Physics 2022 Paper I 50 marks Compulsory Solve

Q1

An electron is moving under the influence of a point nucleus of atomic number Z. Show that the orbit of the electron is an ellipse. 10 marks Show that the mean kinetic and potential energies of non-dissipative simple harmonic vibrating systems are equal. 10 marks An observer on a railway platform observed that as a train passed through the station at 108 km/hr, the frequency of the whistle appeared to drop by 350 Hz. Find the frequency of the whistle. (Velocity of sound in air = 380 m s⁻¹) 10 marks Show that for very small velocity, the equation for kinetic energy, K = Δmc² becomes K = ½m₀v², where notations have their usual meanings. 10 marks A phase retardation plate of quartz has thickness 0·1436 mm. For what wavelength in the visible region will it act as quarter-wave plate? Given that μ₀ = 1·5443 and μᴇ = 1·5533. 10 marks

हिंदी में प्रश्न पढ़ें

एक इलेक्ट्रॉन, परमाणु क्रमांक Z के बिंदु नाभिक के प्रभाव में गतिमान है। दर्शाइए कि इलेक्ट्रॉन की कक्षा एक दीर्घवृत्त है। 10 अंक दर्शाइए कि ऊर्जा-संरक्षी सरल आवर्त कंपन तंत्रों की औसत गतिज और स्थितिज ऊर्जा बराबर हैं। 10 अंक एक रेलवे प्लेटफॉर्म पर एक प्रेक्षक ने देखा कि जैसे ही एक ट्रेन स्टेशन से 108 km/hr की गति से गुजरती है, सीटी की आवृत्ति में 350 Hz की कमी प्रतीत होने लगती है। सीटी की आवृत्ति ज्ञात कीजिए। (वायु में ध्वनि का वेग = 380 m s⁻¹) 10 अंक दर्शाइए कि बहुत कम वेग के लिए गतिज ऊर्जा का समीकरण K = Δmc², K = ½m₀v² हो जाता है, जहाँ संकेतों का अपना सामान्य अर्थ होता है। 10 अंक क्वार्ट्ज की एक कला मंदन प्लेट की मोटाई 0·1436 mm है। दृश्य क्षेत्र में किस तरंगदैर्घ्य के लिए यह एक-चौथाई तरंग प्लेट के रूप में कार्य करेगी? दिया गया है, μ₀ = 1·5443 और μᴇ = 1·5533। 10 अंक

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Approach

This question demands solving five distinct 10-mark problems covering classical mechanics and special relativity. The answer should present each solution as a self-contained unit with clear problem identification, step-by-step working, and final boxed answers. Begin with the orbital mechanics problem (Kepler's first law derivation), followed by the SHM energy theorem, then the Doppler effect calculation, the relativistic kinetic energy approximation, and finally the quarter-wave plate wavelength determination. Conclude with a brief synthesis noting the unifying theme of oscillatory and wave phenomena across the problems.

Key points expected

  • For the electron orbit: Apply Newton's second law with Coulomb force, use polar coordinates, and derive the orbit equation r(θ) = l/(1+εcosθ) showing ε<1 for bound states
  • For SHM energies: Derive ⟨T⟩ and ⟨V⟩ over one complete period using x = Asin(ωt+φ), showing both equal ¼kA² or ¼mω²A²
  • For Doppler effect: Identify this as a moving source problem (observer stationary), apply f' = f[v/(v±vs)] with proper sign convention for approaching/receding train
  • For relativistic KE: Expand γ = (1-v²/c²)^(-½) using binomial theorem, keeping terms up to v²/c² to recover classical result
  • For quarter-wave plate: Use condition (μe-μo)t = λ/4, solve for λ = 4(μe-μo)t, and verify result falls in visible range (400-700 nm)

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies all five physical situations: inverse-square law central force motion, time-averaged energies in SHM, transverse Doppler effect for moving source, low-velocity limit of Lorentz factor, and birefringent phase retardation. Properly selects appropriate coordinate systems and reference frames for each problem.Identifies most physical situations correctly but confuses Doppler effect type (e.g., treats as moving observer) or makes sign errors in orbit equation. May not recognize why ε<1 implies ellipse rather than hyperbola.Fundamental misconceptions such as treating Coulomb force as Hooke's law for orbit, confusing instantaneous with mean energies in SHM, or applying wrong Doppler formula entirely.
Derivation rigour25%12.5Complete mathematical rigour: explicit Binet equation derivation for orbit, proper integration limits for time averages, clear algebraic steps for Doppler frequency difference, valid binomial expansion with remainder justification, and exact wavelength calculation with unit consistency throughout.Derivations mostly complete but skips key steps like explicit integration for ⟨T⟩, assumes small-angle approximations without justification, or presents final formulae without showing intermediate algebra.Missing derivations entirely (stating results without proof), mathematically invalid steps like dividing by zero or incorrect differentiation, or confusion between proper and improper integrals for averaging.
Diagram / FBD15%7.5Clear diagrams for: (a) electron orbit showing r, θ, focus at nucleus with velocity vectors; (b) SHM displacement-time and energy-time graphs; (c) Doppler geometry with train approaching/receding; (d) quartz plate with ordinary/extraordinary axes and optical path difference; all properly labeled with symbols used in derivations.Some diagrams present but incomplete—e.g., orbit diagram without velocity vectors, or missing energy graphs for SHM. Labels may not match notation in text.No diagrams despite visual nature of problems, or diagrams that contradict the physics (e.g., circular orbit for elliptical motion claim, wrong wavefront geometry for Doppler effect).
Numerical accuracy25%12.5Precise calculations: Doppler frequency computed correctly as f = Δf·v/(2vs) = 350×380/(2×30) = 2216.67 Hz with proper unit conversion (108 km/hr = 30 m/s); quarter-wave plate wavelength λ = 4×(1.5533-1.5443)×0.1436×10⁻³ = 517.0 nm (green light, visible). All significant figures appropriate.Correct method but arithmetic errors, or correct final answers with wrong units. May forget to convert km/hr to m/s or mm to m, giving answers off by orders of magnitude.Gross calculation errors, impossible results (e.g., whistle frequency in MHz range, ultraviolet wavelength claimed as visible), or no numerical working shown for calculation-based problems.
Physical interpretation15%7.5Insightful commentary: connects elliptical orbits to Bohr-Sommerfeld quantization and atomic spectra; notes virial theorem (2⟨T⟩ = -⟨V⟩) for inverse-square law; discusses why Doppler drop is asymmetric for finite platform length; explains mass-energy equivalence in classical limit; identifies 517 nm as characteristic mercury line relevant to spectroscopy.Brief correct statements about each result without deeper connections, or correct but generic interpretation (e.g., 'energy is conserved' without specifying what this implies for the problems).No interpretation provided, or physically incorrect statements (e.g., claiming electron spirals into nucleus, or that relativistic mass 'becomes infinite' at low velocities).

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