Why Cats Land on Their Feet

And 76 Other Physical Paradoxes and Puzzles

Mark Levi

• Publisher: Princeton University Press
• Year: 2012
• ISBN:
  9781400841721
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

How to use physical reasoning to solve surprising paradoxes Ever wonder why cats land on their feet? Or what holds a spinning top upright? Or whether it is possible to feel the Earth's rotation in an airplane? Why Cats Land on Their Feet is a compendium of paradoxes and puzzles that readers can solve using their own physical intuition. And the surprising answers to virtually all of these astonishing paradoxes can be arrived at with no formal knowledge of physics.Mark Levi introduces each physical problem, sometimes gives a hint or two, and then fully explains the solution. Here readers can test their critical-thinking skills against a whole assortment of puzzles and paradoxes involving floating and diving, sailing and gliding, gymnastics, bike riding, outer space, throwing a ball from a moving car, centrifugal force, gyroscopic motion, and, of course, falling cats.Want to figure out how to open a wine bottle with a book? Or how to compute the square root of a number using a tennis shoe and a watch? Why Cats Land on Their Feet shows you how, and all that's required is a familiarity with basic high-school mathematics. This lively collection also features an appendix that explains all physical concepts used in the book, from Newton's laws to the fundamental theorem of calculus.

1

FUN WITH PHYSICAL PARADOXES, PUZZLES, AND PROBLEMS

1.1 Introduction

A good physical paradox is (1) a surprise, (2) a puzzle, and (3) a lesson, rolled into one fun package. A paradox often involves a very convincing argument leading to a wrong conclusion that seems right, or to a right conclusion that seems wrong or surprising. The challenge to find the mistake—or explain the surprise—may be hard to resist. A joke heard back in the Cold War years claimed that the West could impede Soviet military R&D efforts by scattering leaflets containing puzzlers and brainteasers over the secret Siberian weapons research facilities. Times have changed, and these same brainteasers now are used in hiring interviews. As a Soviet propagandist would have said: either way, they are a capitalist tool.

Resolving a paradox is not only fun; it also trains intuition, logic, and critical thinking. One becomes a better lie detector by resolving paradoxes. A good paradox also teaches caution and humility by showing us how easy it is to go wrong even in relatively simple matters of elementary physics. It is liberating to know that some very smart people have made mistakes in seemingly simple matters of basic physics. Other fields—such as astronomy, biology, medicine, economics, climate, politics, and media—deal with more complicated objects than physics,¹ offering even more room for mistakes there. In addition, some ‘mistakes’ can be beneficial, at least temporarily.

My main reason in writing this book is to share the fun of imagining how things work. These paradoxes also teach the gist of some physics without the pain of mathematics.²

The puzzles in this book deal with physics—a subject that walks on two legs, one being mathematics, and the other, physical intuition. Unfortunately, in school the subject is often presented with a severe limp.

A musical analogy. If music were taught the way physics often is taught, we would learn the notes but not the melodies they produce. For too many students of physics, the subject is reduced to a collection of formulas that must be matched to a problem at hand. Not surprisingly, many intelligent students are turned off.

Intuition should come first. Exercise of physical intuition is one practical benefit of this book’s puzzles. All too many physics courses give short shrift to intuition, emphasizing instead a search for the formula that fits the situation. Examples in this book go in the opposite direction: I tried for a minimum of formulas and a maximum of intuition. The discussion of the spinning top is an example, where I give a formula-free explanation of why the spinning top stays upright. It takes quite a few years of study in mathematics and physics to learn to write differential equations for the motion of a spinning top and to see how to deduce stability from these equations. And at the end of this long study few students end up with an intuitive understanding of why a spinning top stays up. The most powerful tool—our physical intuition—ends up unused.

1.2 Background

Much (but not all) of this book should be accessible to readers without formal background in physics. All physical concepts used are explained in the appendix. Mathematics in this book does not go beyond algebra, with a couple of exceptions where calculus is used. Even there, the reader who is willing to take a little math on trust should not be snagged by these references.³

Attraction to anything surprising is a basic instinct in most living creatures, or, at least, most mammals. By driving us to explore, the instinct helps us survive—with some exceptions, such as Darwin Prize winners or the heroes of Jackass. The same instinct that drove Einstein to his great discoveries also drives a curious child to see what’s inside a mechanical clock. It even drives puppies and cubs to explore. In some people this instinct is so strong it can survive the educational system.

1.3 Sources

This book grew out of a collection of puzzles I started long ago on my father’s advice, after I showed him one that occurred to me after a high school class on the capillary effect (page 128). Although I invented some of this book’s puzzles,⁴ it is most likely that others thought of them or of something equivalent before I was born. When I know the author or the origin of a puzzle, I make a reference.

Literature. Fortunately, much of the essence of basic physics can be understood, and enjoyed, without (m)any formulas, as some excellent popular books demonstrate. Among these are Walker’s The Flying Circus of Physics, Epstein’s Thinking Physics, Jargodzki and Potter’s Mad about Physics, and Perelman’s classic Physics for Entertainment. Unfortunately, Makovetsky’s delightful book Smotri v koren’ (a loose translation: “Seek the essence”), which sold over a million copies in the former Soviet Union, does not seem to have been translated into English. Minnaert’s The Nature of Light and Color in the Open Air, dedicated to optical phenomena in nature, will never age and will give pleasure to any curious individual lucky enough to open it.

 

¹ This is not a statement on the relative difficulty of various sciences. I am simply referring to the fact that a physicist deals with much simpler objects (e.g., crystals) than a biologist (e.g., a cell).

² I refer to “pain" with tongue in cheek—mathematics is of course indispensable and beautiful to me, at least, since it’s my job.

³ I am referring here to the sling problem on page 93 where the rock reaches infinite speed after one second.

⁴ For example, 2.1, 2.3, 2.4, 3.1, 3.2, 3.5, 3.6, 4.1, 4.2, 4.4–4.6, 5.3–5.8, 6.6, 6.7, 6.10–6.12, 8.2, 8.5, 8.6, 9.4, 11.1, 12.3, 13.2, 14.6, 14.8.

2

OUTER SPACE PARADOXES

2.1 A Helium Balloon in a Space Shuttle

Problem. Two astronauts, Al and Bob, are strapped to the opposite ends in a space capsule, as in Figure 2.1. Al is holding a large helium-filled balloon, and everything is at rest. Now Al pushes the balloon, which begins to drift toward Bob. Which way will the capsule drift, as seen by an observer hovering in space outside the capsule? Since the astronauts are strapped to the walls, let’s consider them part of the capsule.

Figure 2.1. Which way does the capsule move after Al pushes the balloon?

Figure 2.2. Motion as viewed from the capsule’s (and Al’s) reference frame.

A reasonable guess. When Al pushes right, the balloon pushes him back, according to Newton’s “action equals reaction” third law. And since the balloon pushes Al left, he, and the shuttle, will drift left. Is this correct?

Answer. Actually, no: the capsule will drift right as well!

An explanation via center of mass. The center of mass of the entire system (the capsule and the contents) is fixed, because there are no external forces (all the concepts in this sentence are explained in the appendix, (pages 169–73). Now the motion inside the capsule, from Al’s point of view, is sketched in Figure 2.2. The balloon has a lot less mass than the air it displaces, and so, from Al’s point of view, the center of mass moves left. But the center of mass of the whole system is fixed in space, since there are no external forces. Therefore, Al himself, and the capsule, move right from the external observer’s point of view.

Our mistake has been in paying too much attention to the balloon and not enough attention to the more massive air that moves left to replace the moving balloon.

An equivalent explanation via linear momentum. As explained in the appendix (pages 169–73), the fact that the center of mass stays put is equivalent to saying that the linear momentum remains zero. Now from Al’s viewpoint, the displaced air moves left. This signals that Al himself (and the shuttle) are moving right, to cancel the leftward motion of the air and to keep the linear momentum at zero.

Figure 2.3. Water stays in place; the nearly massless shell moves right.

The intuitive feel of all this becomes particularly clear by taking the mass ratio to the extreme, as in Figure 2.3, where helium/air is replaced by helium/water. S...

Table of contents

• Cover
• Title
• Copyright
• Contents
• Acknowledgments
• 1 Fun with Physical Paradoxes, Puzzles, and Problems
• 2 Outer Space Paradoxes
• 3 Paradoxes with Spinning Water
• 4 Floating and Diving Paradoxes
• 5 Flows and Jets
• 6 Moving Experiences: Bikes, Gymnastics, Rockets
• 7 Paradoxes with the Coriolis Force
• 8 Centrifugal Paradoxes
• 9 Gyroscopic Paradoxes
• 10 Some Hot Stuff and Cool Things
• 11 Two Perpetual Motion Machines
• 12 Sailing and Gliding
• 13 The Flipping Cat and the Spinning Earth
• 14 Miscellaneous
• Appendix
• Bibliography
• Index