s, n ≈ 31 oscillations; correct unit handling throughout; proper significant figuresCorrect method but arithmetic errors leading to slightly wrong final values; correct order of magnitudeMajor calculation errors, wrong formulas substituted, or missing units; order of magnitude incorrect
Physical interpretation20%10Explains why rotation becomes primarily about the axis with largest moment of inertia (I₃) for large t in (a); discusses energy dissipation in (b); connects Lorentz invariance to relativity principle and constancy of light speed in (c)Brief mention of physical meaning without elaboration; states results without explaining why they occurPurely mathematical answer with no physical insight; fails to interpret what the results mean physically

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 2025 Paper I