Question

QUESTION 2 [25 marks] Figure 2 depicts a thin plate where its shape is the region, R bounded by the graphs y 3 and yx2. If the density distribution function of a thin plate is given by p(x, y) -1 2y 6x2, show the fastest way to find the centre of mass. Note: show your detailed calculation steps. a) (12 marks) -4 -2 Figure 2 b)x, y)represents a velocity field of a fluid over a surface S defined by z 6- 3x - 2y. If the magnitude of the velocity in the direction of the unit normal vector, h, on S is 3z/V14, compute the flux of Foxy) over the surface S in the first octant oriented upward, using the projection of S on the xy - plane. (13 marks)

Answer 1

Y = 3 & y = x^2 implies x^2 = 3 implies x = plusorminus squareroot 3 integral(x, y) = 1 + 2y + 6x^2 mass of Region R(b_1) = integral^squareroot 3_x = squareroot 3 integral^x^2_y = 3 (1 + 2y + 6x^2)dy dx M = integral^squareroot 3_- squareroot 3 (y + 2y^2/2 + 6x^2y)^x^2_3 dx M = integral^squareroot 3_- squareroot 3 (x^2 + x^4 + 6x^4 - 3 - 9 - 18x^2)dx M = integral^squareroot 3_- squareroot 3 (7x^4 - 17x^2 - 12) dx M = 7x^5/5 - 17x^3/3 - 12x]^squareroot 3_- squareroot 3 = 14(squareroot 3)^5/5 - 3 4(squareroot 3)^3/3 - 24(squareroot 3) M = 43.65 - 38.89 - 41.56 M = -56.81 Now, Moments of lamina M_x = integral^squareroot 3_- squareroot 3 integral^x^2_3 y(1 + 2y + 6x^2)dy dx = integral^squareroot 3_- squareroot 3 (y^2/2 + 2y^3/3 + 6x^2y^2/2)^x^2_3 dx M_x = integral^squareroot 3_- squareroot 3 (x^4/2 + 2x^6/3 + 3x^4 - 9/2 - 2 * 27/3 - 27x^2)dx M_x = integral^squareroot \r\n