Several months ago, my desire to learn physics crystallized from a vague goal to a determination for action. It was a confluence of multiple streams of influence. A lifetime of pop science and sci-fi merged with the can-do attitude from a recently completed math degree. A difficult job market provided the opportunity and incentive to learn a more practical set of skills. Inspired by the book Ultralearning, I decided my ultimate goal would be to learn the equivalent amount of material as is in an undergraduate degree in physics. Naturally, this is a huge undertaking that may require several years to complete fully. So to make it more manageable I will be taking the material one class at a time. With my mind made up I began my search for how to best study physics on my own.
Eventually I stumbled across the excellent guide written by Susan Rigetti: “So You Want to Learn Physics“, which spells out which topics to study from which textbooks and gives a general order for doing so. This guide was exactly what I was looking for so I committed myself to following it in my self study (which is why the title of this post mirrors the title of her guide). I will reference her guide simply as ‘the guide’ throughout this post and many of the following ones. In addition to the guide, I looked at the degree requirements for a number of top tier universities to get a general sense of what is expected of a physics degree. Most of them have the same general layout, which I will discuss in more detail in another post. All the programs I looked at begin with a multi-course sequence covering introductions to mechanics, electromagnetism, and thermodynamics. This lines up with the guide.
It has now been several months and I have completed the material in the first ‘unit’ of the guide: Introduction to Mechanics. This post will be a reflection on my experience along with lessons learned and advice for anyone who wishes to undertake a similar journey. The most basic stats of my experience are these: It took me approximately ninety hours of study over five weeks to complete the material. I took about one hundred pages of notes, including solutions to example problems. The amount of time it would take you to complete the material might vary due to prior experience and mathematical maturity.
If you are wondering: Is it worth it to study physics? The answer is yes, absolutely. Even just from the knowledge I have gained so far I feel more empowered to understand the world around me. A benefit that I did not expect was that seeing how some math topics were applied in physics gave me a better appreciation for why those topics were created and defined the way they were. Physicists and mathematicians approach problems in a different way, with a different mindset. Having both mindsets in my tool belt (or at least being able to switch back and forth) gives me greater confidence to approach and solve problems I encounter in the world.
Prerequisites
As I began to study physics I was just finishing up my undergraduate degree in mathematics, so I did not have to worry about mathematical prerequisites. That said, the main prerequisite you need for the introduction to mechanics material is a solid grasp of basic algebra and some calculus. In my view, you could get through most of the material without calculus but you would be missing a lot of the intuition for why things work the way they do. As far as physics prerequisites go, I had not had a complete course on physics since high school so I do not think you need to know any physics going in.
The main math topics you need to know are trigonometry, algebraic manipulations (you should be able to rearrange expressions to isolate a particular variable), and basic calculus. By basic calculus I mean knowing what a derivative is and how to find one for polynomials and trig functions, and how to solve fairly simple integrals. If you have passed a calculus course or can reference a calculus textbook (any calculus textbook should work, they are all very similar) without much difficulty, you should be fine. There are some brief discussions involving line integrals and cross products (particularly chapters 6, 7, and 10). You can easily find the formula for cross products or you can sidestep them with some trigonometry in most cases. I found that I did use techniques from linear algebra, differential equations, and numerical methods in some places, but that knowledge was by no means required.
Content
The book that is recommended in the guide is “University Physics with Modern Physics” by Young and Freedman. I picked up the 14th edition used for about fifty dollars and it was in great shape. Honestly, I love the book. The book has excellent explanations, colorful illustrations, and worked out example problems. At the end of each chapter there are section exercises and general problems. After spending so much time with advanced mathematics textbooks which are, to put it mildly, terse, it was nice to have a bit of hand holding. The mechanics material makes up the first 14 chapters of the book. Other sections in the book include thermodynamics, electromagnetism, optics, and modern physics. The guide returns to this book for both electrostatics and modern physics. I will also likely work through the sections of this book that aren’t recommended by the guide as well, just because I love the book that much.
I will now go through the fourteen chapters and describe what they are about, my thoughts on the chapter, and my recommendations on how to make studying easier. The first ten chapters represent the core of the material which absolutely must be worked through completely and in order. The last four chapters are more like application chapters. The final chapters were some of my favorites because the problems were interesting and demonstrated how much I had progressed since the beginning of the book.
Overall, mechanics is about how objects move and interact with each other through forces. You will learn how to think about situations in terms of forces, energy, and momentum and gain insight from these perspectives (either separately or in combination). You will see how objects move, rotate, and respond to stress, and apply these concepts to springs, rockets, planets, and other objects. This material sets you up with the knowledge needed for further study in physics, engineering, or science. The large amount of bio and medical examples show how the material can be applied to biology and medicine.
Chapter 1: The first chapter is the foundation of everything. This chapter covers units, measurement conventions, and vectors. In many ways, this chapter forms an introduction to physics and the book itself. You may be tempted to skip this chapter, especially if you have seen these topics before, but you should take your time to make sure that you understand what is presented here. In particular, you should become a master of the rules for significant figures in measurements.
Chapter 2: The second chapter covers motion in a straight line including concepts like position, velocity, and acceleration. The most common examples used for this sort of motion is throwing (or launching) and object straight up in the air and letting it fall back down again. There is a good chance that you will remember some or all of this content from middle school or high school physics, or a calculus class. The key takeaway from this chapter is how the equations for position, velocity, and acceleration relate to each other. There are a couple of equations derived here that are worth memorizing or writing down in a convenient place because you will be referencing them fairly frequently.
Chapter 3: The next chapter expands to cover motion in two and three dimensions. This includes more realistic projectile motion, motion in a circle, and relative velocities. The use of vectors and trigonometry come into full focus in this chapter. You will want to spend as much time as you need working through these problems until you have them down. I do not believe it is necessary to do every single exercise in every chapter (more on this below) but this is a chapter where you should probably do every single exercise.
Chapter 4: Next up is an introduction to Newton’s Laws of Motion. The chapter begins by defining the concept of force and how forces combine. Newton’s Laws then tell us how forces affect the motion of the objects they act on. The book emphasizes drawing free-body diagrams (which show all the forces acting on an object) when beginning each problem and I second that view. A mistake I made several times was thinking that I did not need to draw a free-body diagram and could just keep it all straight in my head. This almost always led to me getting an angle mixed up or a plus switched with a minus. Many of the problems in the first three chapters were one (or sometimes two) step plug-n-chug problems. These problems involve multiple steps and reward spending some planning time setting up each problem (draw the free-body diagrams!).
Chapter 5: Chapter 5 is basically chapter 4 part two. The first two sections of the chapter are all about applying Newton’s Laws to systems (collections of objects) that are either in equilibrium or in motion. A simple version of friction (which had previously been ignored) is introduced. The chapter ends by going over circular motion involving forces and giving a brief summary of the forces of nature. More realistic situations involving weight, friction, tension, and other forces in two or three dimensions can now be analyzed. This chapter was around the time that I started to really feel empowered by what I was learning.
Chapter 6: This chapter is about work and kinetic energy. Work is a force done over a distance, and kinetic energy can be thought of as how much work it would take to stop a moving object. The big result of the chapter is the work-energy theorem, which directly relates work done to a change in the kinetic energy of an object. Although being based off of forces, thinking about the energy in a system can answer questions about it without needing to know the exact forces involved. The concept of work took a little while to click in my mind and I had to review this chapter a few times. This chapters talks a bit about springs, particularly the work that goes into stretching a spring. This is important to pay attention to because it will reappear in chapters 7 and 11.
Chapter 7: This was the chapter where I started having some difficulty and had to slow down. Up till this point I could largely rely on my math skills to carry the day, but here I had to start focusing more on the physics concepts themselves. The chapter introduces potential energy. In many ways it builds on the material from the previous chapter to give a complete way of thinking about systems in terms of their energy, rather than just the forces acting on the components of the system. In a closed system, energy is conserved so keeping track of the kinetic and potential energy system can tell you how the system will behave. The most immediate example of potential energy is gravitational potential energy, but potential energy can come in other forms as well. One example is the elastic potential energy that is stored in a stretched or compressed spring. The concept of energy diagrams are introduced, which are good ways to gain insight into a system with some simple graphing.
Chapter 8: This chapter covers momentum, impulse, and collisions. Momentum and impulse are conceptually similar to kinetic energy and work, respectively. However, while work was force done over a distance, impulse is force done over a time. The chapter presents an impulse-momentum theorem which is similar to the work-energy theorem of chapter 6. Like energy, momentum in a closed system is conserved so keeping track of it provides insight into how the system will behave. This is best exemplified in the chapter’s coverage of collisions. By keeping track of the momentum of the objects colliding, we can model the collision without having to make any reference to the forces explicitly.
Chapter 9: In this chapter, the focus is on the rotation of rigid bodies. Familiar topics such as position, velocity, and acceleration are redefined in an angular setting. Most of the material here should feel familiar from chapters 2 and 3 so I recommend taking a brief moment to review those chapters. You will also learn how to translate between the angular motion and the linear motion of a point on the object. The chapter ends with discussion of the center of mass and moment of inertia calculations.
Chapter 10: Almost a continuation of the previous chapter, chapter 10 is about the dynamics of rotational motion. Basically, this chapter is about how forces affect the rotational motion of an object. This is also the first in the book where the location on an object that force is applied matters to how the object will move. Up till now, the book has more or less been treating objects as point particles. Changing the location or direction that force is applied to an object changes the torque that is applied. I did have some difficulty with the vector descriptions of angular momentum at first, but the key is to keep doing problems until you get used to it. Combining traditional motion with rotational motion gives a much more realistic model of how objects behave and react to forces.
Chapter 11: This chapter covers equilibrium and elasticity. Equilibrium is essentially about having all of the forces on an object being balanced so that there is no motion, linear or rotational. For an object to be in equilibrium there are two conditions which must be satisfied: the net force on the object must be zero, and the net torque on the object must be zero. This involves a deeper description of the center of mass (which was briefly mentioned in chapter 9). The stress and strain portion of this chapter deals with how objects deform in response to forces. At low force levels, object deformations can be modeled with Hooke’s Law which was first introduced in chapter 6 in regards to the elastic potential energy in springs. There are several types of stress and different materials respond to it differently. If you are interested in construction or engineering you will find this section particularly enjoyable.
Chapter 12: Fluid mechanics is a massive subject, worthy of its own courses, but this chapter offers a glimpse of the basics for ideal fluids. This chapter talks about things like density, pressure, Pascal’s Law (which is applied in hydraulics), buoyancy, and fluid flow. Fluid mechanics important in studying the weather and climate, oceanography, and home plumbing. Many of the problems involve fluid flowing through a system of pipes. Solving these problems involve setting up a set of equations that capture the specifics of the pipe network. My best advice for this is to always draw a picture and not skip any terms when setting up the equations.
Chapter 13: This was my favorite chapter of this set of chapters. Chapter 13 is all about gravitation. Up till now in the book, gravity has mostly been assumed to be uniform and given in terms of an acceleration. For example, objects in free fall near the Earth’s surface accelerate downwards at a rate of about 9.8 meters per second squared. However, the actual force exerted by gravity varies by distance and mass so this chapter is all about the more realistic version of gravity. Much of the chapter is about the way things can orbit each other (such as stars and planets). The chapter presents Kepler’s Laws about the orbits of planets, and ends with a brief discussion of black holes. It made me want to go back and play some Kerbal Space Program (so I did).
While all of the topics covered in this book are important and interesting in their own way, things like black holes and binary, rather than friction and angular momentum, are what inspired me to study physics. Being able to understand these topics more deeply than pop science allows was an excellent payoff for all of my hard work to get to this point. Naturally, this chapter is only a taste of a subject that runs much deeper, but it was a satisfying taste nonetheless.
Chapter 14: The final chapter of this section is about periodic motion. This includes things like pendulums, waves, and simple vibrations. I went through this material quickly, since I have seen most of it before and wanted to finish up. I knew that I will be covering periodic motion and waves in more depth later on so I was not worried about missing out. Even when we covered them in the course of my math degree, I was not a huge fan of pendulums. Hopefully when I get to the unit entirely about waves and vibrations I will learn to love this kind of material.
How I Studied
Before I get into the specifics, I would like to talk about some overarching strategies that helped me immensely. The first difficulty of self teaching is that it requires dedication over long periods of time. When you self teach you must be your own accountability and your own support. To hold myself accountable, I started keeping a self study journal. It is a dedicated notebook where I record, in fifteen minute intervals, when and what I studied each day. The goal of not having any empty entries motivated me to study every day until it became a habit. The journal provides additional benefits such as inspiration from seeing how far I have come and spotting patterns in my study habits. Some days, I was tired or felt stuck on a hard concept. On those days I would often watch a relevant lecture from Walter Lewin’s 8.01 course at MIT. These lectures can be found on YouTube and high quality. Most of the lectures include nice demonstrations of the topics at hand. These were fun and relaxing while still being productive. If you feel tired or get stuck, I recommend something similar. Do not try to take notes from the lecture (that’s what the book is for), just enjoy and try to understand. You may feel tempted to skip certain chapters or to do them out of order. Avoid this as much as possible.
The best method I found for studying from this book involved a cycle of skimming, note taking, solving example problems, and finally doing exercises. When I started a new chapter I would take a few minutes to look over the whole chapter. I wanted to know what topics were being covered, what the different sections were about, and what sort of problems I would be solving at the end. I would pick out a few problems and try to imagine how I might solve them, just to get my mind working on it in the background. The next step was to proceed with a careful reading of the chapter while taking notes. How detailed to make your notes is up to you, but I try to get only the important ideas and terms (which are often in bold font in the text). Throughout each chapter are a number of example problems which provide detailed explanations and step by step solutions. Instead of reading these right away, I attempt the solution on my own first. If I get stuck, or just end up with the wrong answer, it is easy to compare my work to theirs to figure out exactly where I went wrong. Even if my solution was correct, the extra insight provided by the explanations was valuable.
Finally, at the end of each chapter there are the exercises and problems. The exercises are organized by section and stay focused on the topic of that section. If the exercise is listed under the section on friction you know it is going to involving calculating some friction without many other bells and whistles. The problems are less structured, often longer, and can involve topics from multiple sections. In an ideal world, one should do all of the exercises and the problems (or at least the odd-numbered ones since they have answers in the back of the book). In the real world, how many questions you answer will depend on your priorities. For most chapters I did not do every single exercise and problem (chapter three being the main exception). My goal was to be confident in my understanding of the material and know I could deal correctly with the concepts when they came up again.
If I felt comfortable in a section I would start with the hardest (typically last) exercise for that section. If I nailed it then that was a good sign I could do fewer of that section’s exercises. Otherwise I would start at the beginning and do the odds in each section until I felt satisfied. I certainly did a larger proportion of the exercises as I got to the later chapters. The order for the problems is less important. My advice is to pick problems that interest you. Many of the problems deal with interesting applications of the topics in the chapter, so I was learning new things even while reading the problems. At the very end of the chapters are passage problems. These are multipart, multiple choice questions designed to mimic problems on standardized exams. I used these problems as a final exam of sorts. When I finished all fourteen chapters I went back and did all of the passage problems. Checking the answers in the back of the book allowed me to grade myself in an unbiased way. Luckily for me, I passed.
Ultimately, the more exercises and problems you solve the better, but you are the judge of how to best spend your time.
Final Thoughts
Choosing to study physics on my own has been a rewarding and empowering experience so far. When I first started, the path ahead seemed impossibly long. I returned to the guide and to other sources of inspiration often to renew my determination. From day to day, the progress seemed slow but page by page I moved forward. In hindsight, the experience does not seem so long. The hours I spent were roughly equivalent to what one might spend on an actual course but since I could move at my own pace it did not take an entire semester to move through the material.
Completing a goal which I set for myself is satisfying, and I have no doubt that this only the beginning of my journey into physics. Whether or not I will ultimately complete the guide and the equivalent of an undergraduate degree remains to be seen. I hope so, but I will take it one course at a time. As I write this, I am most of the way through the electrostatics material from the next unit of the guide. When I finish that I will write a similar post, and edit in a link to it here.
Happy studying, and thanks for reading!