Q6
(a) A regular tetrahedron, formed of six light rods, each of length l, rests on a smooth horizontal plane. A ring of weight W and radius r is supported by the slant sides. Using the principle of virtual work, find the stress in any of the horizontal sides. (15 marks) (b) A particle executes simple harmonic motion such that in two of its positions, velocities are u and v, and the two corresponding accelerations are f₁ and f₂. For what value(s) of k, the distance between the two positions is k(v² - u²)? Show also that the amplitude of the motion is 1/(f₂² - f₁²) [(u² - v²)(u²f₂² - v²f₁²)]^(1/2) (15 marks) (c) (i) Find the second solution of the differential equation xy'' + (x-1)y' - y = 0 using u(x) = -e^{-x} as one of the solutions. (10 marks) (ii) Find the general solution of the differential equation x²y'' - 2xy' + 2y = x³ sin x by the method of variation of parameters. (10 marks)
हिंदी में प्रश्न पढ़ें
(a) लंबाई l की छः हल्की छड़ों द्वारा निर्मित एक सम चतुष्फलक एक चिकने क्षैतिज समतल पर रखा है। W भार तथा r त्रिज्या का एक छल्ला तिर्यक भुजाओं द्वारा आलंबित है। कल्पित कार्य सिद्धांत का उपयोग करते हुए किसी भी एक क्षैतिज भुजा में प्रतिबल ज्ञात कीजिए। (15 अंक) (b) सरल आवर्त गति में एक कण की दो स्थितियों में वेग u और v हैं तथा दो संगत त्वरण f₁ और f₂ हैं। k के किस मान/किन मानों के लिए दोनों स्थितियों के बीच की दूरी k(v² - u²) है? यह भी दर्शाइए कि गति का आयाम 1/(f₂² - f₁²) [(u² - v²)(u²f₂² - v²f₁²)]^(1/2) है। (15 अंक) (c) (i) एक हल के रूप में u(x) = -e^{-x} का उपयोग करते हुए अवकल समीकरण xy'' + (x-1)y' - y = 0 का दूसरा हल ज्ञात कीजिए। (10 अंक) (ii) प्राचल-विचरण विधि का उपयोग कर अवकल समीकरण x²y'' - 2xy' + 2y = x³ sin x का व्यापक हल ज्ञात कीजिए। (10 अंक)
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How this answer will be evaluated
Approach
Solve all four sub-parts systematically, allocating approximately 30% time to part (a) on virtual work and tetrahedron geometry, 30% to part (b) on SHM derivation, 20% to part (c)(i) on reduction of order, and 20% to part (c)(ii) on variation of parameters. Begin each part with clear statement of principles, show complete derivation with intermediate steps, and conclude with boxed final answers including proper units where applicable.
Key points expected
- For (a): Correct geometry of tetrahedron with height h = l√(2/3), proper identification of virtual displacements, and application of principle of virtual work to find tension in horizontal rods as Wl/(3√(l²-3r²))
- For (b): Derivation using v² = ω²(a²-x²) and f = -ω²x to establish k = 1/(2ω²) = 1/(2√(f₁f₂)), and rigorous algebraic manipulation to prove the amplitude formula
- For (c)(i): Application of reduction of order with y = v(x)u(x) to obtain second solution y₂ = x-1 (or equivalent), verifying linear independence via Wronskian
- For (c)(ii): Correct identification of complementary function y_c = c₁x + c₂x², computation of Wronskian W = x², determination of particular integral via variation of parameters yielding y_p = -x² sin x
- Proper handling of singular points at x=0 in both differential equations of part (c), with justification of solution validity
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correct tetrahedron geometry with proper coordinate system and identification of all forces; for (b): correct SHM energy relations and sign conventions for acceleration; for (c): proper identification of singular point x=0 and valid solution domains | Most geometric relations correct but minor errors in force identification or missing domain restrictions for ODEs | Fundamental errors in tetrahedron geometry, incorrect SHM relations, or failure to recognize singular nature of ODEs |
| Method choice | 20% | 10 | Virtual work principle applied correctly in (a); energy/acceleration method chosen appropriately in (b); reduction of order in (c)(i) and variation of parameters in (c)(ii) executed with correct formulas | Correct methods chosen but inefficient approaches or missing standard formulas requiring re-derivation | Incorrect method selection such as using equilibrium instead of virtual work in (a), or undetermined coefficients instead of variation of parameters in (c)(ii) |
| Computation accuracy | 20% | 10 | Algebraic manipulations in (b) leading to exact amplitude formula without errors; correct integration in reduction of order and variation of parameters; accurate handling of trigonometric and exponential integrals | Minor computational slips in coefficients or signs, but overall structure recoverable; one part may have significant calculation error | Major computational errors preventing meaningful final answers, or consistent sign errors throughout |
| Step justification | 20% | 10 | Each virtual displacement justified in (a); physical meaning of k explained in (b); verification of linear independence via Wronskian in (c)(i); clear derivation of Wronskian and integration steps in (c)(ii) | Most steps shown but gaps in justification, particularly for virtual work principle application or variation of parameters formulas | Missing crucial justifications such as why virtual work applies, or unjustified leaps in ODE solution methods |
| Final answer & units | 20% | 10 | All four answers clearly stated: tension in (a) with correct dimensional analysis; k and amplitude in (b) in simplified radical form; second solution y₂ in (c)(i); general solution y = c₁x + c₂x² - x² sin x in (c)(ii) | Correct forms but unsimplified, or missing one final answer; units present but inconsistent | Missing or incorrect final answers; failure to combine complementary and particular solutions in (c)(ii); dimensional inconsistencies |
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