Copyright © 2012 Pearson Education Inc. CHAPTER 1 Units, Physical Quantities & Vectors

Copyright © 2012 Pearson Education Inc. Goals for Chapter 1 To learn three fundamental quantities of physics and the units to measure them To keep track of significant figures in calculations To understand vectors and scalars and how to add vectors graphically To determine vector components and how to use them in calculations To understand unit vectors and how to use them with components to describe vectors To learn two ways of multiplying vectors

Copyright © 2012 Pearson Education Inc. WHAT YOU ALWAYS WANTED TO KNOW BUT WERE AFRAID TO ASK How Science REALLY works

Copyright © 2012 Pearson Education Inc. Taxonomies of “understanding” Nature Model – an explanation that leaves out details. These omissions may be employed because one assumes (ass/u/me) they are negligible; or details left out in early levels of instruction (so you don’t get brain-scorch) and “cleaned up/added” in higher level courses; or there are not current methods available to assess the omission. Proximate explanation – used by the general public, an early partial explanation, or a model Ultimate explanation – an explanation that explains all phenomena, at all levels of understanding (your Grandmother or the Nobel Prize Committee)

Copyright © 2012 Pearson Education Inc. Epistemology – How do we establish “truth”? Applies both to CIVILIZATIONS, and to your own development over YOUR life span Faith – Rely upon individual (tribal leader, shaman, parent) or document Rationalism – What can be formed in the human mind exists in nature Empiricism – What we can measure in the Laboratory, only

Copyright © 2012 Pearson Education Inc. The Nature of Physics Physics is an experimental (empirical) science in which physicists seek patterns that relate the phenomena (plural of phenomenon) of Nature. We make individual measurements called experimental facts or phenonmena. These facts are organized into patterns are called physical laws or principles. A very well established, or widely used, or summative set of laws is called a theory. Often, the theory gives the broadest, deepest explanation of Nature.

Copyright © 2012 Pearson Education Inc. Hierarchical Structure of Explanations THEORIES (COMPREHENSIVE, USUALLY SAY WHY) LAWS/PRINICPLES FACTS/MEASUREMENTS/PHENOMENA Example, “dropping things”

Copyright © 2012 Pearson Education Inc. How Good Is Our Understanding As a freshman, I was taught The Universe has always been here, it will always be here: eternal in time. The laws of Physics we learn here on Earth apply everywhere in the Universe. The laws of Physics are isotropic (the x, y, z number lines are all equivalent). We now understand that Physics is in Domains. Within each Domain, we must apply Physics rules appropriate to THAT domain; these domains are in both space & time

Copyright © 2012 Pearson Education Inc. Spatial/Size Domains ScaleObjectsTheory mNucleus Quarks ? mAtomsQuantum Mechanics to 10 8 mEarthClassical Mechanics* 10 8 to m Kuiper BeltClassical Mechanics to mGalaxies“Dark Matter” to mVisible Universe“Dark Energy” * We will start the course in this domain

Copyright © 2012 Pearson Education Inc. Domains of Physics – by scale

Copyright © 2012 Pearson Education Inc. Domains of Physics – by time (pg of text)

Copyright © 2012 Pearson Education Inc. We now experience four fundamental forces; ‘twas not always that way!

Copyright © 2012 Pearson Education Inc. Evolution of Understanding Scientific MethodParadigms & Revolutions Paradigm – RESTRICTED, mutually agreed upon list of discovery methods and questions that can be addressed 1. Paradigmic State: Experimentalist – Theorist Duality 2. Anomaly: 3. Quashed, or Revolution & New Paradigm Much like a Religion

Copyright © 2012 Pearson Education Inc. Solving problems in physics A problem-solving strategy offers techniques for setting up and solving problems efficiently and accurately. This scheme forms an important skeleton for the course!

Copyright © 2012 Pearson Education Inc. Standards and Units – We Will Impose Empiricism Length, time, and mass are three fundamental quantities of physics. **Next term, electric charge/current The International System (SI for Système International) is the most widely used system of units. In SI units, length is measured in meters, time in seconds, and mass in kilograms. History of unit choice

Copyright © 2012 Pearson Education Inc. Orthogonality These units may be considered as lying along orthogonal axes, in multiple dimensions. They are considered (for now) as INDEPENDENT quantities. ** Warning, this is a model!

Copyright © 2012 Pearson Education Inc. Unit prefixes Table 1.1 shows some larger and smaller units for the fundamental quantities.

Copyright © 2012 Pearson Education Inc. Unit consistency and conversions An equation must be dimensionally consistent. Terms to be added or equated must always have the same units. (Be sure you’re adding “apples to apples.”) Always carry units through calculations. Convert to standard units as necessary. (Follow Problem-Solving Strategy 1.2) Follow Examples 1.1 and 1.2.

Copyright © 2012 Pearson Education Inc. Units are an error checking technique in Physics Math doesn’t care (take account of) units. If you write: A = BC + D/E, not to worry in Math. In Nature, EACH of the terms above must have the same units (M, L, T, or the same combination of these). This is an error checking tool! EX 1-1 Suppose that you are solving a problem involving the volume of a cylinder. You think you remember that it is: V = π R l, where R is the radius of the cylinder and l is it’s length. How would you verify this by unit equivalence? GW 1-1

Copyright © 2012 Pearson Education Inc. Unit Conversion Across “Systems of Measure” Converting between unit systems: English SI, or MKS cgs EX 1-2The piston displacement in a high- performance scooter engine is 2.11 liters (where 1 liter = 10 3 cm 3 ); express this volume in cubic inches. GW 1-2

Copyright © 2012 Pearson Education Inc. Uncertainty and significant figures—Figure 1.7 The uncertainty of a measured quantity is indicated by its number of significant figures. For multiplication and division, the answer can have no more significant figures than the smallest number of significant figures in the factors. For addition and subtraction, the number of significant figures is determined by the term having the fewest digits to the right of the decimal point. Refer to Table 1.2, Figure 1.8, and Example 1.3. As this train mishap illustrates, even a small percent error can have spectacular results!

Copyright © 2012 Pearson Education Inc. Estimates and orders of magnitude An order-of-magnitude estimate of a quantity gives a rough idea of its magnitude. Follow Example 1.4.

Copyright © 2012 Pearson Education Inc. EX 1-3 Suppose one is asked to evaluate the following formula for a, and you have misplaced your calculator: a = b 2 c d 2 / e where: b = 7.25x10 3 ; c = π; d = 8; e = 325. What is a rough estimate of the value of a? GW 1-3

Copyright © 2012 Pearson Education Inc. Theory of Measurement

Copyright © 2012 Pearson Education Inc. “Fundamental” Measurements  Length  Time  Mass  Electric charge (or current)

Copyright © 2012 Pearson Education Inc. What Can We “Really Measure” with Certainty? Wonderland  Alisha is Alice’s virtual self; both carry a standard of length measurement (1 m, their height)  They agree about each other’s measurements when at the interface  As they separate, they begin to disagree as to the “truth’  They carry their measurement instruments with them, which undergo the same potential distortions as the person does. /ThinLens/lens&mirror/lensDemo.html

Copyright © 2012 Pearson Education Inc. Is the Universe Flat (rectilinear, isotropic)? Is Euclidean Geometry Appropriate? Why does the Universe look flat? This was one of the perplexing questions in cosmology for a long time. Today, most astronomers believe in the theory of inflation (and there are pieces of evidence supporting this). According to this theory, the Universe underwent exponential expansion about seconds after the Big Bang. The result was that something of the size of an atom expanded to the size of the solar system by the end of the inflationary epoch.

Copyright © 2012 Pearson Education Inc. If this were the case, irrespective of the original geometry of the Universe, it would appear flat to us. The analogy will be to take a balloon; we can easily see it to be rounded; now blow the balloon to a very large volume and then put a small ant on its surface. The ant will think that it is on a sheet; it cannot detect the curvature. To put this in another way, the distances that we probe are way too small to detect any possible curvature in the Universe.

Copyright © 2012 Pearson Education Inc. Another infinite/finite universe BvbGgza2Q0BHNlYwNzcgRzbGsDdmlkBHZ0aWQDVjExNg-- ?p=walking+on+a+mobius+strip+animation&vid=f4564ac8800e3501c558927e8836a4bb& l=&turl=http%3A%2F%2Fts4.mm.bing.net%2Fth%3Fid%3DV %26pid %3D15.1&rurl=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DpGy- WxLaKl8&tit=Mobius+Strip+II+Animated+Movie&c=0&sigr=11aqbe1rf&&tt=b

Copyright © 2012 Pearson Education Inc. What “Is” the Visible Universe?

Copyright © 2012 Pearson Education Inc. Visible Universe

Copyright © 2012 Pearson Education Inc. Is This Classroom Rectilinear? Plumb Bobs point to the center of the Earth

Copyright © 2012 Pearson Education Inc. Sum of angles in triangle Planar triangle A + B + C = openref.com/tria ngleinternalangl es.html Spherical triangles  > Application: property boundaries

Copyright © 2012 Pearson Education Inc. What is time? PAST AND FUTURE ARE DURATIONS IN TIME Unlike the present we see past and future as measurable durations of hour days months and years. Past historical events, an upcoming meeting, wedding, or other events are all measurable durations or extensions in time, just like a recorded material video recordings, and tracks of. This similarity suggests that past is just a recorded memory, while future is like an unrecorded tape. Future is a projection created by our past experiences which are stored in our memory Personal Story

Copyright © 2012 Pearson Education Inc. “Logical/Common-Sense Tools” Fundamental, Independent Physical Units** Causality Reductionism** Absolute access to Truth** Paradigm’s (what happens if the answer is outside your “box”?) Occam’s Razor** Observer can be non-invasive in measurement** ** Will be violated later in the course

Copyright © 2012 Pearson Education Inc. Uniformity (space & time)* Euclidean Geometry* Human Perception is sufficient*

Copyright © 2012 Pearson Education Inc. Annoying Historical Detail Asking the “correct” question what happens if you don’t 1.“Where is the electron in the atom?” 2.“Does Quantum Mechanics (Physics of the electrons) apply in the nucleus?” 3.3. “Can you violate energy conservation?”

Copyright © 2012 Pearson Education Inc. Nearly All of these “logical” Tools Will Prove Untrustworthy by the End of the Course For everyday life (say within 50 miles of home and back 2000 years), there is no serious difference; these tools work just fine. Beyond that, measurements begin to fail. Using this “fundamental” bag of tools, however, will give us practice and confidence that science works; some of this may be tedious. When that fundamental set of ideas is in place, we can venture into “Alice In Wonderland”

Copyright © 2012 Pearson Education Inc. We will discover that there are “limits” to what we can do and perceive: Because of the finite speed of light, starlight you see now may have originated on stars that are no longer emitting light or even no longer there. Or light has passed you long in the past Because of the Uncertainty Principle, at least at the microscopic (atomic) level, measuring the present forever changes the future and probably erases knowledge of the past Example: finding a ping pong ball in the dark

Copyright © 2012 Pearson Education Inc. What We Will End Up With No one knows what “time” is We always measure time indirectly by measuring a length, and THAT measurement will become slippery Time and Length are interchangeable; they are connected through c Energy and Matter are interchangeable; they are connected through c 2 c = speed of light = 3.0x10 8 m/s

Copyright © 2012 Pearson Education Inc. At this point in time, we are left with only two absolutes:  Speed of light = c* = 3x10 8 m/s =  Entropy (disorderliness) points qualitatively in the direction of positive time (and there are some indications that even this may be a local phenomenon) * Mention Aether Drag Irreversible process: Reversible process:

Copyright © 2012 Pearson Education Inc. IMPORTANT MATH TOOLS FOR ORTHOGONAL SYSTEMS VECTORS

Copyright © 2012 Pearson Education Inc. The Roles of Math & Science Math is a rational endeavor; what the mind conceives is fair game. Analog in nature? TBD. Science is an empirical endeavor (based on measurement), and drags in the subset of Math that best fits experimental data When I introduce Math, be assured it describes measureable nature

Copyright © 2012 Pearson Education Inc. Vectors and scalars A scalar quantity can be described by a single number (often + units). A vector quantity has both a magnitude (scalar) and a direction in space. In print, a vector quantity is represented in boldface italic type, A, or with an arrow over it: A. The magnitude of A is written as A or |A|.   

Copyright © 2012 Pearson Education Inc. Mathematical “legitimacy” of vectors? a.Can we add and subtract them? b.Can we multiply vectors [actually, there will three (3) forms of multiplication]? c.Are addition and multiplication commutative? FYI: 1.Vectors are best learned by “doing”; so I recommend you use the practice applets in this PPT, and then learn from DOING the homework. 2.When we actually USE them, I will go at it as if you know very little about vectors, which will be pretty accurate. This is a core, TRADE skill.

Copyright © 2012 Pearson Education Inc. Drawing vectors—Figure 1.10 Draw a vector as a line with an arrowhead at its tip. The length of the line shows the vector’s magnitude. The direction of the line shows the vector’s direction. Figure 1.10 shows equal-magnitude vectors having the same direction and opposite directions.

Copyright © 2012 Pearson Education Inc. Emphasize: Vectors are “sliders”

Copyright © 2012 Pearson Education Inc. Vector Addition is “commutative” Tails at same point do not preclude addition; you slide them

Copyright © 2012 Pearson Education Inc. Practicing WebTutorials/HeadToTailMethod.htm WebTutorials/Parallelogram.htm

Copyright © 2012 Pearson Education Inc. Adding two vectors graphically—Figures 1.11–1.12 Two vectors may be added graphically using either the parallelogram method or the head-to-tail method.

Copyright © 2012 Pearson Education Inc. Adding more than two vectors graphically—Figure 1.13 To add several vectors, use the head-to-tail method. The vectors can be added in any order.

Copyright © 2012 Pearson Education Inc. Subtracting vectors Figure 1.14 shows how to subtract vectors.

Copyright © 2012 Pearson Education Inc. Addition of two vectors at right angles First add the vectors graphically. Then use trigonometry to find the magnitude and direction of the sum. R 2 = (1 km) 2 + (2 km) 2 tan  = (2 km)/(1 km) Follow Example 1.5.

Copyright © 2012 Pearson Education Inc. Practicing addition/vector-addition_en.html Click on “Show Sum” button. You may have to slide the sum vector to make sense of it.

Copyright © 2012 Pearson Education Inc. Unit vectors—Figures 1.23–1.24 A unit vector has a magnitude of 1 with no units. It carries ALL the direction information. The unit vector î points in the +x-direction, points in the +y- direction, and points in the +z-direction. Any vector can be expressed in terms of its components as A =A x î+ A y + A z. Follow Example 1.9. 

Copyright © 2012 Pearson Education Inc. Components of a vector—Figure 1.17 Adding vectors graphically provides limited accuracy. Vector components provide a general method for adding vectors with precision. Any vector can be represented by an x-component A x and a y- component A y. Use trigonometry to find the components of a vector: A x = Acos θ A y = Asin θ, where θ is measured from the +x-axis, counter-clockwise.

Copyright © 2012 Pearson Education Inc. If you have brain-flatulence Write the basic trig rules cos  = A x /A sin  = A y /A tan  = A y /A x Then do the algebra to isolate what you want to “use” EX: A(sin  = (A y /A)A => A sin  = A y

Copyright © 2012 Pearson Education Inc. Positive and negative components—Figure 1.18 The components of a vector can be positive or negative numbers, as shown in the figure.

Copyright © 2012 Pearson Education Inc. Finding components—Figure 1.19 We can calculate the components of a vector from its magnitude and direction. Follow Example 1.6. Note: b) is a trap

Copyright © 2012 Pearson Education Inc. Calculations using components We can use the components of a vector to find its magnitude and direction:  = tan -1 (A y /A x ) We can use the components of a set of vectors to find the components of their sum: Refer to Problem-Solving Strategy 1.3. Work this on Board

Copyright © 2012 Pearson Education Inc. The First Way Calculators Can S___ You For those of you who refuse to draw diagrams

Copyright © 2012 Pearson Education Inc. The Second Way Calculators Can S___ You Suppose you wish to calculate 3 / If you type: 3 / 4+2 it may calculate 3 / 6 = 0.5 Check your Calculator Manual for “Precedence of Operators” Method 1: USE parentheses when in doubt (3/4)+2 Method 2: Use the symbol 3 / 4 + 2

Copyright © 2012 Pearson Education Inc. Adding vectors using their components—Figure 1.22 Follow Examples 1.7 and 1.8.

Copyright © 2012 Pearson Education Inc. 1. Multiplying a vector by a scalar If c is a scalar, the product cA has magnitude |c|A. Figure 1.15 illustrates multiplication of a vector by a positive scalar and a negative scalar. 

Copyright © 2012 Pearson Education Inc. 2. The scalar product—Figures 1.25–1.26 The scalar product (also called the “dot product”) of two vectors is the scalar Figures 1.25 and 1.26 illustrate the scalar product.

Copyright © 2012 Pearson Education Inc. Calculating a scalar product [Insert figure 1.27 here] In terms of components, Show on Board. Example 1.10 shows how to calculate a scalar product in two ways.

Copyright © 2012 Pearson Education Inc. Finding an angle using the scalar product Example 1.11 shows how to use components to find the angle between two vectors.

Copyright © 2012 Pearson Education Inc. Practicing prod.html You should come back to this in CH 6 One Mathematical use: sign tells us if vectors are relatively parallel or relatively antiparallel or orthogonal

Copyright © 2012 Pearson Education Inc. 3. The vector cross product—Figures 1.29–1.30 The vector product (“cross product”) of two vectors has magnitude = (Asin  ) B = A(Bsin  ) and the right-hand rule gives its direction. See Figures 1.29 and 1.30.

Copyright © 2012 Pearson Education Inc. Calculating the vector product—Figure 1.32 Use C=ABsin  to find the magnitude and the right-hand rule to find the direction. Refer to Example 1.12.

Copyright © 2012 Pearson Education Inc. Practicing ssprod.html You should come back to this in CH 9

Copyright © 2012 Pearson Education Inc. Vector Product is NOT commutative We call this a “pseudo-vector”, so the Math Department doesn’t have caniption fits! REMEMBER, IN THIS ROOM NATURE DRIVES MATH, AND NOT THE REVERSE!

Copyright © 2012 Pearson Education Inc. Differentiating Vectors Because vector dot products contain BOTH a magnitude of each and direction, a Vector can vary in either or both. d( )/dt = d(|A| |B| cos  )/dt = d|A|/dt B cos  + |A| d|B|/dt cos  + |A| |B| (- sin  d  /dt Note: this says the dot product can change/vary when either vector changes length or when the angle between them changes or any combination! We won’t use this for a while, but it is here for completeness