Preliminaries |
23 |
This is an extremely powerful prescription for deeply learning nearly anything. Here is the motivation. 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 understand 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 di culty!
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, di erential calculus, integral calculus, even a smattering of di erential equations. Somebody may have
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: |
|
|
|
|
|
|
α2 + |
R |
α + |
1 |
= 0? |
||
|
|
LC |
||||
|
L |
|
|
|||
• What is: |
|
ρ0 |
|
r |
|
|
Q(r) = |
|
4π Z0 |
r′3dr′? |
|||
R |
||||||
• What is:
d cos(ωt + δ) ? dt
y
A
?
θ
x
?
•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 + Ay yˆ and B = Bxyˆ + By yˆ? |
|
~ |
~ |
• What is the inner/dot product of the two vectors A = Axxˆ + Ay yˆ and B = Bxyˆ + By yˆ?
~
• What is the cross product of the two vectors ~r = rxxˆ and F = Fy yˆ (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 xndx (needed to solve the second one), don’t recall how to di erentiate 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 |
25 |
• Here are the two roots, found with the quadratic formula:
α = |
−L |
± q |
|
|
|
|
|
|
|
|
|
R |
|
R |
2 |
|
1 |
|
||||||
L |
|
2 |
− LC = |
|
|
|
|
|
||||||||||||||||
|
|
R |
|
|
R |
|
|
|
4 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
± |
|
¡2 |
¢ |
|
|
|
|
|
|
−2L ± r 4L − LC |
|||||||||||||
• |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
2 |
|
|
|
|
|
Q(r) = |
R 4π Z0 |
|
r′3dr′ = R |
4π 4 |
¯0 = |
|
R |
|||||||||||||||||
|
|
ρ0 |
r |
|
|
|
|
|
ρ0 |
|
|
r′4 |
r |
ρ0 |
πr4 |
|||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
¯ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
¯ |
|
|
|
|
|
|
|
• |
|
|
d cos(ωt + δ) |
|
|
|
|
|
|
|
¯ |
|
|
|
|
|
|
|
||||||
|
|
= −ω sin(ωt + δ) |
|
|
|
|
|
|
||||||||||||||||
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||||||||
|
|
|
|
|
dt |
|
|
|
|
|
|
|
|
|
|
|||||||||
• |
|
Ax = A cos(θ) |
|
|
|
Ay = A sin(θ) |
|
|
|
|
||||||||||||||
|
|
|
|
|
|
|
|
|
||||||||||||||||
• |
|
~ |
~ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
A + B = (Ax + Bx)xˆ + (Ay + By )yˆ |
|
|
|
|
||||||||||||||||||
• |
|
|
~ |
~ |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|||
|
|
|
|
A · B = AxBx + Ay By |
|
|
|
|
|
|
|
|
||||||||||||
•
~
~r × F = rxxˆ × Fy yˆ = rxFy (xˆ × yˆ) = rxFy zˆ
My strong advice to you, if you are now feeling the cold icy grip of panic because in fact you are signed up for physics using this book and you couldn’t answer any of these questions and don’t even recognize the answers once you see them, is to seek out the course instructor and review your math skills with him or her to see if, in fact, it is advisable for you to take physics at this time or rather should wait and strengthen your math skills first. You can, and will, learn a lot of math while taking physics and that is actually part of the point of taking it! If you are too weak going into it, though, it will cost you some misery and hard work and some of the grade you might have gotten with better preparation ahead of time.
So, what if you could do at least some of these short problems and can remember once learning/knowing the tools, like the Quadratic Formula, that you were supposed to use to solve them? Suppose you are pretty sure that – given a chance and resource to help you out – you can do some review and they’ll all be fresh once again in time to keep up with the physics and still do well in the course? What if you have no choice but to take physics now, and are just going to have to do your best and relearn the math as required along the way? What if you did in fact understand math pretty well once upon a time and are sure it won’t be much of an obstacle, but you really would like a review, a summary, a listing of the things you need to know someplace handy so you can instantly look them up as you struggle with the problems that uses the math it contains? What if you are (or were) really good at math, but want to be able to look at derivations or reread explanations to bring stu you learned right back to your fingertips once again?
Hmmm, that set of questions spans the set of student math abilities from the near-tyro to the near-expert. In my experience, everybody but the most mathematically gifted students can probably benefit from having a math review handy while taking this course. For all of you, then, I provide the following free book online:
Mathematics for Introductory Physics
It is located here:
http://www.phy.duke.edu/ rgb/Class/math for intro physics.php
26 |
Preliminaries |
It is a work in progress, and is quite possibly still somewhat incomplete, but it should help you with a lot of what you are missing or need to review, and if you let me know what you are missing that you didn’t find there, I can work to add it!
I would strongly advise all students of introductory physics (any semester) to visit this site right now and bookmark it or download the PDF, and to visit the site from time to time to see if I’ve posted an update. It is on my back burner, so to speak, until I finish the actual physics texts themselves that I’m working on, but I will still add things to them as motivated by my own needs teaching courses using this series of books.
Summary
That’s enough preliminary stu . At this point, if you’ve read all of this “week”’s material and vowed to adopt the method of three passes in all of your homework e orts, if you’ve bookmarked the math help or downloaded it to your personal ebook viewer or computer, if you’ve realized that your brain is actually something that you can help and enhance in various ways as you try to learn things, then my purpose is well-served and you are as well-prepared as you can be to tackle physics.