Sink or Float?

Sink or Float?: Thought Problems in Math and Physics

Keith Kendig
Volume: 33
Copyright Date: 2008
Edition: 1
Pages: 390
https://www.jstor.org/stable/10.4169/j.ctt6wpwdk
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  • Book Info
    Sink or Float?
    Book Description:

    Sink or Float: Thought Problems in Math and Physics is a collection of problems drawn from mathematics and the real world. Its multiple-choice format forces the reader to become actively involved in deciding upon the answer. The book's aim is to show just how much can be learned by using everyday common sense. The problems are all concrete and understandable by nearly anyone, meaning that not only will students become caught up in some of the questions, but professional mathematicians, too, will easily get hooked. The more than 250 questions cover a wide swath of classical math and physics. Each problem's solution, with explanation, appears in the answer section at the end of the book. A notable feature is the generous sprinkling of boxes appearing throughout the text. These contain historical asides or little-known facts. The problems themselves can easily turn into serious debate-starters, and the book will find a natural home in the classroom.

    eISBN: 978-1-61444-207-3
    Subjects: Mathematics, Physics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Preface
    (pp. vii-xii)
  3. Table of Contents
    (pp. xiii-xiv)
  4. What Do You Think? A Sampler
    (pp. 1-8)
  5. Geometry
    (pp. 9-32)

    Geometry is as old as civilization itself. Monuments from the Late Stone Age such as Stonehenge in England and the pyramids in Egypt already show considerable geometric sophistication. Just as the push toward civilization led to a need for astronomy, it did the same for geometry. High-value land next to the Nile became subdivided into parcels with property lines and taxable areas, and all sorts of tables, rules and formulas were developed to help surveyors and tax-collectors. Thus, the Nile heightened the need for geometry: the river’s annual flooding kept surveyors busy re-establishing property lines.

    The ancient Greeks inherited a...

  6. Numbers
    (pp. 33-44)

    People began to count long before they began to write. In the late 1930s, an archaeologist sifting dirt in what is now the Czech Republic stumbled upon the most ancient mathematical object found so far: a 30,000-year-old wolf bone with 57 notches in it. These were arranged in groups of five, much as in today’s tally-method, suggesting that people may have counted on their fingers then. The relic offers no clue as to what was being counted. Was it days? food supplies? Some 25,000 years later, when trading had grown more widespread, such missing information became important, and filling it...

  7. Astronomy
    (pp. 45-66)

    Astronomy ranks among the oldest of sciences—hardly surprising, since astronomy played such a central role in the rise of civilization. Before the rise, we were clever nomads cooperating in small groups simply to exist, constantly searching for sustenance. Afterwards, we formed larger groups with greater organization and specialization, nomadic life being replaced by farmers planting crops and raising herds which could feed more people. Civilization gave more time for specialization and increased efficiency, and eventually culture.

    The transition hinged on agriculture and that depended on good timing in planting and harvesting crops, leading to a quest for calendars. People...

  8. Archimedes’ Principle
    (pp. 67-84)

    Archimedes’ principle is one of the earliest scientific principles, yet it has a surprisingly wide range of applications. This chapter illustrates some of them. The principle is

    A body partly or completely immersed in a fluid is buoyed up by a force equal to the weight of the displaced fluid.

    Not only was Archimedes the greatest mathematician of antiquity, he was something of a one-man Department of Defense. He was related to King Hiero II of Syracuse, a just, prudent, and generous man as well as a patron of the arts. The good ruler often called upon Archimedes for ideas,...

  9. Probability
    (pp. 85-104)

    The roots of probability go back to prehistoric games of chance and gambling. Today’s dice are descendants of knucklebones from various animals such as sheep, deer, oxen and llamas. These early dice had four sides that were often roughly tetrahedral, and on specimens unearthed from prehistoric American Indian burial mounds we see marked sides and a polish resulting from extensive use. There’s also evidence of extensive gambling by the ancient Greeks and Romans. Sundry gaming tables were scratched or carved into stone corridors of the Roman Coliseum, and even into steps leading to the temple of Venus. Gambling was an...

  10. Classical Mechanics
    (pp. 105-122)

    Newton’s mechanics fundamentally altered man’s vision of the universe, affecting the way we conceive of the physical world. Others before him, such as Copernicus (1473–1543), Kepler and Galileo (1564–1642), had helped clear a long trail of false starts and half truths thattheyhad inherited. Some of the earliest physical “laws” turned out to be utterly wrong. Aristotle, for example, made pronouncements coming from pure thought that were largely untested by physical experiment, and so did Plato (ca. 428–348 BCE), who incorrectly concluded that planets travel in perfect circles. Even during the early decades of the Renaissance...

  11. Electricity and Magnetism
    (pp. 123-142)

    Practical needs stimulated the growth of ancient counting, geometry and astronomy. Not so with electricity and magnetism. Various curious properties uncovered by the ancient Greeks seemed to offer no solutions to the practical problems of the day, and instead became fodder for myth, legend and misconception.

    The ancients had some familiarity with both magnetism and static electricity, although in those early days no distinction was made between the two. Awareness of magnetism came first. The Greek historian Pliny the Elder (23–79 CE) wrote that when the Greek shepherd Magnes chanced upon a lodestone, the iron tip of his staff...

  12. Heat and Wave Phenomena
    (pp. 143-178)

    You’re in your living room on a winter night holding a cup of hot chocolate and enjoying a crackling fire. Soft strains of music mingle with the slow tick-tock of an old grandfather clock. The heat from the cup of chocolate and the crackling fire may seem to have little connection with the music’s sound waves or the clock’s stately pendulum motion, but viewed through the prism of mathematics the two phenomena reveal a brotherhood through the equations governing them.

    Early forays into heat flow and wave phenomena progressed along mostly separate paths. Here’s a brief look.

    Heat. It was...

  13. The Leaking Tank
    (pp. 179-196)

    Take an ordinary empty tin can with no top, punch a small hole in the bottom and stuff a bit of sponge into the hole. Now fill the can with water and let’er drain! In this setup, the rate of draining is nearly proportional to the water height, and proportionality implies that the water height decreases to zero exponentially. The ubiquitous exponential describes a wide range of real world phenomena, and we’ve already met one example in the last chapter, the cooling cup of coffee. There is an analogy between heat and liquid: the cup of coffee cools like liquid...

  14. Linear Algebra
    (pp. 197-216)

    The equation$y = ax$of a line through the origin is simple indeed, but generalizing this to many linear equations in many variables? To those unfamiliar with linear algebra, that might sound like a recipe for a mess! However, linear algebra’s powerful ideas of vectors and matrices allow such a generalization to be written in essentially the same way as$y = ax$. By virtue of compact notation, linear algebra lets us back away and look at a large forest, and a possibly bewildering number of leaves can be much more easily understood and managed.

    We begin by briefly shaking hands with a...

  15. What Do You Think? Answers
    (pp. 217-226)
  16. Geometry Answers
    (pp. 227-244)
  17. Numbers Answers
    (pp. 245-252)
  18. Astronomy Answers
    (pp. 253-266)
  19. Archimedes’ Principle Answers
    (pp. 267-272)
  20. Probability Answers
    (pp. 273-284)
  21. Mechanics Answers
    (pp. 285-294)
  22. Electricity Answers
    (pp. 295-300)
  23. Heat and Wave Phenomena Answers
    (pp. 301-316)
  24. The Leaking Tank Answers
    (pp. 317-322)
  25. Linear Algebra Answers
    (pp. 323-338)
  26. Glossary
    (pp. 339-366)
  27. References
    (pp. 367-368)
  28. Problem Index
    (pp. 369-372)
  29. Subject Index
    (pp. 373-374)
  30. About the Author
    (pp. 375-375)