Exercises: Simulating Sedimentation
Exercise 1: Suppose that 1000 spheres of radius 1 are dropped into a cylinder of radius 11 from random locations above the cylinder. Assume that the spheres fall straight down and stop in position only if they reach the bottom of the cylinder or if they come in contact with another sphere (i.e. k = 2). Approximate the number of the spheres that will reach the bottom of the cylinder.
Exercise 2: Suppose that 100 spheres of radius 1 are dropped into a cylinder of radius 11 from random locations above the cylinder. For each of the cases k = 2.0 and k = 4.0 do the following: run the program given in the text (for n = 100) and create a three-dimensional scatter plot of the final positions of the centers of the spheres. What is noticeably different about the two scatter plots?
Exercise 3: The value of
/4 can be approximated by finding the fraction of times that a point (x, y) placed randomly in the square [-1, 1]×[-1, 1] (the region -1
x
1 and -1
y
1) also lies in the unit circle x2 + y2
1. Approximate the value of
/4 = 0.78539816... in this manner using 100 points placed randomly in this square. With the random placement of 100 points, the best approximation that can be obtained in this manner is 79/100 = .79. With the random placement of 100 or fewer points, the best approximation that can be obtained is 11/14 = .785714..., which can occur only when n is a positive multiple of 14 (and so, in this case, more points does not necessarily produce a better approximation). How close is your estimate to this value?
Exercise 4: Repeat the work for Exercise 3, but use 1000 points placed randomly in [-1, 1]×[-1, 1]. Since we are utilizing more points, and a number that is an integer multiple of the number of points from Exercise 3, we would expect a better approximation. Does this yield a better approximation of
/4? If so, then how much better? With the random placement of 1000 points, the best approximation that can be obtained in this manner is 785/1000 = .785. With the random placement of 1000 or fewer points, the best approximation that can be obtained is 355/452 = .78539823..., which can occur only when n is a positive multiple of 452. How close is your estimate to this value?
Exercise 5: Consider the function f(x) = e-cos(x). For 0
x
1, this function satisfies 0
f(x)
1. Thus the area of the region below the graph of y = f(x) and above the x-axis is the same as the fraction that this area represents in the square [0, 1]×[0, 1]. This area can then be approximated by randomly generating points (x, y) in [0, 1]×[0, 1], and finding the fraction of these points that satisfy y
e-cos(x). Approximate this area in this manner using 100 randomly generated points.
Last modified: Thu Sep 27 2001