Preliminaries 31

OK, so now you are ready to learn physics. Your math skills are buffed and honed, you’ve

practiced the method of three passes, you understand that success depends on your full engagement

and a certain amount of hard work. In case you missed the previous section (or are unused to

actually reading a math-y textbook instead of minimally skimming it to extract just enough “stuff”

to be able to do the homework) I usually review its content on the first day of class at the same

time I review the syllabus and set down the class rules and grading scheme that I will use.

It’s time to embark upon the actual week by week, day by day progress through the course

material. For maximal ease of use for you the student and (one hopes) your instructor whether or

not that instructor is me, the course is designed to cover one chapter per week-equivalent, whether or

not the chapter is broken up into a day and a half of lecture (summer school), an hour a day (MWF),

or an hour and a half a day (TTh) in a semester based scheme. To emphasize this preferred rhythm,

each chapter will be referred to by the week it would normally be covered in my own semester-long

course.

A week’s work in all cases covers just about exactly one “topic” in the course. A very few are

spread out over two weeks; one or two compress two related topics into one week, but in all cases

the homework is assigned on a weekly rhythm to give you ample opportunity to use the method of

three passes described in the first part of the book, culminating in an expected 2-3 hour recitation

where you should go over the assigned homework in a group of three to six students, with a mentor

handy to help you where you get stuck, with a goal of getting all of the homework perfectly correct

by the end of recitation.

That is, at the end of a week plus its recitation, you should be able to do all of the week’s

homework, perfectly, and without looking or outside help. You will usually need all three passes, the

last one working in a group, plus the mentored recitation to achieve this degree of competence! But

without it, surely the entire process is a waste of time. Just finishing the homework is not enough,

the whole point of the homework is to help you learn the material and it is the latter that is the real

goal of the activity not the mere completion of a task.

However, if you do this – attempt to really master the material – you are almost certain to do

well on a quiz that terminates the recitation period, and you will be very likely to retain the material

and not have to “cram” it in again for the hour exams and/or final exam later in the course. Once

you achieve understanding and reinforce it with a fair bit of repetition and practice, most students

will naturally transform this experience into remarkably deep and permanent learning.

Note well that each week is organized for maximal ease of learning with the week/chapter review

first. Try to always look at this review before lecture even if you skip reading the chapter itself until

later, when you start your homework. Skimming the whole week/chapter guided by this summary

before lecture is, of course, better still. It is a “first pass” that can often make lecture much easier

to follow and help free you from the tyranny of note-taking as you only need to note differences in

the presentation from this text and perhaps the answers to questions that helped you understand

something during the discussion. Then read or skim it again right before each homework pass.

Preliminaries 27

Homework for Week 0

Problem 1.

Skim read this entire section (Week 0: How to Learn Physics), then read it like a novel, front to

back. Think about the connection between engagement and learning and how important it is to try

to have fun in a physics course. Write a short essay (say, three paragraphs) describing at least one

time in the past where you were extremely engaged in a course you were taking, had lots of fun in

the class, and had a really great learning experience.

Problem 2.

Skim-read the entire content of Mathematics for Introductory Physics (linked above). Identify

things that it covers that you don’t remember or don’t understand. Pick one and learn it.

Problem 3.

Apply the Method of Three Passes to this homework assignment. You can either write three short

essays or revise your one essay three times, trying to improve it and enhance it each time for the

first problem, and review both the original topic and any additional topics you don’t remember in

the math review problem. On the last pass, write a short (two paragraph) essay on whether or not

you found multiple passes to be effective in helping you remember the content.

Note well: You may well have found the content boring on the third pass because it was so

familiar to you, but that’s not a bad thing. If you learn physics so thoroughly that its laws become

boring, not because they confuse you and you’d rather play World of Warcraft but because you know

them so well that reviewing them isn’t adding anything to your understanding, well damn you’ll do

well on the exams testing the concept, won’t you?

24 Preliminaries

told you that you can go ahead and take physics having gotten C’s in introductory calculus, perhaps

in a remedial course that you took because you had such a hard time with precalc or because you

failed straight up calculus when you took it.

They lied.

Sorry to be blunt, but that’s the simple truth. Here’s a list of a few of the kinds of things you’ll

have to be able to do during the next two semesters of physics. Don’t worry just yet about what

they mean – that is part of what you will learn along the way. The question is, can you (perhaps

with a short review of things you’ve learned and knew at one time but have not forgotten) evaluate

these mathematical expressions or solve for the algebraic unknowns? You don’t necessarily have to

be able to do all of these things right this instant, but you should at the very least recognize most

of them and be able to do them with just a very short review:

• What are the two values of α that solve:

α

2 +

R

L

α +

1

LC = 0?

• What is:

Q(r) = ρ0

R

4π

Z r

0

r

′3

dr′

?

• What is:

d cos(ωt + δ)

dt ?

θ

x

y

?

?

A

• What are the x and y components of a vector of length A that makes an angle of θ with the

positive x axis (proceeding, as usual, counterclockwise for positive θ)?

• What is the sum of the two vectors A~ = Axxˆ + Ayyˆ and B~ = Bxyˆ + Byyˆ?

• What is the inner/dot product of the two vectors A~ = Axxˆ + Ayyˆ and B~ = Bxyˆ + Byyˆ?

• What is the cross product of the two vectors ~r = rxxˆ and F~ = Fyyˆ (magnitude and direction)?

If all of these items are unfamiliar – you don’t remember the quadratic formula (needed to solve

the first one), can’t integrate x

ndx (needed to solve the second one), don’t recall how to differentiate

a sine or cosine function, don’t recall your basic trigonometry so that you can’t find the components

of a vector from its length and angle or vice versa, and don’t recall what the dot or cross product

of two vectors are, then you are going to have to add to the burden of learning physics per se the

burden of learning, or re-learning, all of the basic mathematics that would have permitted you to

answer these questions easily.

Here are the answers, see if this jogs your memory:

Preliminaries 23

This is an extremely powerful prescription for deeply learning nearly anything. Here is the moti-

vation. Memory is formed by repetition, and this obviously contains a lot of that. Permanent (long

term) memory is actually formed in your sleep, and studies have shown that whatever you study right

before sleep is most likely to be retained. Physics is actually a “whole brain” subject – it requires

a synthesis of both right brain visualization and conceptualization and left brain verbal/analytical

processing – both geometry and algebra, if you like, and you’ll often find that problems that stumped

you the night before just solve themselves “like magic” on the second or third pass if you work hard

on them for a short, intense, session and then sleep on it. This is your right (nonverbal) brain

participating as it develops intuition to guide your left brain algebraic engine.

Other suggestions to improve learning include working in a study group for that third pass (the

first one or two are best done alone to “prepare” for the third pass). Teaching is one of the best

ways to learn, and by working in a group you’ll have opportunities to both teach and learn more

deeply than you would otherwise as you have to articulate your solutions.

Make the learning fun – the right brain is the key to forming long term memory and it is the seat

of your emotions. If you are happy studying and make it a positive experience, you will increase

retention, it is that simple. Order pizza, play music, make it a “physics homework party night”.

Use your whole brain on the problems – draw lots of pictures and figures (right brain) to go with

the algebra (left brain). Listen to quiet music (right brain) while thinking through the sequences

of events in the problem (left brain). Build little “demos” of problems where possible – even using

your hands in this way helps strengthen memory.

Avoid memorization. You will learn physics far better if you learn to solve problems and un-

derstand the concepts rather than attempt to memorize the umpty-zillion formulas, factoids, and

specific problems or examples covered at one time or another in the class. That isn’t to say that you

shouldn’t learn the important formulas, Laws of Nature, and all of that – it’s just that the learning

should generally not consist of putting them on a big sheet of paper all jumbled together and then

trying to memorize them as abstract collections of symbols out of context.

Be sure to review the problems one last time when you get your graded homework back. Learn

from your mistakes or you will, as they say, be doomed to repeat them.

If you follow this prescription, you will have seen every assigned homework problem a minimum

of five or six times – three original passes, recitation itself, a final write up pass after recitation, and

a review pass when you get it back. At least three of these should occur after you have solved all of

the problems correctly, since recitation is devoted to ensuring this. When the time comes to study

for exams, it should really be (for once) a review process, not a cram. Every problem will be like

an old friend, and a very brief review will form a seventh pass or eighth pass through the assigned

homework.

With this methodology (enhanced as required by the physics resource rooms, tutors, and help

from your instructors) there is no reason for you do poorly in the course and every reason to expect

that you will do well, perhaps very well indeed! And you’ll still be spending only the 3 to 6 hours

per week on homework that is expected of you in any college course of this level of difficulty!

This ends our discussion of course preliminaries (for nearly any serious course you might take,

not just physics courses) and it is time to get on with the actual material for this course.

Mathematics

Physics, as was noted in the preface, requires a solid knowledge of all mathematics through calculus.

That’s right, the whole nine yards: number theory, algebra, geometry, trigonometry, vectors, differ-

ential calculus, integral calculus, even a smattering of differential equations. Somebody may have