(a) Consider a large stationary cylinder of inner radius R. A smaller solid cylinder of radius r rolls without slipping inside the larger cylinder. Determine the equation of motion of the smaller cylinder. 10

(b) Derive the expression for the gravit

Question

(a) Consider a large stationary cylinder of inner radius R. A smaller solid cylinder of radius r rolls without slipping inside the larger cylinder. Determine the equation of motion of the smaller cylinder. 10

(b) Derive the expression for the gravitational self-energy of a uniform solid sphere of mass M and radius R. 10

(c) A particle of rest mass 1 kg and velocity of magnitude 0·9c collides with a particle of mass 2 kg at rest. After collision the two particles coalesce and form a single particle of mass M and velocity V. Determine M and V. 10

(d) In a double slit Fraunhofer diffraction experiment, the slit width is 0·12 mm and the spacing between the two slits is 0·48 mm. The distance of the screen from the slits is 1·5 m. If the wavelength of the light used is 600 nm, determine (i) the missing orders of the interference maxima, and (ii) the distance between the central maxima and the first minima. 10

(e) A light beam of wavelength 600 nm produced by a 20 mW laser source is incident on a plane mirror. Determine :
(i) number of photons per second striking the surface of the mirror.
(ii) force exerted by the light beam on the mirror. 10

UPSC Answer Check · Accepted Answer

Begin with clear statement of physical principles for each sub-part. For (a), set up Lagrangian with constraint; for (b), integrate gravitational potential energy; for (c), apply relativistic energy-momentum conservation; for (d), combine single-slit diffraction envelope with double-slit interference; for (e), use photon energy and radiation pressure concepts. Allocate approximately 20% time each to (a), (b), (c), (d), and (e) combined, with (e)(i) and (e)(ii) sharing that final 20%. Present derivations step-wise with final boxed answers for numerical parts.
- (a) Rolling constraint: arc length relation (R-r)θ = rφ; correct Lagrangian with kinetic energy of CM plus rotational energy; equation of motion as simple harmonic oscillator with period depending on √(R-r)/g
- (b) Gravitational self-energy: assemble sphere shell by shell; integration of -GM(r)dm/r from 0 to R; final result U = -3GM²/5R with correct handling of negative sign
- (c) Relativistic collision: calculate γ = 1/√(1-0.9²) ≈ 2.294; conserve total energy and momentum; solve for M and V with M > 3kg due to kinetic energy conversion to mass
- (d) Missing orders: condition d/a = 4 implies interference maxima at n=4,8,12... coincide with diffraction minima; first minima distance using y = λD/d for interference pattern
- (e)(i) Photon flux: N = Pλ/(hc) ≈ 6.03 × 10¹⁶ photons/second; (e)(ii) Force: F = 2P/c = 2×20mW/c ≈ 1.33 × 10⁻¹⁰ N for perfect reflection

Dimension	Weight	Max marks	Excellent	Average	Poor
Concept correctness	20%	10	Correctly identifies all physical principles: pure rolling constraint in (a), shell theorem in (b), invariant mass in relativistic collisions for (c), intensity modulation in (d), and photon momentum transfer in (e); no conceptual errors across any sub-part	Identifies most principles correctly but confuses classical with relativistic mass in (c) or misses that missing orders require both conditions simultaneously in (d)	Fundamental errors such as treating collision as classical in (c), using non-relativistic kinetic energy, or applying Huygens' principle incorrectly in (d)
Derivation rigour	20%	10	Complete step-by-step derivations: explicit Lagrangian formulation for (a), proper shell-by-shell integration limits in (b), clear Lorentz factor calculation and algebraic solution in (c), combined intensity formula derivation in (d), and explicit photon energy to number conversion in (e)	Correct final formulas but skips key steps like constraint equation derivation in (a) or assumes result without integration in (b); algebra present but condensed	Jumps to final answers without derivation, missing critical steps like the γ calculation in (c) or the factor of 2 in radiation pressure for (e)(ii)
Diagram / FBD	15%	7.5	Clear diagrams for (a) showing inner cylinder, contact point, angles θ and φ with forces; for (d) depicting intensity envelope with missing orders marked; coordinate systems labeled with all relevant parameters	Basic sketch for (a) without angle labeling or force components; mentions intensity pattern in (d) but no visual representation	No diagrams despite clear need for geometric visualization in (a) and (d); or incorrect diagrams showing wrong geometry
Numerical accuracy	25%	12.5	All numerical values correct with proper significant figures: M ≈ 4.36 kg and V ≈ 0.526c for (c); missing orders n = 4, 8, 12... and y ≈ 1.875 mm for (d); N ≈ 6.0 × 10¹⁶ s⁻¹ and F ≈ 1.33 × 10⁻¹⁰ N for (e)	Correct methodology but arithmetic errors like wrong γ value or unit conversion mistakes (mm to m in d); final answers within 10% of correct value	Order of magnitude errors, incorrect unit conversions, or missing factors of c; answers without units or with wrong dimensions
Physical interpretation	20%	10	Interprets results physically: explains why motion in (a) is simple harmonic equivalent to pendulum; discusses negative binding energy significance in (b); notes mass increase in (c) demonstrates E=mc²; explains why certain orders missing in (d); relates photon flux to classical intensity in (e)	Brief comment on one or two results but misses deeper connections like equivalence of (a) to physical pendulum or significance of M > m₁+m₂ in (c)	No interpretation provided; purely mathematical treatment without explaining what results mean physically

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