Physics 2022 Paper I 50 marks Calculate

Q2

Consider two frames of reference S and S' having a common origin O. The frame S' is rotating with respect to the fixed frame S with a uniform ω⃗ = 3aₓ rad s⁻¹. A projectile of unit mass at position vector r⃗ = 7aₓ + 4aᵧ m is moving with v⃗ = 14aᵧ m s⁻¹. Calculate in the rotating frame S' the following forces on the projectile: (i) Euler's force (ii) Coriolis force (iii) Centrifugal force 15 marks A particle P of mass m₁ collides with another particle Q of mass m₂ at rest. The particles P and Q travel at angles θ and φ, respectively, with respect to the initial direction of P. Derive the expression for the maximum value of θ. 15 marks Obtain the system matrix for a thick lens and derive the thin lens formula. 20 marks

हिंदी में प्रश्न पढ़ें

दो निर्देश तंत्र S और S′ हैं, जिनका उभयनिष्ठ मूलबिंदु O है। S′ तंत्र एकसमान ω⃗ = 3aₓ rad s⁻¹ से स्थिर तंत्र S के सापेक्ष घूम रहा है। स्थिति सदिश r⃗ = 7aₓ + 4aᵧ m पर इकाई द्रव्यमान का एक प्रक्षेप्य v⃗ = 14aᵧ m s⁻¹ के साथ गतिमान है। घूर्णन तंत्र S′ में प्रक्षेप्य पर निम्नलिखित बलों की गणना कीजिए: (i) ऑयलर बल (ii) कोरिऑलिस बल (iii) अपकेन्द्री बल 15 अंक द्रव्यमान m₁ का एक कण P, विरामावस्था में स्थित द्रव्यमान m₂ के दूसरे कण Q से टकराता है। कण P और Q, P की प्रारंभिक दिशा के सापेक्ष क्रमशः कोण θ और φ पर प्रगमन करते हैं। θ के अधिकतम मान के लिए व्यंजक व्युत्पन्न कीजिए। 15 अंक मोटे लेंस का तंत्र आव्यूह ज्ञात कीजिए और पतले लेंस का सूत्र प्राप्त कीजिए। 20 अंक

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Directive word: Calculate

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How this answer will be evaluated

Approach

Begin with a brief statement of the rotating frame transformation equations and collision kinematics principles. Allocate approximately 30% time to Part 1 (rotating frame forces with explicit vector calculations), 30% to Part 2 (collision geometry derivation with momentum conservation), and 40% to Part 3 (thick lens matrix method leading to thin lens limit). Present derivations step-by-step with clear identification of final expressions, and conclude with a verification check for the thin lens limit.

Key points expected

  • Part 1: Correct identification that Euler's force is zero for uniform ω, and calculation of Coriolis force (-2mω⃗×v⃗') and centrifugal force (-mω⃗×(ω⃗×r⃗)) with proper vector cross products
  • Part 1: Explicit computation showing v⃗' = v⃗ - ω⃗×r⃗ = 14aᵧ - 12aᵧ = 2aᵧ m/s, yielding Coriolis force = -12a_z N and centrifugal force = 84a_x + 48a_y N
  • Part 2: Application of momentum conservation in x and y directions and kinetic energy conservation (elastic collision) to establish tanθ = sin(2φ)/(m₁/m₂ + cos(2φ))
  • Part 2: Derivation of maximum θ condition leading to sinθ_max = m₂/m₁ when m₁ > m₂, with physical constraint that scattering is impossible if m₁ < m₂
  • Part 3: Construction of system matrix M = R₂T(d)R₁ for thick lens with two refractions and translation, using refraction matrix R = [[1, 0], [(n₁-n₂)/n₂R, n₁/n₂]] and translation matrix T = [[1, d/n], [0, 1]]
  • Part 3: Reduction to thin lens formula 1/f = (n-1)(1/R₁ - 1/R₂) by taking d→0 limit and identifying focal length from matrix element C = -1/f

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies that Euler force vanishes for constant ω; distinguishes between absolute and relative velocity in Coriolis calculation; applies correct elastic collision conditions; recognizes thick lens matrix as product R₂T(d)R₁ in proper orderIdentifies correct force formulas but confuses v⃗ with v⃗' in Coriolis calculation; applies momentum conservation but misses energy conservation for elastic case; attempts matrix method with incorrect element orderingApplies Coriolis formula with wrong sign convention or uses v⃗ instead of v⃗'; treats collision as inelastic without justification; uses thin lens formula directly without matrix derivation
Derivation rigour20%10Shows complete vector algebra for cross products with unit vector notation; derives tanθ expression systematically eliminating velocities; carries matrix multiplication explicitly with d→0 limit clearly shownStates final formulas without showing intermediate cross product steps; jumps to maximum θ condition without derivation; presents final thin lens result with minimal matrix stepsGives numerical answers without any derivation; states θ_max formula without proof; writes thin lens formula without matrix method
Diagram / FBD15%7.5Clear free-body diagram showing ω⃗, r⃗, v⃗' directions and resulting fictitious force vectors; collision diagram with before/after momentum vectors and angles θ, φ labeled; thick lens ray diagram showing principal planes and matrix element physical meaningSketch showing coordinate axes and rotation direction; simple scattering diagram with angles marked; basic lens diagram without principal plane identificationNo diagrams; or incorrect force directions in sketch; missing angle labels in collision diagram
Numerical accuracy25%12.5Correct numerical values: Euler = 0; Coriolis = -12a_z N (or 12 N in -z direction); Centrifugal = 84a_x + 48a_y N; correct algebraic final expressions for θ_max and thin lens formula with proper sign conventionsCorrect method but arithmetic errors in cross products (e.g., wrong sign on Coriolis); correct θ_max expression but missing m₁ > m₂ condition; correct matrix form but error in final focal length expressionMajor calculation errors in vector products; incorrect final numerical values; wrong formula for maximum scattering angle; incorrect thin lens formula sign or form
Physical interpretation20%10Explains why Coriolis force is perpendicular to both ω and v'; interprets θ_max condition physically (backscattering when m₁ >> m₂); explains significance of principal planes in thick lens and how thin lens limit removes separationBrief comment on direction of fictitious forces; states that lighter projectile scatters more; mentions that thick lens becomes thin lens when d→0No physical interpretation; purely mathematical manipulation without explaining what results mean; no connection between matrix elements and optical parameters

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