FZX Physics Lecture Notes reserved FZX: Personal Lecture...

FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved

FZX: Personal Lecture Notes from Daniel W. Koon

St. Lawrence University Physics Department

CHAPTER 1

Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu.

1. Mathematics Review Algebra Exponents Trigonometry Geometry Problem-Solving Units of Measure Unit Conversion

page 1 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


FZX, Chapter 1: MATHEMATICS REVIEW Many people think of physics as a foreign language, and in some ways it is. To fully understand the physics in this course, you need to be conversant in math, the language in which much of physics is stated. There are three areas of math that you will need for this course: algebra, trigonometry, geometry. We will outline these below, as well as saying some things about problem solving, units of measure, and unit conversion. First, however, let’s consider why physics is perceived as such a difficult subject. Certainly part of the answer lies in the math needed to analyze specific problems, but students tend to have a lot of problems with the concepts of physics as well as the specifics of doing the math to solve a particular problem. One of the reasons physics can be difficult is that we have all been studying physics since we were born (and possibly even before that). That would be okay if we had drawn the right conclusions from our billions of physical observations, but we have all drawn some conclusions that are not valid and that will have to be changed in this course. Because we have believed some of these ‘false theorems’ for so long, they have become part of our common sense, and our common sense is something that is notoriously difficult to change. BOGUS EXAMPLE: Is the Earth flat? As an example, consider what we have observed about the ‘motion’ of the Sun in the course of the day. We’ve all noticed pretty early on that the Sun is in the Eastern sky in the morning and seems to travel to the West as the day wears on. It was clear to us, as it was to our ancestors millenia ago, that the Sun is moving, relative to the Earth. So why do we believe that the Earth moves around the Sun? At some point, some grownup or older sibling (or someone) told us that the Earth is round, that it spins about an axis, and as it spins, the Sun appears to move in the sky. Did you believe this immediately? Probably not. If you were like me, you probably came up with many good, common-sense reasons not to believe this new theory. At the risk of belaboring the point, let me outline them: First, the Earth is clearly flat. Look around you: you see mountains and valleys, or you see flat plains. It is hard, looking around you to believe that the Earth is round. Second, imagine that the Earth were round. Our schoolbooks say that the opposite point on the globe is inhabited. We are clearly on the top of this supposedly spherical Earth, because objects fall down, and we do not fall off the globe. The people on the other side of the Earth would therefore fall off the Earth because they are upside down. Third, if we accept that the Earth is round and that it is the size that our schoolbooks tell us it is, and accept that it must turn once around its axis every day, then this means that we are spinning around that axis at up to 1000mph (700mph in Northern New York State). Okay, why don’t we fly off the Earth if it spins this fast? And how come we don’t feel this motion? We obviously can feel the difference between sitting in a car that’s moving and one that’s not. Okay, have I convinced you that the Sun is actually moving around the Earth? Probably not. Why not? As I said before, we are very stubborn about our common sense, and the burden of proof to dislodge anything we have believed for so long is very heavy. This is a good thing, because it prevents a lot of people from believing the New York City cabbie who tries to sell them the Brooklyn Bridge, but in this course, we will exert a lot of energy trying to dislodge various notions which you hold dear. It is important that you are aware of this before we begin, which is why I spent so much time on this bogus example. ALGEBRA: I can split up the most important facts we need from algebra into three items: what I call the ‘Golden Rule’ of algebra, the Quadratic equation, and the rules for dealing with exponents. What I mean by Golden Rule can be stated this way: ‘Do unto one side (of an equation) what you would do unto the other side.’ If you have an equation (two mathematical expressions which are combined by an equal sign), you can perform any operation on the left-hand-side, provided you do the same to the right-hand-side. Some specific examples are the following: If A = B , then A + C = B + C, [Addition] A - C = B - C, [Subtraction] A C = B C, [Multiplication] A / C = B / C, (C≠0) [Division] A2 = B2, [Exponentiation] and so on.



It is important that you do to an entire side of the equation what you do to the entire other side of the equation. Consider the following: If A + C = B, then (A + C)2 = B2. But A2 + C2 does not equal B2! The most common type of equation we will encounter in this course is the linear equation, in which A x + B = C, [Linear equation] in which A, B, and C are constants (quantities that are known) and ‘x’ is the unknown, the quantity we wish to find. These equations are relatively easy to solve, and I won’t discuss them here. A more difficult type of equation to solve is the quadratic equation, which is an equation that can be written as A x2 + B x + C = 0, [Quadratic equation] where A, B, and C are constants, and ‘x’ is again unknown. This equation cannot be solved, as the linear equation can, by applying the Golden Rule until ‘x’ is on one side of the equation and the other side contains only constants. As long as A, B, and C are all nonzero, the only way to solve this equation is by the Quadratic Formula:

x B B ACA

=− ± −2 4

2 [Quadratic formula]

Notice that it is possible for the stuff under the square root to be negative. This means that there is no physically-meaningful solution for x. We will see some examples of this in the future. Also notice that, because of the plus-or-minus sign, ±, there are generally two possible solutions for x, unless the stuff under the square root is zero (Then there’s only one solution.) or negative (Then you’re out of luck: there are no solutions.). Knowing when to use the Quadratic Formula is not always easy. Usually a quadratic equation does not appear in front of us in the form we wrote above (Ax2+Bx+C=0). In the following exercises, you will want to put some equations into the quadratic form by judicious use of the Golden Rule:



EXPONENTS Exponents serve as a handy shorthand for describing repeated multiplication. If we add the same number several times, we can use multiplication to serve as a shorthand for this complicated operation: 2 + 2 + 2 + 2 + 2 = 5 x 2. [ Definition of multiplication ] In a similar way, if we multiply the same number several times, we can use exponents as a sort of shorthand for a more complicated operation: 2 x 2 x 2 x 2 x 2 = 25. [ Definition of exponent ] There are a number of rules that govern how we deal with exponents. If any of these don’t make sense, go back to this ‘definition of exponent’ to prove them. If we multiply two exponential expressions THAT HAVE THE SAME BASE:

An x Am = An+m If an exponential expression is raised to some power: (An)m = Anm. The inverse of taking an expression to the ‘nth’ power is to take it to the ‘-nth’ power: (An)-n = 1 = (A-n)n, page 4 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


and A-n = 1/An. Notice that the following expressions cannot be simplified by the rules given above: An + Bn [ Cannot be simplified ] An x Bm. [ Cannot be simplified ]

TRIGONOMETRY Trigonometry can be thought of as the study of triangles. In physics we will mostly be concerned with right triangles, a small but vocal segment of the triangle community. The major results we need to recall from trig for this course are the Pythagorean Theorem, the definitions of just three trigonometric functions: sinθ, cosθ, and tanθ, and a unit for measuring angles known as the radian. To apply the Pythagorean Theorem, we need to begin with a right triangle, a triangle in which one of the angles is a 90o angle. The hypotenuse is the side of the triangle opposite this angle (the only side which doesn’t touch that corner). Let’s call the hypotentuse ‘C’. If the other two ‘legs’ of the triangle are labelled ‘A’ and ‘B’, then A2 + B2 = C2. [Pythagorean theorem (right triangles only)] Now for the definitions of the trigonometric functions. Consider the angle θ in a right triangle. θ can be any of the two angles which is not the right angle. The three sides of the triangle can be described in terms of θ. One of the sides is, of course, the hypotenuse, and would still be the hypotenuse regardless of which angle we were looking at. Another of the legs is opposite this angle -- that is, it does not touch that angle. We call it the ‘opposite’ side. The third side touches the angle we are considering as well as touching the right angle. We call this the ‘adjacent’ angle. Reread this last paragraph while drawing a picture. Be sure to label everything that’s mentioned in the paragraph -- sides and angles. For angle θ, we define three functions in terms of the lengths of the three sides: sinθ = opposite/hypotenuse [ ‘SINE’ function ] cosθ = adjacent/hypotenuse [ ‘COSINE’ function ] tanθ = opposite/adjacent = sinθ/cosθ [ ‘TANGENT function ]

There are other trig functions, but we won’t need them in this course. The definitions above work for angles between 0o and 90o, for which we can draw a right triangle. It is important to realize that we can define the three functions more generally, so that we can talk about angles greater than 90o or less than 0o. For these angles, some of these functions will have negative values. We generally measure angles in degrees, with 360o in a complete circle. Why 360o? For some reason, that’s the number of units the ancient Babylonians used, and it has stuck

with us, just as 60 seconds in a minute, 60 minutes in an hour, and 24 hours in a day -- all subdivisions of time that we owe to the ancients’ religious obsession with the number 6. So the choice of 360o in a circle is an arbitrary choice selected by an ancient people. There is a more natural way of slicing up the pie. (By the way, some ancient peoples divided the calendar into 360 days with five or six leapdays, so maybe 360 is not such an arbitrary division. I’ll leave you to figure out why we use a ‘base-10’ system of numbers -- another fairly arbitrary choice.) page 5 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


We can think of any angle as a fraction of a circle, a slice of a pie. If we measure the length of the circular arc at the opposite side of the slice from the angle we are considering, and divide by the radius (so that the size of the circle is irrelevant), then we have a measure of angle which is natural and unitless. However, we will use the dimensionless units of ‘radians’ for such a measure, in order to remind ourselves that we are measuring an angle.

θ = s/r, [ Angle measurement in radians ] where ‘s’ is the length of the arc of the circle opposite the angle and ‘r’ is the radius of the circle. In using radians as measurement, we can see that an entire circle (360o) is 2π radians big. This leads us to a simple unit conversion factor:

1180=

radπ

o

[ Conversion between degrees and radians ]

If your calculator ordinarily displays the letters ‘DEG’, then chances are that it is capable of doing trig calculations in either degrees or radians. Check your instruction manual for how to switch the calculator between ‘degree mode’ and ‘radian mode’. (There is another way of measuring angles, using a unit called ‘grad’, but no physicists I know ever use it.)



GEOMETRY: Geometry, like algebra and trigonometry, has a rich and interesting existence independent of physics. Just as with the other two subjects, we will borrow only a few items out of geometry to use in this course, particularly equations for perimeters, areas, and volumes of various two- and three-dimensional shapes which occur frequently in our study of the Universe around us. Let’s start with two-dimensional objects. For each shape we will list a formula for both the ‘perimeter’ -- the distance around the shape -- and the area. PERIMETER: AREA: Circle: C = 2πr A = πr2 Rectangle: P = 2l + 2w A = l w Square: P = 4s A = s2 Triangle: P = a + b + c A =½ bh [FORMULAE FOR TWO-DIMENSIONAL OBJECTS] For the circle above, the perimeter is called the circumference, and ‘r’ is the radius of the circle. For the rectangle, ‘l’ and ‘w’ are the length and width. For the square, ‘s’ is the length of one side. For the triangle, ‘a’, ‘b’, and ‘c’ are the lengths of the three sides of the triangle. In the expression for the area of the triangle, ‘b’ is the base, and ‘h’ the height of the triangle, where the height must be perpendicular to the base. Three-dimensional shapes occur all around us, and a lot of them resemble the most simple three-dimensional objects from geometry. For these, we need both the surface area and the volume. SURFACE AREA: VOLUME: Sphere: SA = 4πr2 V = 4

3π r3

Rectangular box: SA = 2 lw + 2wh + 2 lh V = lwh Cylinder: SA = 2πr2 + 2πrh V = πr2h [FORMULAE FOR THREE-DIMENSIONAL OBJECTS] The sphere’s radius is ‘r’; the rectangular box has a length, width, and height of ‘l’, ‘w’, ‘h’; and the cylinder has a height ‘h’, with a cross-section that is a circle of radius ‘r’. Be especially aware of the similarity between the two equations for the sphere. These are the easiest two equations to mix up.

PROBLEM SOLVING: The following is an approach which I have found a useful checkoff list to make sure that one is not skipping steps when attacking a problem. There are a number of steps that students tend to leave out when doing problems, and I have found page 7 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


that if you consciously go through this list of steps, you are much less likely to forget them. Also, if you have this set of guidelines in front of you, you can set up a problem, sometimes even if you are uncertain of how to proceed. This may sound odd to hear from a professional teacher of physics, but a lot of things I teach -- like the process listed below -- are taught to keep you from having to think too hard. The less time you spend pondering relatively simple questions -- like ‘which do I use here: sine or cosine?’ -- the more time you have to devote to more important questions. I call this problem-solving process listed below RDTESNR, which is an acronym which stands for the different steps in the approach. The steps are... READ the problem. Well, duh! Of course you have to read the problem. You may have to reread the problem several times, each time mining more information from it. Don’t assume that you are done with this step once you have moved on to the next step, which is... DRAW a DIAGRAM. Some of us are more visual than others and need to see a thing before we feel we understand it. In physics, if you can’t draw a situation, it is unlikely that you really understand it. If the problem involves actions taking place at two different times, draw a ‘before’ and ‘after’ picture for each time. It is not enough to draw your diagram: you must also LABEL it. Take all the information given in the problem, and TRANSLATE that information into symbols. If you are given that an object is 12m tall, write ‘L=12m’ on your diagram next to the object. Translating the word problem into an algebra problem is the most crucial step in this problem: it allows you to assess what quantities are known and what are unknown. Next, write the appropriate EQUATION that combines the knowns and unknowns. Too many people are in too big a hurry to get to this step that they leave out the first few steps: they are more likely to miss any subtle variations that distinguish this problem from others, and less likely to understand what they are doing. Next, SOLVE this equation for the unknown quantity. I highly recommend that you not plug in numbers into your equation until you have done this step. If you need to do several steps of algebra to get your answer, it is much easier to carry around (and possibly multiply both sides of the equation by) ‘L’ than by ‘1.2453x1024kg.m2/s2’. (especially if you need to invoke the quadratic equation!) Once you have the unknown quantity on one side of the equation, plug in the NUMBERS on the other side. Finally, RECHECK your answer to see if it makes physical sense. If you get the length of an object to be -13m, you’ve probably gone wrong somewhere. If you are satisfied that you have gone through the motions, and don’t really care whether you got the right answer, then by all means, don’t bother with this step. If it helps to make up your own mnemonic to remember the steps of this method, do it. The time you spend memorizing these steps now is thinking time you will save later trying to remember which step comes next. UNITS OF MEASURE: When we measure a thing, we compare it to standards that we know. In measuring a walked distance, we might ask how many strides we must take to travel that distance. In essence, we ask how many ‘strides’ fit inside the distance to be measured. Similarly we could ask how many ‘yards’ or ‘meters’ fit inside. In this course, we will use a set of standards for our measurement of distance, time, and other quantites. This set of units is called the ‘Systeme Internationale’ -- ‘SI’ -- or the ‘MKS’ system. ‘MKS’ stands for three of the units that make up this system -- meters, kilograms, and seconds. Almost anything you can measure is measured by comparing it to some standard. In communicating to someone the quantity you measured, you must tell them how many of the units fit into the thing you measured, but you must also say what the unit was. Later, when we multiply and divide quantities, we will see that the units associated with them multiply and divide as well. This is good, because you will usually know what the units of your final answer should be. If you carry your units through your calculation to the bitter end, and if the units turn out wrong, this is probably a sign that the number associated with those units is also wrong. Think of carrying your units all the way through as being like a ‘buddy system’. If your partner gets pulled in by the tide, you are his only hope of getting help to save him. (For some reason most students do not like to carry their units. They seem to feel that the time they save outweighs the benefit of double-checking their calculation. This is a false economy. It’s like going out for a soda while your buddy goes surfing alone. You wouldn’t want that buddy to do that to you.)




When measuring very large or very small quantities, we sometimes modify our units a bit by using prefixes to stand in for multiples of 10. The most common of these are actually multiples of either a thousand or a thousandth, and the most commonly used are given below: f = femto 10-15 p = pico 10-12 n = nano 10-9 G = giga 109 μ = micro 10-6 M = mega 106 m = milli 10-3 k = kilo 103 [COMMONLY USED FACTORS OF 1000] d = deci 10-1 c = centi 10-2 [OTHER COMMON PREFIXES] UNIT CONVERSION: Sometimes the units you are given or that you calculate an answer in are not the most appropriate units. Sometimes you may want to convert your answer from MKS units to units you are more familiar with, in order to have a rough idea how big something is. The easiest way to do this is to use what we call a ‘unit conversion factor’. This is nothing more than multiplying your quantity by the number 1.

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