Physics 2021 Paper I 50 marks Compulsory Solve

Q1

(a) A particle moving in a central force field describes the path r = ke^αθ, where k and α are constants. If the mass of the particle is m, find the law of force. (10 marks) (b) A capillary tube having 1·0 mm diameter, 20 cm in length is fitted horizontally to a vessel in which alcohol is kept fully up to the neck. Density of alcohol is 8 × 10² kg/m³. The depth of the centre of the capillary tube below the surface of alcohol is 40 cm. Find the amount of alcohol that will flow out of the capillary tube in 10 minutes. Coefficient of viscosity of alcohol is 0·0012 Ns/m². (10 marks) (c) An observer detects two explosions, one that occurs near him at a certain time and another that occurs 2·0 ms later 100 km away. Another observer finds that the two explosions occur at the same place. What time interval separates the explosions to the second observer? (10 marks) (d) A thin film of petrol of thickness 9 × 10⁻⁶ cm is viewed at an angle 30° to the normal. Find the wavelength(s) of light in visible spectrum which can be viewed in the reflected light. The refractive index of the film μ = 1·35. (10 marks) (e) A mass m is suspended by two springs having force constants k₁ and k₂ as shown in the figure. The mass m is displaced vertically downward and then released. If at any instant t, the displacement of the mass m is x, then show that the motion of the mass is simple harmonic motion having frequency f = 1/(2π) √[1/m (k₁k₂)/(k₁+k₂)] (10 marks)

हिंदी में प्रश्न पढ़ें

(a) एक कण का पथ केंद्रीय बल क्षेत्र में r = ke^αθ से वर्णित किया जाता है, जहाँ कि k और α नियतांक हैं । यदि कण का द्रव्यमान m है, तो बल का नियम ज्ञात कीजिए । (10 अंक) (b) एक 1·0 mm व्यास और 20 cm लंबाई वाली केशिकीय नलिका एक बर्तन, जिसमें 8 × 10² kg/m³ घनत्व का एल्कोहॉल भरा है, से क्षैतिज दिशा में जुड़ी है । केशिकीय नलिका के केंद्र की गहराई एल्कोहॉल के पृष्ठ से 40 cm नीचे है । 10 मिनट में केशिकीय नलिका से बहने वाले एल्कोहॉल की मात्रा ज्ञात कीजिए । एल्कोहॉल का श्यानता गुणांक 0·0012 Ns/m² है । (10 अंक) (c) एक प्रेक्षक दो विस्फोट देखता है, पहला जो कि उसके पास किसी समय पर होता है तथा दूसरा जो कि 2·0 ms बाद 100 km दूर होता है । एक दूसरा प्रेक्षक पाता है कि दोनों विस्फोट एक ही स्थान पर होते हैं । दूसरे प्रेक्षक के लिए विस्फोटों का समय अंतराल क्या होगा? (10 अंक) (d) 9 × 10⁻⁶ cm पतले पेट्रोल की परत (फिल्म) को लम्बवत् दिशा से 30° कोण पर देखा जाता है । परावर्तित प्रकाश के उन तरंगदैर्घ्य(यों) को ज्ञात कीजिए जो कि दृश्य वर्णक्रम (स्पेक्ट्रम) में आते हैं । पेट्रोल की परत (फिल्म) का अपवर्तनांक μ = 1·35 है । (10 अंक) (e) चित्र में दर्शाए अनुसार एक द्रव्यमान m, दो स्प्रिंगों जिनका बल नियतांक k₁ और k₂ है, से लटकाया गया है । द्रव्यमान m को उर्ध्वाधर दिशा में नीचे की तरफ थोड़ा-सा विस्थापित करके छोड़ दिया जाता है । यदि किसी समय t पर द्रव्यमान m का विस्थापन x हो, तो दिखाइए कि द्रव्यमान की गति सरल आवर्त गति है जिसकी आवृत्ति f = 1/(2π) √[1/m (k₁k₂/(k₁+k₂))] है । (10 अंक)

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Approach

Solve each sub-part sequentially with clear section headers. For (a), derive the central force law using Binet's equation; for (b), apply Poiseuille's law for viscous flow; for (c), use Lorentz transformation for time intervals; for (d), apply thin film interference conditions; for (e), derive the equivalent spring constant for series combination. Allocate approximately 2-2.5 minutes per mark, ensuring all five parts receive proportional attention with brief physical interpretation after each numerical result.

Key points expected

  • (a) Apply Binet's equation for central force motion: d²u/dθ² + u = -F(1/u)/(mh²u²) where u=1/r, substitute r=ke^(αθ), and derive F ∝ 1/r³
  • (b) Use Poiseuille's formula Q = πpr⁴/(8ηL) with hydrostatic pressure p = ρgh, calculate volume flow rate, then find total volume in 10 minutes
  • (c) Apply Lorentz transformation: in frame S' moving with velocity v where Δx'=0, find γ and then calculate Δt' = γ(Δt - vΔx/c²)
  • (d) Apply thin film interference condition for reflected light: 2μtcosr = (2n+1)λ/2 for destructive interference or 2μtcosr = nλ for constructive, using Snell's law to find r from given angle of incidence
  • (e) Show springs in series give 1/k_eq = 1/k₁ + 1/k₂, derive equation of motion md²x/dt² = -k_eqx, and obtain the given frequency expression
  • For (c), identify that second observer sees explosions at same place implies finding proper time interval using spacetime interval invariance

Evaluation rubric

DimensionWeightMax marksExcellentAveragePoor
Concept correctness20%10Correctly identifies Binet's equation for (a), Poiseuille's law for (b), Lorentz transformation for (c), thin film interference conditions for (d), and equivalent spring constant for series combination for (e); no conceptual errors in any sub-partCorrectly identifies most physical principles but has minor errors such as using wrong interference condition (constructive vs destructive) in (d) or incorrect pressure head calculation in (b)Major conceptual errors like using Bernoulli instead of Poiseuille for viscous flow in (b), or treating springs as parallel instead of series in (e), or using Galilean transformation in (c)
Derivation rigour20%10Complete mathematical derivations with all intermediate steps shown: proper substitution in Binet's equation, correct integration for flow rate, full Lorentz transformation algebra, Snell's law application before interference condition, and clear derivation of equivalent spring constantDerivations mostly complete but skips some intermediate steps or has minor algebraic errors that don't affect final answer structureMissing crucial derivation steps, jumps to final formulas without showing substitution, or makes significant algebraic errors leading to wrong functional forms
Diagram / FBD15%7.5Clear diagram for (e) showing mass suspended by two springs in series with displacement x indicated; ray diagram for (d) showing incident, reflected and refracted rays with angles marked; schematic for (b) showing capillary tube orientation and pressure headBasic diagram for (e) showing springs and mass but missing force labels or displacement direction; minimal or no diagrams for other partsNo diagrams despite (e) explicitly mentioning 'as shown in figure'; missing free body diagram makes derivation unclear
Numerical accuracy25%12.5All numerical calculations correct: force law coefficient in (a), volume ~3.14×10⁻⁶ m³ or correct SI value in (b), time interval ~1.73 ms or exact relativistic result in (c), visible wavelengths ~540 nm, 430 nm etc. in (d), final frequency formula verified in (e)Correct order of magnitude in most answers with minor calculation errors; correct unit conversions but arithmetic mistakes in final stepsWrong orders of magnitude, missing unit conversions (cm to m, mm to m, minutes to seconds), or calculation errors exceeding 50% deviation
Physical interpretation20%10Interprets (a) result as inverse-cube force characteristic of certain field configurations; explains (b) result in terms of balance between viscous resistance and driving pressure; notes time dilation/length contraction connection in (c); identifies colors in visible spectrum for (d); discusses physical meaning of reduced effective stiffness in (e)Brief physical comments after some calculations but missing interpretation for 1-2 sub-parts or superficial treatmentPurely mathematical treatment with no physical interpretation; fails to comment on why results make physical sense or their real-world significance

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