Physics 2024 Paper I 50 marks Derive

Q2

(a) (i) Briefly discuss the Kepler's laws of planetary motion. 5 marks (ii) Show that the escape velocity V_e on the surface of the Earth is given by V_e = √(2gR), where g = 9·8 m/s² and R is the radius of the Earth. 5 marks (iii) Two satellites A and B of same mass are orbiting the Earth at altitudes R and 5R, respectively, where R is the radius of the Earth. Assuming their orbits to be circular, calculate the ratios of their kinetic and potential energies. 5 marks (b) Show that the angular momentum of a rigid body consisting of n particles of masses m_i, i = 1, 2, 3, ..., n, rotating with an instantaneous angular velocity ω about an axis passing through the origin O of the coordinate system OXYZ is given by L = I·ω, where I is known as the inertia tensor. 20 marks (c) A weakly damped harmonic oscillator consisting of spring-mass system has the following parameters: Mass m = 0·25 kg, Spring constant k = 100 N m⁻¹, Damping coefficient γ = 1 N s m⁻¹. A periodic force F = 5cos ωt (newton) is applied to the system. Determine (i) the amplitude of the oscillator at resonance and (ii) the Q-value of the oscillator. 15 marks

हिंदी में प्रश्न पढ़ें

(a) (i) ग्रहीय गति के केप्लर के नियमों की संक्षेप में चर्चा कीजिये। 5 अंक (ii) दर्शाइये कि पृथ्वी की सतह पर पलायन वेग V_e = √(2gR) है, जहाँ g = 9·8 m/s² और R पृथ्वी की त्रिज्या है। 5 अंक (iii) समान द्रव्यमान के दो उपग्रह A और B पृथ्वी के चारों ओर क्रमशः ऊँचताओं R और 5R पर परिक्रमण कर रहे हैं, जहाँ R पृथ्वी की त्रिज्या है। उनकी कक्षाओं को वृत्तीय मानकर उनकी गतिज ऊर्जाओं और स्थितिज ऊर्जाओं के अनुपातों की गणना कीजिये। 5 अंक (b) दर्शाइये कि निर्देशांक तंत्र OXYZ के मूलबिंदु O से होकर गुजरते अक्ष के परितः एक तात्क्षणिक कोणीय वेग ω से घूर्णन करते एवं द्रव्यमानों m_i, i = 1, 2, 3, ..., n वाले n कणों से बने एक दृढ़ पिंड का कोणीय संवेग L = I·ω होता है, जहाँ I जड़त्व प्रदिश है। 20 अंक (c) स्प्रिंग-द्रव्यमान निकाय के एक अल्प अवमंदित आवर्ती दोलक के निम्नलिखित प्राचल मान हैं: द्रव्यमान m = 0·25 kg, स्प्रिंग स्थिरांक k = 100 N m⁻¹, अवमंदन गुणांक γ = 1 N s m⁻¹। इस निकाय पर एक आवर्ती बल F = 5cos ωt (न्यूटन) प्रयुक्त किया जाता है। (i) अनुनाद पर दोलक का आयाम और (ii) दोलक का Q-मान ज्ञात कीजिये। 15 अंक

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Directive word: Derive

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How this answer will be evaluated

Approach

This question demands rigorous derivation and calculation across five sub-parts. Allocate approximately 15% (7-8 minutes) to part (a)(i)-(iii) combined, 50% (25 minutes) to part (b) as it carries 20 marks requiring full tensor derivation, and 35% (17-18 minutes) to part (c) for damped oscillator calculations. Structure with clear headings for each sub-part, stating given data before derivations, and conclude with physical significance of results.

Key points expected

  • Part (a)(i): State all three Kepler's laws with correct mathematical forms (elliptical orbits, equal areas in equal times, T² ∝ r³) and mention Newton's later theoretical foundation
  • Part (a)(ii): Derive V_e = √(2GM/R) from energy conservation, then substitute g = GM/R² to obtain V_e = √(2gR) ≈ 11.2 km/s
  • Part (a)(iii): Calculate orbital velocities at r = 2R and r = 6R, then find KE_A/KE_B = 6 and PE_A/PE_B = 3 (using PE = -GMm/2r for bound orbits or -GMm/r)
  • Part (b): Define position vectors r_i, velocities v_i = ω × r_i, derive L = Σm_i(r_i × (ω × r_i)), expand using vector triple product, identify I_αβ = Σm_i(r_i²δ_αβ - x_iαx_iβ) as inertia tensor components
  • Part (c): Calculate ω₀ = √(k/m) = 20 rad/s, damping ratio β = γ/(2m) = 2 s⁻¹, resonance amplitude A_res = F₀/(γω₀) = 1 m, and Q = ω₀/(2β) = 5

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly states Kepler's three laws with proper terminology (focus, semi-major axis, areal velocity); uses correct sign convention for gravitational PE (negative); identifies inertia tensor as symmetric 3×3 matrix; distinguishes underdamped condition (β < ω₀) for part (c)States laws correctly but misses areal velocity definition or confuses PE sign; recognizes tensor form but misidentifies components; calculates resonance frequency as ω₀ but confuses amplitude formulaConfuses Kepler's laws with Newton's laws; treats PE as positive or uses mgh; fails to recognize tensor nature of I; applies undamped resonance formula or ignores damping condition
Derivation rigour25%12.5Complete energy conservation derivation for escape velocity with explicit limits; full vector derivation for L = I·ω using index notation or matrix form with Levi-Civita symbols; proper driven damped oscillator solution with amplitude resonance condition dA/dω = 0Correct final formulas with gaps in derivation steps; states L = Σr_i × p_i but skips to tensor form without expansion; uses standard resonance amplitude formula without derivationStates results without derivation; confuses angular momentum of particle with rigid body; applies wrong resonance condition (velocity resonance ω = ω₀ vs amplitude resonance ω = √(ω₀² - 2β²))
Diagram / FBD15%7.5Clear elliptical orbit diagram for Kepler's first law with Sun at focus; energy diagram showing PE, KE, E vs r for escape condition; labeled coordinate system for rigid body rotation; amplitude-frequency response curve showing resonance peak for part (c)Sketch of circular orbit instead of ellipse; missing energy diagram; basic rotation diagram without coordinate axes; schematic of spring-mass-damper without response curveNo diagrams despite clear geometric/physical content; irrelevant diagrams; completely missing visual representation for 20-mark tensor derivation
Numerical accuracy25%12.5Correct ratios KE_A:KE_B = 6:1 and PE_A:PE_B = 3:1 (or 6:1 for total energy); exact values A_res = 1.0 m and Q = 5 with proper units; consistent use of r = R+h for orbital radii (2R and 6R)Correct formulas with arithmetic errors; uses r = h instead of R+h for altitude; correct Q but wrong resonance amplitude due to formula error; off by factors of 2 in energy ratiosMajor calculation errors (wrong powers of 10, confused m with cm); ignores given data; completely wrong numerical answers despite correct formulas; unit errors throughout
Physical interpretation15%7.5Explains why lower orbit has higher KE (virial theorem connection); discusses physical meaning of inertia tensor (moment of inertia depends on axis); interprets Q = 5 as moderate quality factor with sharp but not extreme resonance; relates escape velocity to cosmic velocities (ISRO launch considerations)Brief comment on energy ratios; states tensor is 'like moment of inertia'; mentions resonance without discussing sharpness; generic statement about satellite orbitsNo physical interpretation; purely mathematical treatment; misinterprets energy ratios (e.g., higher orbit more kinetic energy); fails to connect Q-factor to damping or bandwidth

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