Q7
(a) Verify Stokes' theorem for $\vec{F} = x\hat{i} + z^2\hat{j} + y^2\hat{k}$ over the plane surface : $x + y + z = 1$ lying in the first octant. 20 (b) Solve the following initial value problem by using Laplace's transformation $\frac{d^2y}{dt^2} - 3\frac{dy}{dt} + 2y = h(t)$, where $$h(t) = \begin{cases} 2, & 0 < t < 4, \\ 0, & t > 4, \end{cases} \quad y(0) = 0, \quad y'(0) = 0$$ 15 (c) Suppose a cylinder of any cross-section is balanced on another fixed cylinder, the contact of curved surfaces being rough and the common tangent line horizontal. Let $\rho$ and $\rho'$ be the radii of curvature of the two cylinders at the point of contact and $h$ be the height of centre of gravity of the upper cylinder above the point of contact. Show that the upper cylinder is balanced in stable equilibrium if $h < \frac{\rho\rho'}{\rho+\rho'}$. 15
हिंदी में प्रश्न पढ़ें
(a) स्टोक्स प्रमेय को $\vec{F} = x\hat{i} + z^2\hat{j} + y^2\hat{k}$ के लिए प्रथम अष्टांशक में स्थित समतल पृष्ठ : $x + y + z = 1$ पर सत्यापित कीजिए। 20 (b) लाप्लास रूपांतरण का उपयोग करके निम्नलिखित प्रारंभिक मान समस्या : $$\frac{d^2y}{dt^2} - 3\frac{dy}{dt} + 2y = h(t), \text{ जहाँ } h(t) = \begin{cases} 2, & 0 < t < 4, \\ 0, & t > 4, \end{cases} \quad y(0) = 0, \quad y'(0) = 0$$ को हल कीजिए । 15 (c) माना किसी भी अनुप्रस्थ-काट का एक बेलन दूसरे स्थिर बेलन पर संतुलित है, जहाँ वक्रीय पृष्ठों का संपर्श रूक्ष है तथा उभयनिष्ठ स्पर्श-रेखा क्षैतिज है । माना दोनों बेलनों के स्पर्श बिंदु पर उनकी वक्रता त्रिज्याएं $\rho$ तथा $\rho'$ हैं और संपर्श बिंदु से ऊपरी बेलन के गुरुत्व केंद्र की ऊँचाई $h$ है । दर्शाइए कि स्थायी साम्य में ऊपरी बेलन संतुलित है यदि $h < \frac{\rho\rho'}{\rho+\rho'}$ । 15
Directive word: Verify
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How this answer will be evaluated
Approach
Verify Stokes' theorem in part (a) by computing both surface and line integrals; solve the IVP in part (b) using Laplace transforms with proper handling of the piecewise forcing function; prove the stability condition in part (c) using virtual work and energy methods. Allocate approximately 40% time to part (a) as it carries 20 marks, 30% each to parts (b) and (c). Structure each part with clear statement of method, step-by-step execution, and final verification or conclusion.
Key points expected
- For (a): Correct parameterization of the triangular boundary C in the first octant with vertices (1,0,0), (0,1,0), (0,0,1) and computation of curl F = (2y-2z)i - 0j + 0k
- For (a): Evaluation of surface integral ∫∫(curl F)·n̂ dS over the plane x+y+z=1 and line integral ∮F·dr around the triangular boundary, showing both equal 1/2
- For (b): Correct application of Laplace transform to the piecewise function h(t) using Heaviside step function: L{h(t)} = 2(1-e^{-4s})/s
- For (b): Proper partial fraction decomposition of Y(s) = 2(1-e^{-4s})/[s(s-1)(s-2)] and inversion to obtain y(t) for 0<t<4 and t>4 with continuity at t=4
- For (c): Setting up virtual displacement analysis with upper cylinder's center of gravity at height h above contact, using radii of curvature ρ and ρ'
- For (c): Deriving the stability condition by requiring the potential energy to be minimum, leading to h < ρρ'/(ρ+ρ') = 1/(1/ρ + 1/ρ')
- For (c): Physical interpretation that the equivalent radius of curvature for the combined surfaces must exceed the height of the center of gravity
Evaluation rubric
| Dimension | Weight | Max marks | Excellent | Average | Poor |
|---|---|---|---|---|---|
| Setup correctness | 20% | 10 | For (a): correctly identifies the triangular boundary and unit normal n̂=(i+j+k)/√3; for (b): properly expresses h(t) using Heaviside function and states correct initial conditions; for (c): accurate free-body diagram with correct geometric relationships between ρ, ρ' and h | Minor errors in boundary identification or normal vector direction; partially correct Heaviside setup; basic geometry correct but missing some curvature relationships | Wrong surface or boundary chosen; fails to use Heaviside for piecewise function; incorrect geometric setup with wrong center of gravity position |
| Method choice | 20% | 10 | For (a): chooses efficient parameterization using two variables; for (b): selects convolution theorem or direct inversion appropriately; for (c): applies virtual work principle or energy method rather than force balance alone | Standard methods applied correctly but not optimally; some unnecessary steps or missed shortcuts | Wrong method entirely (e.g., Gauss theorem instead of Stokes); attempts variation of parameters for Laplace problem; uses only force balance without energy consideration |
| Computation accuracy | 20% | 10 | For (a): curl F, surface integral and three line integrals all computed without arithmetic errors, final verification exact; for (b): correct partial fractions 1/s - 2/(s-1) + 1/(s-2) and accurate inversion; for (c): correct differentiation of potential energy and algebraic simplification to final inequality | Minor sign errors or coefficient mistakes that don't propagate catastrophically; one incorrect partial fraction term; algebraic slips in final simplification | Major computational errors in curl or line integrals; wrong characteristic equation roots; algebraic errors leading to wrong stability condition |
| Step justification | 20% | 10 | Each step explicitly justified: orientation of boundary for Stokes' theorem, choice of contour for Laplace inversion, physical reasoning for stability criterion; cites relevant theorems (Stokes, final value theorem, minimum potential energy principle) | Most steps shown but some gaps in reasoning; occasional 'it can be shown that' without justification; missing physical interpretation | Bare calculations without justification; jumps between steps; no physical insight provided for stability condition |
| Final answer & units | 20% | 10 | For (a): explicit verification that both integrals equal 1/2; for (b): complete piecewise solution with verification of initial conditions and continuity; for (c): clean derivation of h < ρρ'/(ρ+ρ') with clear statement of stability condition | Correct final answers but poorly presented; missing verification in (a); incomplete piecewise definition in (b); correct inequality but no simplification | Wrong numerical values; missing pieces of solution; incorrect inequality direction; no boxed or highlighted final answers |
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