How to Succeed in Physics (and reduce your workload) Kyle Thomas, Luke Bruneaux, Veritas Tutors

How to Succeed in Physics (and reduce your workload) Kyle Thomas, Luke Bruneaux, Veritas TutorsTable of Contents HOW TO SUCCEED IN PHYSICS (and reduce your workload) Introduction............................................................................................. 7 MATH CONCEPTS.................................................................... 8 The ideas behind the symbols................................................................ 9 ALGEBRA..................................................................................... 9 Techniques for simultaneous equations............................................... 11 Principle 1: you need as many equations as there are unknowns...... 11 The substitution method....................................................................... 12 The elimination method........................................................................ 13 Word problems...................................................................................... 14 Elementary word problem technique: solving for one unknown variable................................................................14 Units and physics................................................................................... 18 Principle 2: Units should be analyzed algebraically............................ 18 S-o/h C-a/h T-o/a................................................................................... 21 Trig, meet algebra. Algebra, meet trig................................................. 21 Using trig functions to find unknown sides.......................................... 21 Using Inverse Trigonometric functions to solve triangles.................... 22 Pythagorean theorem............................................................................ 23 The second use of trig functions: Percentages, rotations, and waves....................................................... 23 WAVES....................................................................................... 25 Interchangeability of f, T, and ω........................................................ 25 The wave equation................................................................................ 25 Geometry............................................................................................... 21 Demystifying π....................................................................................... 28 Demystifying radians............................................................................. 29 Calculus (optional section).................................................................... 30 No need to fear..................................................................................... 30 A derivative is a rate of change............................................................ 30 A derivative is the slope on a graph..................................................... 31 An integral is the area under a curve.................................................... 32 Vectors and scalars................................................................................ 33 Visualizing vectors................................................................................. 35 Vectors and trigonometry...................................................................... 35How To Succeed in Physics | 5 Principle 3: Whenever you have a problem involving vectors, you should:.....................................................................................................36 Cross products and dot products......................................................... 36 Signs: positive and negative................................................................. 37 One last word........................................................................................ 37 PHYSICS CONCEPTS.............................................................. 38 Human intuitions................................................................................... 39 KINEMATICS.............................................................................. 41 Basic quantities...................................................................................... 41 Velocity and acceleration are vectors................................................... 41 Kinematics equations............................................................................ 41 Principle 4: Make sure you know what every variable represents ..... 42 Forces are vectors.................................................................................. 45 Forces are accelerations........................................................................ 45 Forces: causes and descriptions........................................................... 45 Superposition and free body diagrams................................................ 46 Equilibrium problems............................................................................ 47 Centripetal forces.................................................................................. 47 ENERGY..................................................................................... 49 The mystery of energy........................................................................... 49 Conservation of energy......................................................................... 50 Energy’s many forms.............................................................................. 51 MOMENTUM.............................................................................. 53 Momentum is a vector........................................................................... 53 Momentum, impulse, and the relationship to force............................. 53 Collisions: The confluence of energy and momentum........................ 54 ROTATION.................................................................................. 56 Principle 5: Rotation is just like everything else, but in a circle.......... 56 From linear dynamics to rotational dynamics and back again............. 56 The first piece of the rotational puzzle: Circumference, distance, and radians.................................................. 57 Rotational analogues............................................................................. 58 Rotational kinematics............................................................................ 58 The second piece of the puzzle: Moment of inertia (I)........................ 59 Why moment of inertia?........................................................................ 59 Torque (τ)................................................................................................ 60 Conclusion............................................................................................. 61How To Succeed in Physics | 6 Introduction This will not work in physics. You should aim to study like this: This guide will teach you a number of concepts and techniques to help you ace your physics class with less overall work. We place a strong emphasis on understanding concepts, not just memorizing formulas and example problems. This is not only a better way to learn physics, but it is actually an immensely more efficient way to learn physics, and will make the MCAT (or SAT II or GRE, etc.) much easier as well, if that’s on your horizon. First, we need to make a note here about the relationship between study habits and physics education. As you work your way through the course, each chapter will build upon the last. As a result, if you do not keep up in the beginning, you will be in serious trouble by the middle of the class because you will have no foundation for learning the new material. By the time you get to the final exam, you will likely be completely lost. In lots of courses, students tend to take it easy at the beginning, and then cram at the end: Time spent studying Progression through the quarter Time spent studying Progression through the quarter If you are diligent at the beginning and lay a solid foundation, you will probably actually end up studying like this: Time spent studying Progression through the quarter Those of you with a solid calculus background will realize that total time studying is equal to the area under that line. You will also notice that the last graph has the smallest area, and therefore is the way to get through the course with the least work, and is also the way to assure the most success. We repeat: lay strong foundations! For more help ensuring you have mastered all of the necessary concepts for your course, sign up for the Veritas online physics crash course at Math ConceptsHow To Succeed in Physics | 8 Math Concepts The ideas behind the symbols This book is going to cover math concepts that may seem very familiar to you; you might think about glossing over them. However, since physics uses math in its own special way, we believe you’ll benefit from the review. Much of math education teaches you how to manipulate symbols (for example, “x” or “ϴ”). Since it’s all theoretical, math professors tend to ignore what all the symbols mean; all that matters is how the symbols work with each other on the page. Physics is the opposite. Math is only useful to a physicist if the symbols are tied to some real world concept. In class, however, most physics professors assume that everyone already understands what the symbols mean, and zoom right over the explanation. This math review is structured to remind you how to perform some of the symbol manipulation involved in math, but is much more focused on teaching what these math equations mean in the context of real physical examples. Connecting math to real world concepts will help you not just in physics, but also in any pursuit that requires mathematical thinking (so basically, almost any job). In any case, it is crucial to making physics easy and intuitive, and cutting your workload considerably.How To Succeed in Physics | 9 Algebra We will not go over all of the basics of algebra but rather how it will be used to solve physics problems. While these might appear very simple to you, it’s important that you know them cold. These trusty concepts should always be in your tool belt. The most basic algebraic concept is that anything can be done to an equation as long as it is done to both sides, and this is far and away the most used technique in physics algebra. The following boxes give a few general principles you should be familiar with. Distributive Property a (b+c) = ab + ac and - (b + a) = -b -a Associative Property Commutative Property a + ( b + c) = (a + b) + c a + b = b + a and ab = ba FOIL: First Outside Inside Last (a + b) (c + d) = ac + ad + bc + bdHow To Succeed in Physics | 10 Fraction Math (a + b) c x y + abcd ad (a/b) (c/d) -a b a b = a c = + b c xb + ay yb = = a b a = -b = - bc x d c a b BUT = ad bc a + b + c + d ad ≠ b + cHow To Succeed in Physics | 11 Techniques for simultaneous equations Simultaneous equations are multiple algebraic equations that share unknown variables. What does that mean? Let’s look at an example: x – y = 2 5x + 4y = 19 Both equations have an x and a y. The thing to remember here is that x and y are related to each other in both ways. Let me illustrate with a word problem: x represents apples y represents oranges You have two more apples than oranges. x-y = 2 Apples cost 5¢ and oranges cost 4¢. You paid 19¢ total for all your apples and orange. 5x + 4y = 19 How many apples and oranges do you have? x = ? y = ? or 2 apples and 0 oranges! 41 apples and 39 oranges; The whole point of simultaneous equations is to solve problems like this. And we can. Principle 1: In order to solve a problem, you need as many equations as there are unknowns. How many unknowns were in the example? Two: apples (x) and oranges (y). What if we had just one equation and lost the second? x-y = 2 We know that we have two more apples than oranges. But that could mean 5 apples and 3 oranges;How To Succeed in Physics | 12 Which one is it? We need the second equation to figure it out. Principle 1 is true of one unknown (we only need one equation), three unknown (three equations), or 100 unknowns (100 equations!). Hopefully you won’t have to solve that one. So let’s solve the example above. Since we have two equations, we will be able to solve for two unknown variables. There are two techniques to solve multiple equations, the substitution method, and the elimination method. The substitution method In the substitution method, we rearrange one equation and plug the result into the second equation. This is most easily seen as an example, so let’s take the above equations, 5x + 4y = 19, and x – y = 2: Start with the first equation: it’s simpler. x – y = 2 Let’s rearrange this for x. x = 2 + y OK. Now we can plug (2+y) wherever we see x in the other equation. 5x + 4y = 19 5(2 + y) + 4y = 19 Let’s simplify and do some algebra. 10 + 5y + 4y = 19 10+9y =19 9y = 9 y =1 OK! We have one orange. But how do many apples do we have? Since x – y = 2 We can replace y with 1 to get x x – 1 = 2 And simply solve x = 3 Three apples and one orange. In other terms: x = 3 y = 1How To Succeed in Physics | 13 The elimination method In the elimination method we essentially do simple arithmetic, but use the entire equation. The goal here is to eliminate one of the variables so that we are left with a single one-variable equation. This is also easiest to see in an example, so let’s pick two new equations: Example: 4x + 3y = 10 2x + 3y = 6: We’re going to line these up and subtract like you did in grade school 4x + 3y = 10 - (2x + 3y = 6) 2x + 0 y= 4 We can perform arithmetic on each variable and constants: x: 4-2 = 2 y: 3-3 = 0 constants: 10- 6 =4 See how we produce a third equation with 0y? We can solve that for x. 2x + 0 y= 4 2x = 4 x =2 Then we plug x=2 back into one of the original equations: 4(2) + 3y = 10 8 + 3y = 10 y = 2/3 We have our answer x = 2 and y = 2/3. That’s the goal in the elimination method: find a way to get rid of one variable (either x or y). It’s a little bit trickier, but you could also have done this with x by multiplying the second equation by 2: 2*(2x + 3y) = 2* (6) 4x + 6y = 12 Why does this help? Because now we can subtract out x: 4x + 3y = 10 - (4x +6 y = 12) 0x + -3 y= -2 We get a new equation -3 y = -2 or y = ⅔ like above-- which is a good thing! When we plug y back into the first equation, we retrieve x: 4x + 3 (2/3) = 10 4 x = 8 x=2 It checks out!How To Succeed in Physics | 14 Word problems In your tenure as a physics student, you will be be asked to solve word problems - a lot of them. Word problems are notoriously painful, and for good reason: they ask you to think, and thinking is often painful. But if you get good at word problems, they can be fun - like a good workout. We will start by discussing approaches that should help get you started. Here we will discuss the most basic techniques using algebraic principles. Elementary word problem technique: solving for one unknown variable You’ve got a paragraph in front of you with a lot of labels and numbers. You’re pretty sure there’s a vaguely worded question in there. How do you figure it out? 1. First off, draw a picture and label everything. This will give you a clear idea of what is going on and what you need to account for. Even if there is already a figure in the book, draw your own so you can mark it up. 2. Immediately write all defined variables off to the side of the page. This way you can keep track of what you know and what you don’t. 3. Figure out the missing variable that the problem is asking you to solve for. This can be really tricky but ask yourself: what is the writer asking me for? 4. Find an equation that contains all of the known variables and the unknown variable. Look at a list of formulas for the chapter. For most simple word problems, you can find one that will solve the problem immediately. For more difficult problems, you’ll need multiple equations (see below). 5. Once you have an equation that relates all the variables together, solve for the missing variable. 6. Plug in the numbers and carry out the calculation. Keep all the units (kg, m/s, etc). together too. 7. Box your answer. Graders love being told where to look.How To Succeed in Physics | 15 Let's say you are given the following problem: A spaceship launches with a constant acceleration. After 2 seconds the ship has gone up 6 meters. A) What is the acceleration? B) What is its velocity at this point? Step 1 Write all variables off to the side - we know t is 2 seconds - we know ∆x is 6 meters - we also know the initial velocity v0 Step 3 Find an equation with all of the variables you are given and the unknown. Because you do not know the equations yet, I will list them here: A) v =v0 + at B) ∆x = v0t + 1/2 at2 C) v2 = v0 2 + 2a∆x is 0 meters per second because the ship started at rest So we will put all of these in a list to the right... Step 2 Figure out what variable the problem is asking us to solve for. We are actually being asked to solve for two, a, and v. Generally, we will solve for them in sequential order. a∆x = 6m v0 = 0 m/s To solve for acceleration we should use equation B) because it contains our three known variables listed above (t, Δx, v0) and "a" so we can solve the equation with only one unknown To solve for final velocity we should use equation A) for the same reason So your paper will look like this: A) ∆x = v0 t + 1/2 at2 ∆x = 1/2 at2 a = 2∆x/t2 because v0 = 0, v0 t = 0t and so it drops out of the equation Plug in your numbers and you get a = 3 m/s (This is now one of our variables) B) v = v0 + at Again, v0 drops out v = at Plug in your numbers and you get v = 6 m/s a = 3 m/s t = 2s ∆x = 6m v0 = 0 m/sHow To Succeed in Physics | 16 Technique for multiple unknowns: Multiple unknowns require multiple equations So you’ve got a problem with two unknowns in it. We solve these word problems using basically the same technique as above, except here we will need to find multiple equations. 1. Remember our first principle: We need as many equations as we have unknowns. 2. Try to limit the number of variables! Every new variable introduced by an equation will require us to add another equation. Thus if you pick an equation that contains an additional unknown, you will need another equation, and you will not have gained anything. 3. The equations that we choose will have to have at least one variable in common, or else they cannot be linked together. 4. To solve our set of equations we will use either the substitution method or the elimination method. Typically you will use the substitution method because the elimination method only tends to be helpful if both equations have the same variables. Let’s use a simple example to demonstrate: You have twice as many jelly beans as you do M & Ms. You have 36 total pieces of candy. How many of each do you have? 1. Define unknown variables. What is the problem asking for? Number of jelly beans (let’s call that J) and number of M&M’s (let’s call that M): J = ? M = ? 2. Write an equation using the information in the problem. We know that total candy is J + M, and that we have 36 pieces so: J + M = 36. 3. Check how many unknowns and how many equations you have. If you don’t have enough equations, look for a way to generate