ch05_07 - 496 CHAPTER 5 . Applications of the Definite...

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496 CHAPTER 5 . . Applications of the Definite Integral 5-66 y b a x a ± w b ± w 49. For tennis rackets, a large second moment (see exercises 47 and 48) means less twisting of the racket on off-center shots. Compare the second moment of a wooden racket ( a = 9 , b = 12 ,w = 0 . 5), a midsize racket ( a = 10 , b = 13 = 0 . 5) and an oversized racket ( a = 11 , b = 14 = 0 . 5). 50. Let M be the second moment found in exercise 48. Show that dM da > 0 and conclude that larger rackets have larger second moments. Also, show that d w > 0 and interpret this result. EXPLORATORY EXERCISES 1. As equipment has improved, heights cleared in the pole vault have increased. A crude estimate of the maximum pole vault possible can be derived from conservation of energy princi- ples. Assume that the maximum speed a pole-vaulter could run carrying a long pole is 25 mph. Convert this speed to ft/s. The kinetic energy of this vaulter would be 1 2 m v 2 . (Leave m as an unknown for the time being.) This initial kinetic energy would equal the potential energy at the top of the vault minus what- ever energy is absorbed by the pole (which we will ignore). Set the potential energy, 32 mh , equal to the kinetic energy and solve for h . This represents the maximum amount the vaulter’s center of mass could be raised. Add 3 feet for the height of the vaulter’s center of mass and you have an estimate of the maximum vault possible. Compare this to Sergei Bubka’s 1994 world record vault of 20 ± 1 3 4 ±± . 2. An object will remain on a table as long as the center of mass of the object lies over the table. For example, a board of length 1 will balance if half the board hangs over the edge of the table. Show that two homogeneous boards of length 1 will balance if 1 4 of the first board hangs over the edge of the table and 1 2 of the second board hangs over the edge of the first board. Show that three boards of length 1 will balance if 1 6 of the first board hangs over the edge of the table, 1 4 of the second board hangs over the edge of the first board and 1 2 of the third board hangs over the edge of the second board. Generalize this to a pro- cedure for balancing n boards. How many boards are needed so that the last board hangs completely over the edge of the table? L 2 L 4 . . . 5.7 PROBABILITY The mathematical fields of probability and statistics focus on the analysis of random pro- cesses. In this section, we give a brief introduction to the use of calculus in probability theory. It may surprise you to learn that calculus provides insight into random processes, but this is in fact, a very important application of integration. We begin with a simple example involving coin-tossing. Suppose that you toss two coins, each of which has a 50% chance of coming up heads. Because of the randomness involved, you cannot calculate exactly how many heads you will get on a given number of tosses. But you can calculate the likelihood of each of the possible outcomes. If we denote heads by H and tails by T, then the four possible outcomes from tossing two coins are HH, HT, TH and TT. Each of these four outcomes is equally likely, so we can say that each has
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ch05_07 - 496 CHAPTER 5 . Applications of the Definite...

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