Introduction to Mechanics for Engineersby Andy Ruina & Rudra PratapPreface 0 What is mechanics?

To the student

A brief study guide. Students who understand and follow the advice in this section tend to learn more and get better grades.

Nature’s rules are so strict that, to the extent that you know the rules, you can make reliable predictions about how Nature, the set of all things, behaves. In particular, most objects of concern to engineers obediently follow a subset of Nature’s rules called the laws of Newtonian mechanics^†^†margin: Also called ‘classical’ or ‘Newton-Euler’ mechanics . So, if you learn the laws of mechanics, as this book should help you to do, you will be able to make quantitative predictions about how things stand, move, and fall. And you will gain intuition about the mechanics part of Nature’s rules.

How to use this book

Here is some general guidance.

Check your own understanding

Most likely you want a decent grade by successfully getting through the homework assignments and exams. You will naturally get help by looking at examples and samples in the text or lecture notes, by looking up formulas in the front and back covers of this book, and by asking questions of friends, teaching assistants and professors. What good are books, notes, classmates or teachers if they don’t help you do the homework? All the examples and sample problems in this book, for example, are just for this purpose.

But watch out. Too much use of help from books, notes and people can lead to self deception

^†^†margin: “Exams are harder than homework.” Some struggling students say “I can do the homework problems. I just can’t do the exams. The exams are harder and trickier.” These students may be fooling themselves. Actually, most exams are not trickier than the homework. Many students get through a homework problem with help. And, then they think they know how to do the problem. But, often, they don’t. Without help, they sometimes can’t even do a simplified version of that same problem.

After you finish a problem using such help you should, at least sometimes, check that you have actually learned to solve the problem.

To see if you have learned to do a problem, do it again, justifying each step, without looking up even one small (‘oh, I almost knew that’) thing.

If you find that you can’t do a problem totally alone, then good! You now have two learning opportunities. First, you can learn the missing skill or idea. But more deeply, by getting stuck after you have been able to get through with help, you can learn things about your learning process. Often the real source of difficulty isn’t a key formula or fact, but something more subtle, high level or abstract (so called ‘meta-cognition’). We hope you can learn some of these useful, and more subtle, ideas from the general text discussions here.

Read the parts of the book that are at your level

You might be science and math school-smart, mechanically inclined, and already especially interested in mechanics. Or you might be reluctantly taking this class to fulfill a requirement.

In either case, we’ve tried to write this book for you. The sections start with generally accessible introductory material and include simple examples. The early sample problems in each section are also easy. But, for the more motivated students with better backgrounds, we also have discussions of the theory and other more advanced applications and asides to challenge them. If you are a nerd, please be patient with the slow introductions and the calculations that go line by line without skipping steps. On the other hand, if you are just trying to get through a course using this book, don’t get hung up by every side discussion about history or theory. Stick with the earlier easier parts.

Calculation strategies and skills

We try to demonstrate a systematic approach to solving problems. But it’s impossible to reduce all mechanics problem solutions to one clear recipe (despite the generally applicable recipe on the inside back cover).

Suppose a recipe existed to solve all statics and dynamics problems. Then someone could write a computer program that followed the recipe. The course you are taking could be cancelled

^†^†margin: Actually, computers can do mechanics. This book presents some methods which computers can handle well. Once a problem has been reduced to a precise mechanical model, a computer code could take over. Such might be called a ‘finite-element’ program or a ‘rigid-body’ dynamics program.

And your mind could be freed from mechanics problem solutions. Mechanics could be like long division, and you could be freed from the skills taught here with a calculator

^†^†margin: Analogy with long division. Since the mid 1970s, division by a 3 (or more) digit number is not done by pencil-and-paper long-division but with a calculator or computer. Nonetheless, understanding division (that, for example, it is inverse multiplication, or that division by zero is bad, or which number to divide by which in real problems) is necessary. And such knowledge comes better by practice manipulating numbers in one’s head and on paper. Similarly, it is useful to know mechanics solution methods well, even for problems that can be solved by a computer package.

There is an art to solving mechanics problems and understanding their solutions. This applies to homework problems and also engineering design problems. Art and insight, as opposed to application of a fixed precise algorithm, is what makes engineering require humans and not just computers.

We hope you learn some of this art. For starters, here are some tips.

Understand the question

It is tempting to start writing equations and quoting principles when you first see a problem. However, it is usually worth a few minutes (and sometimes a few hours) to try to

Get an intuitive sense of a problem before jumping to equations.

Before you draw any sketches or write equations, think: does the problem make sense? What information has been given? What are you trying to find? Is what you are trying to find determined by what is given? What physical laws make the problem solvable? What extra information do you think you need? What information have you been given that you don’t need? You should first get a general sense of the problem to steer you through the technical details.

Some students find they can read every line of sample problems yet cannot do test problems, or, later on, cannot do applied design work effectively. This failing may come from following details without spending time, thinking and gaining an overall sense of the problems.

Think through your solution strategy

For problem solutions you read, like those in this book, someone had to think about the order of work. You also have to think about the order of your work. You will find some tips in the text and samples. But it is your job to own the material, to learn how to think about it your own way, to become an expert in your own style, and to do the work in the way that makes things most clear to you.

The order of calculation is often backwards from the order of thinking

When working out how to solve a problem, you often start ‘backwards’, with general principles, then look at terms you need to know. If these are not given, then you think how to figure those from other terms, and so on. On the other hand, when you go to calculate an answer you have to start with the information given and work your way ‘forwards’ into the equation which has your answer from the information given

^†^†margin: A tree analogy. Say you are interested in how energy gets stored in a tree. The energy flows to the roots from the trunk. The branches feed the trunk, the twigs feed the branches, and the leaves feed the twigs with energy from the sun. But the flow goes the opposite way, from the leaves on down to the roots. But if you try to invent a tree by starting at the leaves, with no knowledge of the root, you could easily get lost and connect leaves to electric wires or gas pipes — all nonsense. There’s no point in connecting the leaves to anything until you have a sense of the whole tree from the roots up.

. To find the net worth of a corporation you add the value of the various divisions. To get the value of a division you add up the values of the factories. For each factory you add up the value of the pieces of machinery. But to get an actual corporate value you have to start by evaluating the pieces of machinery in each factory and working from the known towards the answer. Beware that

A polished calculation, especially an algorithmic recipe or computer program, is often written in the inverse order of the thinking that went into making it.

Real problem-solving goes both ways. You think about what you need in order to calculate what you want. But you also think about what you can calculate easily from what is plainly given to you. You reach from the unknown towards the known details. And you work with known details towards answers of any kind, wanted or not. And you thus hunt out, building from details and simultaneously reaching back from the goal, a route leading all the way from the known details to the goal.

Look for equations containing unknowns. Don’t look for formulas that evaluate unknowns.

In elementary science and math we often learn formulas like

V=LWH,\qquad d=\frac{1}{2}at^{2},\hbox to36.135pt{\hfil and\hfil}x=\frac{-b\pm% \sqrt{b^{2}-4ac}}{2a}

to find $V, d,$ or $x$ . So, it is common wishful thinking for beginners to hope for a formula that generates the sought unknown in terms of given quantities. Rather, you should

Find relations that contain variables of interest; don’t worry about whether they are on the right or left side of an equation. Don’t worry about whether the variables are packed with others or are isolated.

Most often, you will not know a formula that has the thing you want on the left and every known quantity is on the right. You will have, say,

$\displaystyle V$	$\displaystyle=$	$\displaystyle LWH$	when you want to find $W$ from $V, L,$ and $H$ ,
$\displaystyle d$	$\displaystyle=$	$\displaystyle\frac{1}{2}at^{2}$	when you want to find $t$ from $a$ and $d$ , and
$\displaystyle ax^{2}+bx+c$	$\displaystyle=$	$\displaystyle 0$	when you want to find $x$ from $a, b,$ and $c$ .

Once you get enough equations for the number of unknowns you have, the only problem left is math

^†^†margin: For this and other courses, you should be good at solving math problems with your pencil and with a computer. But you should distinguish between the task of setting up a math problem and the task of solving it. Solving often takes most of the time and most of the space on your paper, but it’s not where your thoughts should start. The important new material for you in this book concerns setting up the math problems that arise in mechanics. Of course you should develop your skills at solving equations too, but that’s not the main new content here.

Here are ways of thinking that may help.

1)

Pretend you know a math and computer genius. She is helpful but doesn’t know any mechanics. Your first, and main, task is to write things down so she could finish up for you. She doesn’t want to help? Then realize that finishing up without her is a separate job for you. You will do this later when you take off your mechanics hat and put on your math-genius hat.
2)

Be an egotist. Pretend you are omniscient and know everything. Then write down true statements about those things; equations that contain terms that omniscient-you already know: “If I knew $x, y$ and $z$ the following equation would be true.” Then relax your ego a bit. Count equations and unknowns to see if you, or at least your math genius friend, could solve for some of the things you previously pretended to know.

Thinking outside the books

Filename:tfigure8-rel-ang-vel

Figure 1: Thinking outside of the books. A famous puzzle asks: using four contiguous straightline segments connect all nine dots that are in a square

3\times 3

array. The only solution has segments extending outside the “box” of 9 points. Hence the expression “thinking outside of the box”.

It is fun to puzzle out how things work. It’s satisfying to do calculations that make realistic predictions. Mechanics is interesting in its own right and, interesting or not, it feels good to take pride in new skills. We wrote this book because we want to help you learn the subject if you are interested, and get through it if you must. But we don’t know the sure path through your resources (say a path with 4 straight segments, see fig. 1) that will get you to deeper understanding.

We do know that to learn deeply you need to

think outside of the confines of your usual study resources.

That is, think when you are relaxed, away from the pressures of books, notes, pencils or paper, say when you are walking, showering or lying down. These are the places where you naturally work out life problems, but they are good places to work out mechanics problems too.

Having an animated mechanics discussion with friends is also good. You should enjoy your inner nerd socially. Are your friends turned off by tech-talk? There are billions of people out there, you should be able to find one or two who would like to talk shop with you.

Math skills. We assume you start with the normal skills of, say, a typical successful engineering student. There are some things we assume you know, and some things that we teach here as needed. Success in mechanics depends especially on your being comfortable with some computations.

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Vectors especially dot and cross products, are introduced here from scratch in the context of mechanics. So, if you already have these down pat, fine. If, like most students, you don’t, this book is for you.
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Elementary linear algebra. This book shows how to set up algebraic equations in matrix form for computer solution. The key fact we use is this: once you have rewritten a problem as a set of linear algebraic equations, it is generally easy (almost trivial) to solve them on a computer. A previous or concurrent course in linear algebra would add perspective, but previous mastery of linear algebra is not needed.
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Differential Equations. Concepts for differential equations are presented here, as needed. A previous or con-current course in ordinary differential equations (ODEs) would add re-inforcement and perspective, but is not needed. But you have to get solid on some ODE basics.
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Computers. For some examples and homework problems we assume you have access to, and basic facility with, computation. Nowadays, useful mechanics calculations often depend on your ability to plot, to solve matrix equations and to solve differential equations numerically. As of this writing, the most popular choice for such seems to be Matlab, followed by Python, C++, Mathematica, Julia (up and coming), Octave, and Maple. With more or less effort, one can get by with any program that is good at repetitive arithmetic calculations, for example, Excel. But, one way or another, in this millennium, if you want to do mechanics well you need to know how to use a computer to help you.

You should indulge in getting as comfortable as you can with these things as you learn mechanics. They are useful for mechanics and they are useful in their own right. You will not regret time you spend mastering these computation skills.