Physics 2021 Paper I 50 marks Derive

Q4

(a) (i) Define moment of inertia and radius of gyration of a body of mass M rotating about an axis. State and prove Parallel Axis theorem on moment of inertia. (15 marks) (ii) A sphere of mass 0·5 kg rolls on a smooth surface without slipping with a constant velocity of 3·0 m/s. Calculate its total kinetic energy. (5 marks) (b) The radius of the Earth is 6·4 × 10⁶ m, its mean density is 5·5 × 10³ kg/m³ and the universal gravitational constant is 6·66 × 10⁻¹¹ Nm²/kg². Calculate the gravitational potential on the surface of the Earth. (10 marks) (c) What is damped harmonic oscillation? Write the equation of motion and obtain the general solution for this oscillation. Discuss the cases of dead beat, critical damping and oscillatory motion based on the general solution. What would be the logarithmic decrement of the damped vibrating system, if it has an initial amplitude 30 cm, which reduces to 3 cm after 20 complete oscillations? (20 marks)

हिंदी में प्रश्न पढ़ें

(a) (i) एक अक्ष के चारों तरफ घूर्णन करते हुए एक द्रव्यमान M के लिए जड़त्व आघूर्ण और परिभ्रमण त्रिज्या को परिभाषित कीजिए। जड़त्व आघूर्ण के समांतर अक्ष प्रमेय का उल्लेख कीजिए और इसे सिद्ध कीजिए। (15 अंक) (ii) एक 0·5 kg द्रव्यमान का गोला चिकने पृष्ठ पर बिना फिसले 3·0 m/s के एकसमान वेग से लुढ़क रहा है। इसकी कुल गतिज ऊर्जा की गणना कीजिए। (5 अंक) (b) पृथ्वी की त्रिज्या 6·4 × 10⁶ m है, इसका माध्य घनत्व 5·5 × 10³ kg/m³ और सर्वभौम (सार्वत्रिक) गुरुत्वीय नियतांक 6·66 × 10⁻¹¹ Nm²/kg² है। पृथ्वी के पृष्ठ पर गुरुत्वाकर्षण विभव की गणना कीजिए। (10 अंक) (c) अवमंदित संनादी (हार्मोनिक) दोलन क्या है ? इस दोलन गति के लिए समीकरण लिखिए और उसका सर्वमान्य हल प्राप्त कीजिए । सर्वमान्य हल पर आधारित रुद्ध विस्पंदन क्रांतिक अवमंदन और दोलन गति की चर्चा कीजिए । एक अवमंदित दोलनी निकाय का लघुगणकीय अपक्षय क्या होगा यदि इसका प्रारंभिक आयाम 30 cm है जो कि पूरे 20 दोलन के बाद 3 cm हो जाता है ? (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

Begin with precise definitions for (a)(i), then rigorously prove the Parallel Axis theorem with clear mathematical steps. For (a)(ii), apply rolling motion kinetic energy formula (translational + rotational). In (b), derive gravitational potential using Earth's mass from density data. For (c), derive the damped harmonic oscillator equation, solve the characteristic equation, and classify damping regimes. Conclude with logarithmic decrement calculation. Allocate time proportionally: ~30% to (a)(i), ~10% to (a)(ii), ~20% to (b), and ~40% to (c) given its 20 marks.

Key points expected

  • For (a)(i): Correct definition of moment of inertia as I = Σmᵢrᵢ² and radius of gyration as k where I = Mk²; complete proof of Parallel Axis theorem I = I_cm + Md² with clear diagram showing axis displacement
  • For (a)(ii): Correct application of rolling kinetic energy E = ½Mv² + ½Iω² with I = (2/5)MR² for solid sphere and v = Rω condition, yielding E = (7/10)Mv² = 3.15 J
  • For (b): Derivation of gravitational potential V = -GM/R using M = (4/3)πR³ρ, yielding V ≈ -6.25 × 10⁷ J/kg or equivalent calculation with proper sign convention
  • For (c): Correct equation of motion m(d²x/dt²) + b(dx/dt) + kx = 0 or equivalent; general solution derivation with characteristic roots; classification of underdamped (ω' = √(ω₀²-γ²)), critically damped (γ = ω₀), and overdamped/dead beat (γ > ω₀) cases
  • For (c) continued: Correct calculation of logarithmic decrement δ = (1/n)ln(A₀/Aₙ) = (1/20)ln(30/3) = 0.115 or equivalent using damping factor relations

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Precise definitions of moment of inertia, radius of gyration, gravitational potential, and damped oscillation; correct physical interpretation of negative gravitational potential; accurate distinction between damping regimes with correct condition on damping coefficient vs natural frequencyGenerally correct definitions with minor errors in sign conventions or incomplete distinction between damping cases; gravitational potential formula correct but sign may be confusedConfused definitions (e.g., moment of inertia with angular momentum); missing negative sign in gravitational potential; incorrect conditions for damping regimes
Derivation rigour25%12.5Complete rigorous proof of Parallel Axis theorem with clear coordinate setup; systematic derivation of damped oscillator characteristic equation, three cases analyzed with mathematical justification; all steps logically connectedCorrect approach to derivations but with skipped steps or insufficient justification for key substitutions; characteristic equation solved but case analysis superficialMissing derivations or incorrect mathematical steps; no attempt at proving theorems; characteristic equation not properly solved
Diagram / FBD10%5Clear diagram for Parallel Axis theorem showing parallel axes and distance d; free-body diagram for damped oscillator with restoring, damping, and inertial forces labeled; rolling sphere with velocity and angular velocity vectorsBasic diagram present but poorly labeled or missing force components; axes shown but distance d not clearly indicatedNo diagrams despite clear need; or diagrams with fundamental errors (e.g., wrong force directions)
Numerical accuracy25%12.5Exact values: (a)(ii) KE = 3.15 J or 7/10 × 4.5 = 3.15 J; (b) V ≈ -6.25 × 10⁷ J/kg (accept -6.2 to -6.3 × 10⁷); (c) δ = ln(10)/20 ≈ 0.115; proper significant figures and units throughoutCorrect formulas but arithmetic errors; order of magnitude correct in (b) but calculation imprecise; logarithmic decrement formula correct but arithmetic errorOrder of magnitude errors; missing factors (e.g., 7/10 in rolling KE); incorrect formula for logarithmic decrement; no unit checking
Physical interpretation20%10Clear physical meaning of radius of gyration as equivalent point mass distance; interpretation of negative gravitational potential as binding energy; physical significance of each damping regime (e.g., critical damping for fastest return to equilibrium—relevant to galvanometers, shock absorbers); logarithmic decrement as measure of energy loss per cycleSome physical interpretation present but superficial; damping regimes described mathematically without physical contextPurely mathematical treatment with no physical insight; no connection to real-world applications or physical meaning of quantities

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