Physics 2023 Paper I 50 marks Derive

Q2

(a) (i) Derive the expressions for gravitational potentials at a point (I) outside the spherical shell, (II) inside the spherical shell. 10 marks (ii) Calculate the escape velocity of a body of mass 10 kg from the surface of Moon (g_Moon = 1/6 g_Earth). Mass of Moon = 7·3 × 10^22 kg Radius of Moon = 1·7 × 10^6 m 10 marks (b) Obtain condition for achromatism of two thin lenses separated by a finite distance. If the dispersive powers of the materials of the two lenses are 0·020 and 0·028, their focal lengths are 10 cm and 5 cm, respectively. Calculate the separation between them in order to form achromatic combination. 15 marks (c) (i) The quantities of rotatory motion are analogous to those of translatory motion. Write the corresponding equations of translatory and rotatory motion. 5 marks (ii) Describe the theorems of perpendicular and parallel axes in case of a plane lamina. 10 marks

हिंदी में प्रश्न पढ़ें

(a) (i) गुरुत्वीय विभव के लिए किसी बिंदु पर व्यंजक व्युत्पन्न कीजिए, जबकि बिंदु (I) गोलीय कोश के बाहर हो, (II) गोलीय कोश के अंदर हो । 10 (ii) 10 kg द्रव्यमान के किसी पिंड के पलायन वेग की गणना चंद्रमा के तल से कीजिए $$\left(g_{\text{चंद्रमा}} = \frac{1}{6} g_{\text{पृथ्वी}}\right)$$ चंद्रमा का द्रव्यमान = $7.3 \times 10^{22}$ kg चंद्रमा की त्रिज्या = $1.7 \times 10^6$ m 10 (b) एक निश्चित दूरी पर स्थित दो पतले लेंसों के लिए अवर्णकता की शर्त को प्राप्त कीजिए । यदि दो लेंसों के पदार्थों की परिक्षेपण क्षमता 0·020 और 0·028 हो, तथा इनकी फोकस दूरियाँ क्रमशः 10 cm और 5 cm हैं, तो इनके बीच की दूरी परिकलित कीजिए जिससे कि ये एक अवर्णक संयुग्म बना सकें । 15 (c) (i) घूर्णन गति की मात्राएँ स्थानांतरण गति की मात्राओं के अनुरूप होती हैं । घूर्णन एवं स्थानांतरण गतियों के संगत समीकरणों को लिखिए । 5 (ii) एक समतल स्तरीका के लिए लम्बवत और समांतर अक्षों के प्रमेयों का वर्णन कीजिए । 10

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Directive word: Derive

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How this answer will be evaluated

Approach

Begin with clear statement of objectives for each sub-part. For (a)(i), derive gravitational potentials using shell theorem with proper integration limits; for (a)(ii), calculate escape velocity using energy conservation with given Moon data. For (b), derive achromatism condition using dispersive power and focal length relations, then compute separation. For (c)(i), present analogy table between translatory and rotatory quantities; for (c)(ii), state and prove both axis theorems with diagrammatic illustration. Allocate approximately 35% time to part (a), 30% to part (b), and 35% to part (c) based on marks distribution.

Key points expected

  • (a)(i) Derivation of V_out = -GM/r for point outside spherical shell using integration of ring elements or Gauss's law analogy
  • (a)(i) Derivation of V_in = -GM/R (constant) for point inside spherical shell showing potential is independent of position
  • (a)(ii) Calculation of escape velocity v_esc = √(2GM/R) = √(2g_moon × R_moon) ≈ 2.38 km/s with proper unit conversion
  • (b) Derivation of achromatism condition: d/f₁ + (1-d/f₁)/f₂ = 0 or ω₁/f₁ + ω₂/f₂ = 0 for separated lenses, leading to d = (ω₁f₁ + ω₂f₂)/(ω₁ + ω₂)
  • (b) Numerical calculation: d = (0.020×10 + 0.028×5)/(0.020+0.028) = (0.20+0.14)/0.048 = 7.08 cm
  • (c)(i) Complete analogy table: displacement θ↔s, angular velocity ω↔v, angular acceleration α↔a, torque τ↔F, moment of inertia I↔m, angular momentum L↔p
  • (c)(ii) Statement and proof of perpendicular axis theorem: I_z = I_x + I_y for planar lamina
  • (c)(ii) Statement and proof of parallel axis theorem: I = I_cm + Md² with proper diagram showing axis translation

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly applies shell theorem for gravitational potential, uses energy conservation for escape velocity, understands dispersive power and focal length relations for achromatism, and accurately states both axis theorems with proper conditionsMinor errors in theorem statements or formula applications; understands basic concepts but confuses conditions (e.g., finite vs. contact lens separation) or makes sign errors in potential expressionsFundamental misconceptions such as treating potential inside shell as variable, using wrong escape velocity formula, confusing ω with 1/ω in achromatism, or stating incorrect axis theorem relationships
Derivation rigour25%12.5Complete mathematical rigour: explicit integration for shell potential with proper limits, clear energy conservation steps for escape velocity, systematic derivation of achromatism condition from focal length formula, and rigorous proof of both axis theorems with all steps justifiedCorrect final expressions but skips key steps or assumes results without proof; incomplete integration setup or missing algebraic steps in achromatism derivation; states theorems without proper derivationMissing derivations entirely or presents circular reasoning; jumps to conclusions without mathematical justification; incorrect calculus applications or algebraic errors throughout
Diagram / FBD15%7.5Clear diagrams: spherical shell with point P at distance r>R and r<R showing ring elements; lens system showing separation d and focal points; lamina with x, y, z axes labeled for perpendicular axis theorem, and parallel axes showing CM and displaced axis with distance dDiagrams present but inadequately labeled or missing key features; rough sketches without proper geometric relationships shown; missing one required diagramNo diagrams or completely irrelevant figures; diagrams that misrepresent physical situations (e.g., wrong axis orientation, incorrect lens types)
Numerical accuracy20%10Precise calculations: escape velocity with correct substitution (v_esc = √(2×6.67×10⁻¹¹×7.3×10²²/1.7×10⁶) ≈ 2.38 km/s or via g_moon = 1.62 m/s²), and exact separation d = 7.08 cm with proper significant figures and unit handlingCorrect method but arithmetic errors or wrong powers of ten; approximate answers without proper significant figures; unit conversion errors (km/s vs m/s)Completely wrong numerical answers due to formula misuse; missing calculations despite data provided; orders of magnitude errors indicating conceptual misunderstanding
Physical interpretation20%10Insightful interpretation: explains why potential is constant inside shell (no net force, work done is zero), compares Moon's escape velocity to Earth's (~2.4 km/s vs 11.2 km/s) explaining atmospheric retention, discusses practical achromatism applications in telescopes, and relates moment of inertia to mass distribution in rotating bodiesBrief mention of physical significance without elaboration; standard comparisons without insight; misses opportunity to connect to real-world applicationsPurely mathematical treatment with no physical insight; incorrect interpretations of results; fails to explain why derived results matter physically

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