ENGR 233 Term Test 2 - Teacher: Suraj Joshi - Summer 2021 PDF

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Summer 20211Concordia University Department of Mechanical, Industrial and Aerospace Engineering ENGR 233: Applied Advanced Calculus Term Test 2Tuesday, 15 June 2021 Duration: 1 hr Maximum Marks: 50Note: All questions are compulsory and carry equal marks.Q1. Test if the force field ��⃗(��,��)=(2�� + ...

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Summer 2021 Concordia University Department of Mechanical, Industrial and Aerospace Engineering ENGR 233: Applied Advanced Calculus Term Test 2 Tuesday, 15 June 2021

Duration: 1 hr

Maximum Marks: 50

Note: All questions are compulsory and carry equal marks. Q1. Test if the force field 𝐹 (𝑥, 𝑦) = (2𝑥 + 𝑒 −𝑦 )𝑖 + (4𝑦 − 𝑥𝑒 −𝑦 )𝑗 is path independent and find the work done in moving a particle along the curve 𝑦 = 𝑥4 + ln(𝑥 + 1) from the origin to the point (1,1) in ℝ2 . Q2. Find the polar moment of inertia using rectangular coordinates for a lamina formed by the intersection of two parabolas 𝑥 = 𝑦 2 + 2 and x = 6 − 𝑦 2 , where the density at any point 𝑃 in the lamina is inversely proportional to the square of the distance from the origin. Q3. Find the surface area of the end cap of the solid formed by the portion of the cylinder 𝑥 2 + 𝑦 2 = 2𝑦 that lies inside the sphere 𝑥 2 + 𝑦 2 + 𝑧 2 = 4 above the x-y plane. Q4. Use the Green’s Theorem to evaluate the line integral ∮𝐶 (2𝑥𝑦 − 𝑥2 )𝑑𝑥 + (𝑥 + 𝑦 2 )𝑑𝑦 , where C is the boundary of the region determined by the graphs of 𝑥 = 𝑦 2 and 𝑦 = 𝑥 2 .

Q5. Find the work done using polar coordinates in moving a particle in the force field 𝐹 = (𝑥 − 𝑦 3 )𝑖 + 𝑥 3 𝑗 along the left half of the circle 𝑥 2 + 𝑦 2 = 1.

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