Question

elcise .1S.3 pp69i-698. Figure 8.85 shows an old cylindrical bore- hole that has been filled in part with silt and in part with water. Before the hole can be redrilled the water has to be pumped to the surface. We wish to estimate the work (en- ergy) required for this purpose water? Ground level 50 mAy Silt Figure 8.83 (a) As a first approximation, assume that the silting has been uniform - as indicated in Figure 8.85 and that the water thus forms a right-circular cone of base radius 5 m and height 50 m. An element of work ΔW is the element of mass Δm times g times the height (h) raised. Here, g is the acceleration due to gravity, mass in kg and height in m. The height raised is h 50 -y. We require to calculate the element of mass. Consider an element of revolution about the y-axis. For this geometry, we use the disk method although we are integrating along y. Here, r is the radius (in m) and the element of volume Δ V is in m3. (i) Write down in terms of y the radius of revolution, r. Hence, we can deduce that the volume of the small element of water shown is 10 (Continued over the page.)

Answer 1

Given diagram is Given data is height = 50 m radius = 5 m Delta W = mgh h = 50 - y (a) (i) Delta V = pi n^2 Delta y Delta V = pi (y/10)^2 Delta y Because n = 5 times y/50 = y/10 (ii) density = 1 kg/lit = 1000 kg/m^3 Therefore Delta m = k Delta V = 1000 times pi times y^2/(10)^2 times Delta y Delta m = 10 pi y^2 Delta y (iii) Delta W = Delta mgh Integrating integral Delta W = integral 10 pi y^2 Delta y g(50 - y) W_1 = 10 pi g integral^50_0 y^2 (50 - y) dy = 10 pi g [50 y^3/3 - y^4/4]^50_0 \r\n = 10 times pi times g times 50^4 times 1/12 = 5208333.3 pi g (b) Using Simpson's rule integral^b_a f(x) dx = Delta x/3 [f(x_0) + f(x_n) + 4 f(x_1) + f(x_3) +] + 2 [f(x_2) + f(x_4)