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 CONTENTS Introduction Fermi's Piano Tuner Problem How Old is Old? If the Terrestrial Poles were to Melt... Sunlight Exerts Pressure Falling Eastward What if an Asteroid Hit the Earth Using a Jeep to Estimate the Energy in Gasoline How do Police Radars really work? How "Fast" is the Speed of Light? How Long is a Light Year? How Big is a Trillion? "Seeing" the Earth, Moon, and Sun to Scale Of Stars and Drops of Water If I Were to Build a Model of the Cosmos... A Number Trick Designing a High Altitude Balloon Pressure in the Vicinity of a Lunar Astronaut Space Suit due to Outgassing of Coolant Water Calendar Calculations Telling Time by the Stars - Sidereal Time Fields, an Heuristic Approach The Irrationality of The Irrationality of The Number (i)i Estimating the Temperature of a Flat Plate in Low Earth Orbit Proving that (p)1/n is Irrational when p is a Prime and n>1 The Transcendentality of Ideal Gases under Constant Volume, Constant Pressure, Constant Temperature and Adiabatic Conditions Maxwell's Equations: The Vector and Scalar Potentials A Possible Scalar Term Describing Energy Density in the Gravitational Field A Proposed Relativistic, Thermodynamic Four-Vector Motivational Argument for the Expression-eix=cosx+isinx Another Motivational Argument for the Expression-eix=cosx+isinx Calculating the Energy from Sunlight over a 12 hour period Calculating the Energy from Sunlight over actual full day Perfect Numbers-A Case Study Gravitation Inside a Uniform Hollow Sphere Further note on Gravitation Inside a Uniform Hollow Sphere Pythagorean Triples Black Holes and Point Set Topology Additional Notes on Black Holes and Point Set Topology Field Equations and Equations of Motion (General Relativity) The observer in modern physics A Note on the Centrifugal and Coriolis Accelerations as Pseudo Accelerations - PDF File On Expansion of the Universe - PDF File

## Introduction

Many of the articles on this Web site are versions of the Fermi Problem described in the first section. Others are essays - some short, some long. Some are merely attempts to come to terms with basic concepts, such as the 'size' of the speed of light or the number 'one trillion'. Others discuss more advanced concepts. The last few essays involve college-level physics, and might be difficult for those who have not taken at least some introductory courses. The essays discussing the gravitational field energy density and the thermodynamic four-vector are speculative and invite comments from you, the reader.

The energy density article was written to fill a gap, which I noted in books on Special Relativity. Of the three classical problems of General Relativity, two (the deflection of starlight and the gravitational red shift) are routinely presented as exercises with a discussion of similarities and differences with General Relativity; the third (the rotation of perihelion) seems never to be touched at this level. This article is an attempt to rectify this situation. The four-vector article arises from the observation that components of a four-vector satisfy a continuity equation. Einstein used this idea in his early writing on Special Relativity when introducing the current density four-vector. I have raised the question as to whether similar logic might apply to thermodynamics when the heat conduction equation and the thermal diffusion equation are combined to form a continuity equation.

The earlier pieces are nowhere near so involved, and require only a little number skill and, possibly, some high school algebra. Enrico Fermi, the theoretical physicist of Manhattan Project fame, knew only too well that physicists are often confronted by situations in which they are forced to reason from minimal information. He, therefore, taught his students how to think about complicated sounding problems by using everyday knowledge. The 'Piano Tuner' is typical: "If 3,000,000 people live in Chicago, then how many of them are piano tuners?"

There are many cases in science, and even in everyday life, when we encounter seemingly insolvable problems such as this. So the problems presented here provide an opportunity to practice with Fermi's approach. Some of the problems are of my own invention. Others came from students or people I met when conducting public lectures. Some are simple, others, complex. Each problem has, of course, a rigorous solution, although the solutions presented here make no pretense at rigor! Consider them 'back-of-the-envelope' calculations - estimates, ballpark solutions. In so doing, you will gain valuable insight into a technique much used by professionals.

And most of all, remember to have fun!

- Joe Kolecki