Units, Physical Quantities and Vectors


 Hilda Hill
 1 years ago
 Views:
Transcription
1 Chapter 1 Units, Physical Quantities and Vectors 1.1 Nature of physics Mathematics. Math is the language we use to discuss science (physics, chemistry, biology, geology, engineering, etc.) Not all of the mathematical ideas were (so far ) applied to sciences, but it is quite remarkable to see how very abstract mathematical concepts (one way or another) find their way into science. It was even conjectured (relatively recently) that may be all of the mathematics has a physical realization somewhere. There are even evidences for that (i.e. multiverse theories), but it is perhaps too speculative to discuss here. Physics. Physics is built on top of mathematics and serves as a foundation for other sciences. Whether it bridges all of mathematics or only some branches of mathematics we will never know. In fact one can never prove anything in physics. This should not be surprising given that even pure mathematics su ers from incompleteness (Godel s theorems). The best that we can do is to use the language of mathematics to construct physical theories to describe the world around us. The description might not be perfect, but it is remarkable how well we can do. So one can think of physics as a toolbox of ideas which can be used in building scientific models in chemistry, biology, geology, astronomy, engineering, etc. Thus the more tools are in your toolbox the better scientist you will be. In other words the more physics you know the better chemist, biologist, geologist, engineer you will be. But if you really like is to design new tools then you should think about becoming a physicist. 1
2 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS Solving physics problems Concepts. In most disciplines the more material you can memorized the better your final grade will be. This is not the case in physics. In fact it is not the material that you have to learn, but the socalled physical concepts and physical laws that you have to understand. (If you will decide to become a physicists, then in few years you might learn that there is actually only a single very deep concept, known as the least action principle, and almost everything follows from it.) So the main issue in physics is to learn physics concepts (e.g. Newton laws, conservation of energy, momentum, etc.). And learning how to learn physics concepts will be your first and perhaps most di cult task. I will try to help you with that, but it will be a lot more e cient if you also help each other. Please discuss concepts, problems, solutions, etc. with each other. Problems. The only way to evaluate if you really understand concepts is to solve problems. Understanding how someone else (e.g. classmates, professors, etc.) solves problems is good, but definitely not enough. Yes, you might have hard time solving problems at first, but it is absolutely essential to learn how to solve problems on your own. The book provides useful strategies and I encourage you to try to apply these strategies, but I am afraid only practice can make you an expert. For your own record keep track of how many problems you solved by yourself in each chapter (without anyone helping or googling answers). I bet this number will be strongly correlated with your grade on exams as well as your final grade. 1.3 Standards and Units Units. Physics is an experimental science and experiments involves measurements of, for example, time, length, mass, etc. To express the results of measurements we use units. Units of length, units of time units of mass. There are some standard units just because historically we decided that these units (e.g. seconds, meters, kilograms) are convenient in physical experiments. Not everyone (e.g. American and British vs. European) agrees what should be the standard units and the history of such debates is quite interesting. One the other hand we should always be able to convert from one system of units to another given the dictionary. Time. What is time and arrow of time is a deep philosophical question. We might discuss it a bit in the last day of classes in context of the second law of thermodynamics. At this point all the we have to remember that time is measured in units of seconds, milliseconds, microseconds, nanoseconds and
3 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 3 here is the dictionary: 1 nanosecond 1 ns =10 9 s 1 microsecond 1 µs =10 6 s 1 millisecond 1 ms =10 3 s. (1.1) Length. Despite of the fact that length and time appear to us very di erently, there is a very deep connection (symmetry) between them. We might discuss is briefly when we discuss gravitation, but you might not understand it completely until you take a course in special relativity (PHYS 2021). Meanwhile let just remember the conversion dictionary for units of length 1 nanometer 1 nm =10 9 m 1 micrometer 1 µm =10 6 m 1 millimeter 1 mm =10 3 m 1 centimeter 1 cm =10 2 m 1 kilometer 1 km =10 3 m. (1.2) Mass. Mass is also something very familiar to us in everyday life, but also has very deep properties connecting it so length and time. We might mention it briefly in connection to blackholes, but you would not appreciate until you take a course in general relativity (PHYS 5551) or a math course in di erential geometry. As far as the conversion of units goes we have the following dictionary 1 microgram 1 µg =10 6 g =10 9 kg 1 milligram 1 mg =10 3 g =10 6 kg 1 gram 1 g =10 3 kg. (1.3) Other units can be formed from seconds, meters and kilograms, for example, the unit of speed units of volume or units of force kg m s 2 m s m 3 Newton.
4 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS Converting Units Dimension. Any physical quantity expressed in units of TIME, LENGTH or MASS is said to have dimensions of time, length or mass respectively. More generally one can have physical quantities which have mixed dimensions. For example if d has units of LENGTH and t has units of TIME, then quantity v = d t (1.4) has units of LENGTH/TIME. Evidently Eq. (1.4) has quantities with the same dimension (i.e. LENGTH/TIME) on both sides of the equation. So if you are given that then according to Eq. (1.4) d = 30 m t = 5 s (1.5) v = d t = 30 m 5 s =6 m s. (1.6) This must be true for any equation that you write. Checking that the quantities on both sides of equation have the same dimension is a quick, but very important test that you could do whenever you setup a new equation. If the dimension is not the same than you are doing something wrong. Conversion. Sometimes you will need to convert from one system of units to another. This can always be done with the help of conversion dictionary. For the case of conversion from standard system of units to British system of units the dictionary is: 1 in = 2.54 cm 1 pound = Newtons (1.7) Then, if we are given a quantity in units of speed, then we can convert it from one system of units to another, 360 in cm h =360in h 1 in 4 A 1 h 3600 s B =0.254 cm s. (1.8) Note that although we have converted the units, the dimension of both sides remains the same, i.e. LENGTH/TIME.
5 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS Uncertainty and Significant Figures Experimental Measurements. Measurements are always uncertain, but it was always hoped that by designing a better and better experiment we can improve the uncertainty without limits. It turned out not to be the case. There is a famous uncertainty principle of quantum mechanics, but you will only learn it next year in (PHYS 2021) if you decide to take it. From our point of view uncertainty is nothing but uncertainty in measurements. This (as well as significant figures) will be discussed in your lab course. 1.6 Estimates and Order of Magnitude Theoretical Estimates. Similarly to uncertainties in experimental measurements, theoretical predictions are never exact. We always make simplifying assumption and thus the best we can hope for is an estimate for the physical quantities to be measured. A useful tool in such estimates is known as orderofmagnitude estimate (also know as outcome of backoftheenvelope calculations ). Such estimates are often done using the socalled dimensional analysis  i.e. just use the known quantities to form a quantity with the dimension of the quantity that you are looking for. 1.7 Vectors and Vector Addition Vector. Some physical quantities are describe by a single real number. We call these quantities  scalar quantities or scalars. Examples are temperature, mass, density, etc. T = 36.6 C m = 78 kg fl = 1.05 g m 3. (1.9) Other quantities also have a direction associated with them and thus are describe by three real numbers. Ą =(A x,a y,a z ). (1.10)
6 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 6 We call these quantities  vector quantities or vectors. For example, position, velocity, acceleration, etc. r = (1 m, 2 m, 3 m) v = (4 m s, 5 m s, 6 m s ) ą = (7 m s, 8 m 2 s, 9 m ). (1.11) 2 s2 There are even more complicated physical quantities  called tensor quantities or tensors  but we will not discuss them in this course. Apart from the number of real numbers which describe these physical quantities, the scalars and vectors (and also tensors) change very di erently under the change of coordinates. Graphical representation. When dealing with vectors it is often useful to draw a picture. Here is how it is done: Vectors are nothing but straight arrows drawn from one point to another. Zero vector is just a vector of zero length  a point. Length of vectors is the magnitude of vectors. The longer the arrow the bigger the magnitude. It is assumed that vectors can be parallel transported around. If you attach beginning of vector Ą to end of another vector B then the vector Ą + B is a straight arrow from begging of vector Ą to end of vector B.
7 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 7 Coordinates. The space around us does not have axis and labels, but we can imagine that these x, y and z axis or the coordinate system to be there. This makes it possible to talk about position of, for example, point particles using their coordinates  real numbers. Since one needs three real numbers to specify position it is a vector. Similarly, velocity, acceleration and force are all vectors. Symmetries. You might complain that there is arbitrariness in how one chooses coordinate system and you would be right. However, it turns out that the physically observable quantities do not depend on the choice of coordinate systems and thus one can choose it to be whatever is more convenient. Moreover, this symmetry is an extremely deep property which gives rise to conservation laws that we will learn in this course. For example the arbitrariness of choosing the x, y and z coordinates gives rise to momentum conservation. And the arbitrariness in choosing time coordinate gives rise to Energy conservation.
8 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS Components of Vectors Magnitude. The length of vector or magnitude is a scalar quantity or in components For example the length of vector is Ą = Ą = A (1.12) (A x,a y,a z )= Ò A 2 x + A 2 y + A 2 z. (1.13) Ą =(3, 4, 5) (1.14) Ą = Ô = Ô (1.15) Direction. One can also find direction of vector using trigonometric identities. For example, in two dimensions Ą =(A x,a y, 0) (1.16) the angle of the vector measure from xaxis is given by tan = A y A x or =arctan A y A x. (1.17)
9 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 9 Addition. Two vectors can be added together to get a new vector an in component form C = Ą + B (1.18) (C x,c y,c z )=(A x,a y,a z )+(B x,b y,b z )=(A x + B x,a y + B y,a z + B z ). (1.19) For example, the sum of two vectors Ą = (3, 4, 5) B = (6, 7, 8) (1.20) is (3, 4, 5) + (6, 7, 8) = (9, 11, 13). (1.21)
10 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS Unit Vectors Unit vectors. Is a vector that has magnitude one. Its is usually denoted with a hat : û = Ò u 2 x + u 2 y + u 2 z =1. (1.22) For example or simply (1/ Ô 2, 1/ Ô 2, 0) = (1/ Ô 3, 1/ Ô 3, 1/ Ô 3) = (1, 0, 0) = Ô =1 (0, 1, 0) = Ô =1 Ò 1/2+1/2+0=1 Ò 1/3+1/3+1/3 =1 (1.23) (0, 0, 1) = Ô =1. (1.24) In fact the last three vectors are so important that there are special letters reserved to denote these vectors î = (1, 0, 0) ĵ = (0, 1, 0) ˆk = (0, 0, 1). (1.25) Multiplication / division by scalar. Any vector can be multiplied by a scalar to obtain another vector, In components from Ą = C B. (1.26) (A x,a y,a z )=C(B x,b y,b z )=(CB x,cb y,cb z ). (1.27)
11 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 11 Thus one can make a unit vector out of any vector Q R Â = Ą Ą = (A x,a y,a z ) A Ò = a x A y A z Ò, Ò, Ò b. A 2 x + A 2 y + A 2 z A 2 x + A 2 y + A 2 z A 2 x + A 2 y + A 2 z A 2 x + A 2 y + A 2 z For example if then Â = Ą =(2, 3, 4) (1.28) A 2 Ô 29, 2 Ô 29, B 2 Ô. (1.29) 29 Components. Note that any vector can be written in components in two equivalent ways: just because Ą =(A x,a y,a z )=A x î + A y ĵ + A z ˆk (1.30) A x î + A y ĵ + A z ˆk = Ax (1, 0, 0) + A y (0, 1, 0) + A z (0, 0, 1) Product of vectors = (A x, 0, 0) + (0,A y, 0) + (0, 0,A z ) = (A x,a y,a z ) (1.31) Scalar (or dot) product. Dot product is a multiplication between two vectors which produces a scalar: In components Ą B = B Ą = C (1.32) Ą B =(A x,a y,a z ) (B x,b y,b z )=A x B x + A y B y + A z B z = AB cos( ). (1.33)
12 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 12
13 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 13 One can also derive multiplication table for unit vectors î î =(1, 0, 0) (1, 0, 0) = = 1 ĵ ĵ =(0, 1, 0) (0, 1, 0) = = 1 ˆk ˆk =(0, 0, 1) (0, 0, 1) = = 1 î ĵ =(1, 0, 0) (0, 1, 0) = = 0 ĵ ˆk =(0, 1, 0) (0, 0, 1) = = 0 ˆk î =(0, 0, 1) (0, 0, 1) = = 0. (1.34) But then we can also apply the second component representation of vectors (given by Eq. (1.30)) to check that Eq. (1.33) is indeed correct Ą B = (A x î + A y ĵ + A z ˆk) (Bx î + B y ĵ + B z ˆk) = = A x î B x î + A x î B y ĵ + A x î B z ˆk + +A y ĵ B x î + A y ĵ B y ĵ + A y ĵ B z ˆk + +A z ˆk Bx î + A z ˆk By ĵ + A z ˆk Bz ˆk = A x B x + A y B y + A z B z. (1.35) Vector (or cross) product. Cross product is a multiplication between two vectors which produces a vector: In components Ą B = B Ą = C (1.36) Ą B =(A x,a y,a z ) (B x,b y,b z ) (A y B z A z B y,a z B x A x B z,a x B y A y B x )= C. (1.37) One can also derive multiplication table for unit vectors î î =(1, 0, 0) (1, 0, 0) = (0, 0, 0) ĵ ĵ =(0, 1, 0) (0, 1, 0) = (0, 0, 0) ˆk ˆk =(0, 0, 1) (0, 0, 1) = (0, 0, 0) î ĵ =(1, 0, 0) (0, 1, 0) = (0, 0, 1) = ˆk = ĵ î ĵ ˆk =(0, 1, 0) (0, 0, 1) = (0, 0, 1) = î = ˆk ĵ ˆk î =(0, 0, 1) (0, 0, 1) = (0, 0, 1) = ĵ = î ˆk. (1.38) But then we can also apply the second component representation of vectors
14 CHAPTER 1. UNITS, PHYSICAL QUANTITIES AND VECTORS 14 (given by Eq. (1.30)) to check that Eq. (1.37) is indeed correct Ą B = (A x î + A y ĵ + A z ˆk) (Bx î + B y ĵ + B z ˆk) = = A x î B x î + A x î B y ĵ + A x î B z ˆk + +A y ĵ B x î + A y ĵ B y ĵ + A y ĵ B z ˆk + +A z ˆk Bx î + A z ˆk By ĵ + A z ˆk Bz ˆk = (A y B z A z B y ) î +(A z B x A x B z ) ĵ +(A x B y A y B x )(1.39) ˆk. Note that the magnitude of cross product is given by Ą B = AB sin but the direction is determined by righthand rule:
REVIEW OVER VECTORS. A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example.
REVIEW OVER VECTORS I. Scalars & Vectors: A scalar is a quantity that is defined by its value only. This value can be positive, negative or zero Example mass = 5 kg A vector is a quantity that can be described
More informationChapter 1 Units, Physical Quantities, and Vectors
Chapter 1 Units, Physical Quantities, and Vectors 1 The Nature of Physics Physics is an experimental science. Physicists make observations of physical phenomena. They try to find patterns and principles
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationPHYSICS 1.2. BLACKBOARD COURSE (Developed by Dr. V.S. Boyko) PHYS SCI Core. 4 cl hrs, 2 lab hrs, 4 cr
BLACKBOARD COURSE (Developed by Dr. V.S. Boyko) PHYS 1433 PHYSICS 1.2 SCI Core 4 cl hrs, 2 lab hrs, 4 cr Basic concepts and principles of mechanics, heat and sound for liberal arts students and technology
More informationLab 2: Vector Analysis
Lab 2: Vector Analysis Objectives: to practice using graphical and analytical methods to add vectors in two dimensions Equipment: Meter stick Ruler Protractor Force table Ring Pulleys with attachments
More informationGeneral Physics 1. Class Goals
General Physics 1 Class Goals Develop problem solving skills Learn the basic concepts of mechanics and learn how to apply these concepts to solve problems Build on your understanding of how the world works
More informationAP Physics 1 and 2 Lab Investigations
AP Physics 1 and 2 Lab Investigations Student Guide to Data Analysis New York, NY. College Board, Advanced Placement, Advanced Placement Program, AP, AP Central, and the acorn logo are registered trademarks
More informationLab: Vectors. You are required to finish this section before coming to the lab. It will be checked by one of the lab instructors when the lab begins.
Lab: Vectors Lab Section (circle): Day: Monday Tuesday Time: 8:00 9:30 1:10 2:40 Name Partners PreLab You are required to finish this section before coming to the lab. It will be checked by one of the
More informationVectors. Physics 115: Spring Introduction
Vectors Physics 115: Spring 2005 Introduction The purpose of this laboratory is to examine the hypothesis that forces are vectors and obey the law of vector addition and to observe that the vector sum
More informationProf. Rupak Mahapatra
Prof. Rupak Mahapatra Today s class Go through the webpage/syllabus policies grading scheme what s (still relatively) new with 218 general strategies Chapter 0 math review scientific notation algebra simultaneous
More informationA.P. Physics/Physics Notes Chapter 1 Physics is the study of the nature of matter and energy and its interactions in the
A.P. Physics/Physics Notes Chapter 1 Physics is the study of the nature of matter and energy and its interactions in the fields of mechanics (kinematics and dynamics), electricity and magnetism, waves
More informationPHYS 1111L LAB 2. The Force Table
In this laboratory we will investigate the vector nature of forces. Specifically, we need to answer this question: What happens when two or more forces are exerted on the same object? For instance, in
More informationChapter 1 Introduction to Physics. Copyright 2010 Pearson Education, Inc.
Chapter 1 Introduction to Physics Physics and the Laws of Nature Units of Length, Mass, and Time Dimensional Analysis Significant Figures Converting Units OrderofMagnitude Calculations Scalars and Vectors
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More information2 Session Two  Complex Numbers and Vectors
PH2011 Physics 2A Maths Revision  Session 2: Complex Numbers and Vectors 1 2 Session Two  Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar
More informationEquilibrium of Concurrent Forces (Force Table)
Equilibrium of Concurrent Forces (Force Table) Objectives: Experimental objective Students will verify the conditions required (zero net force) for a system to be in equilibrium under the influence of
More informationUnified Lecture # 4 Vectors
Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,
More information3. KINEMATICS IN TWO DIMENSIONS; VECTORS.
3. KINEMATICS IN TWO DIMENSIONS; VECTORS. Key words: Motion in Two Dimensions, Scalars, Vectors, Addition of Vectors by Graphical Methods, Tail to Tip Method, Parallelogram Method, Negative Vector, Vector
More informationAP PHYSICS C Mechanics  SUMMER ASSIGNMENT FOR 20162017
AP PHYSICS C Mechanics  SUMMER ASSIGNMENT FOR 20162017 Dear Student: The AP physics course you have signed up for is designed to prepare you for a superior performance on the AP test. To complete material
More informationSwapping Units 1) X = = 2) X = = 3) X = = 4) X = = Name: Period: MULTIPLYING BY ONE
Name: Period: Swapping Units Swap Meet is a game about trading one thing for another. In the game, we are not trying to end up with stuff that is worth MORE than what we started with. Instead, we simply
More informationSection 1 Tools and Measurement
Section 1 Tools and Measurement Key Concept Scientists must select the appropriate tools to make measurements and collect data, to perform tests, and to analyze data. What You Will Learn Scientists use
More informationUnits, Physical Quantities, and Vectors
Chapter 1 Units, Physical Quantities, and Vectors PowerPoint Lectures for University Physics, Thirteenth Edition Hugh D. Young and Roger A. Freedman Lectures by Wayne Anderson Goals for Chapter 1 To learn
More informationVector has a magnitude and a direction. Scalar has a magnitude
Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude
More informationSection 1.1. Introduction to R n
The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to
More informationPHYSICAL QUANTITIES AND UNITS
1 PHYSICAL QUANTITIES AND UNITS Introduction Physics is the study of matter, its motion and the interaction between matter. Physics involves analysis of physical quantities, the interaction between them
More informationProblem Set 6: Solutions
UNIVERSITY OF ALABAMA Department of Physics and Astronomy PH 15 / LeClair Spring 009 Problem Set 6: Solutions Solutions not yet completed: Halliday, Resnick, & Walker, problems 9.48, 9.53, 9.59, and 9.68
More informationUnits and Vectors: Tools for Physics
Chapter 1 Units and Vectors: Tools for Physics 1.1 The Important Stuff 1.1.1 The SI System Physics is based on measurement. Measurements are made by comparisons to well defined standards which define the
More informationAP Physics Applying Forces
AP Physics Applying Forces This section of your text will be very tedious, very tedious indeed. (The Physics Kahuna is just as sorry as he can be.) It s mostly just a bunch of complicated problems and
More informationSUMMING VECTOR QUANTITIES USING PARALELLOGRAM METHOD
EXPERIMENT 2 SUMMING VECTOR QUANTITIES USING PARALELLOGRAM METHOD Purpose : Summing the vector quantities using the parallelogram method Apparatus: Different masses between 11000 grams A flat wood, Two
More informationLecture Outline Chapter 1. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc.
Lecture Outline Chapter 1 Physics, 4 th Edition James S. Walker Chapter 1 Introduction to Physics Units of Chapter 1 Physics and the Laws of Nature Units of Length, Mass, and Time Dimensional Analysis
More informationDimensions, Units and Conversions
Dimensions, Units and Conversions Science depends on measured numbers, most of which have units. The following sections should help you understand what units are, how they are used, and how they are converted
More informationPhysics Lab Report Guidelines
Physics Lab Report Guidelines Summary The following is an outline of the requirements for a physics lab report. A. Experimental Description 1. Provide a statement of the physical theory or principle observed
More informationAccuracy, Precision and Uncertainty
Accuracy, Precision and Uncertainty How tall are you? How old are you? When you answered these everyday questions, you probably did it in round numbers such as "five foot, six inches" or "nineteen years,
More informationScientific Notation. Measurement Systems. How to Write a Number in Scientific Notation
Scientific Notation The star Proxima Centauri is 23,400,000,000,000 miles away from Earth. If we could travel in a spaceship at 5,000 miles/hour, it would take over 534,000 years to get there. Scientific
More informationScientific Notation: Significant Figures:
Honors Chemistry Riverside STEM Academy Mrs. Hampton All of the material from Chapter 3 of our text book, Prentice Hall Chemistry, has been covered in our 8 th grade science course. Please review this
More informationName: Date: CRN: Scientific Units of Measurement & Conversion Worksheet
Name: Date: CRN: Scientific Units of Measurement & Conversion Worksheet Principle or Rationale: Scientific measurements are made and reported using the metric system and conversion between different units
More informationName: Date: CRN: Bio 210A Scientific Units of Measurement & Conversion Worksheet
Name: Date: CRN: Bio 210A Scientific Units of Measurement & Conversion Worksheet Principle or Rationale: Scientific measurements are made and reported using the metric system and conversion between different
More informationAP Physics C Summer Assignment
AP Physics C Summer Assignment Welcome to AP Physics C! It is a college level physics course that is fun, interesting and challenging on a level you ve not yet experienced. This summer assignment will
More informationNewton s Third Law, Momentum, Center of Mass
Team: Newton s Third Law, Momentum, Center of Mass Part I. Newton s Third Law Atomic Springs When you push against a wall, you feel a force in the opposite direction. The harder you push, the harder the
More informationWorked Examples from Introductory Physics Vol. I: Basic Mechanics. David Murdock Tenn. Tech. Univ.
Worked Examples from Introductory Physics Vol. I: Basic Mechanics David Murdock Tenn. Tech. Univ. February 24, 2005 2 Contents To the Student. Yeah, You. i 1 Units and Vectors: Tools for Physics 1 1.1
More informationGeneral Physics I (aka PHYS 2013)
General Physics I (aka PHYS 2013) PROF. VANCHURIN (AKA VITALY) University of Minnesota, Duluth (aka UMD) OUTLINE CHAPTER 1 CHAPTER 2 CHAPTER 3 CHAPTER 4 SECTION 1.1: THE NATURE OF PHYSICS Mathematics is
More informationClicker Registration
Reminders Website: https://ulysses.phys.wvu.edu/plasma/?q=physics101 Lecture slides online after class, as well as other stuff Join the Facebook group discussion page: www.facebook.com/groups/645609775590472
More informationChapter 1 An Introduction to Chemistry
1 Chapter 1 An Introduction to Chemistry 1.1 What Is Chemistry, and What Can Chemistry Do for You? Special Topic 1.1: Green Chemistry 1.2 Suggestions for Studying Chemistry 1.3 The Scientific Method 1.4
More informationChapter Test B. Chapter: Measurements and Calculations
Assessment Chapter Test B Chapter: Measurements and Calculations PART I In the space provided, write the letter of the term or phrase that best completes each statement or best answers each question. 1.
More informationPHYSICS Unit 1A Kinematics 1
PHYSICS 2204 Unit 1A Kinematics 1 SI Units Metric Prefixes Factor Prefix Symbol 10 12 tera T 10 9 giga G 10 6 mega M 10 3 kilo k 10 2 hecto h 10 0 Standard unit  101 deci d 102 centi c 103 milli
More informationCHAPTER 3 NEWTON S LAWS OF MOTION
CHAPTER 3 NEWTON S LAWS OF MOTION NEWTON S LAWS OF MOTION 45 3.1 FORCE Forces are calssified as contact forces or gravitational forces. The forces that result from the physical contact between the objects
More informationE X P E R I M E N T 8
E X P E R I M E N T 8 Torque, Equilibrium & Center of Gravity Produced by the Physics Staff at Collin College Copyright Collin College Physics Department. All Rights Reserved. University Physics, Exp 8:
More informationAP1 Oscillations. 1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false?
1. Which of the following statements about a springblock oscillator in simple harmonic motion about its equilibrium point is false? (A) The displacement is directly related to the acceleration. (B) The
More information2016 Arizona State University Page 1 of 15
NAME: MATH REFRESHER ANSWER SHEET (Note: Write all answers on this sheet and the following graph page.) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27.
More informationPhysics 53. Rotational Motion 1. We're going to turn this team around 360 degrees. Jason Kidd
Physics 53 Rotational Motion 1 We're going to turn this team around 360 degrees. Jason Kidd Rigid bodies To a good approximation, a solid object behaves like a perfectly rigid body, in which each particle
More informationPart I. Basic Maths for Game Design
Part I Basic Maths for Game Design 1 Chapter 1 Basic Vector Algebra 1.1 What's a vector? Why do you need it? A vector is a mathematical object used to represent some magnitudes. For example, temperature
More informationMechanics 1. Revision Notes
Mechanics 1 Revision Notes July 2012 MECHANICS 1... 2 1. Mathematical Models in Mechanics... 2 Assumptions and approximations often used to simplify the mathematics involved:... 2 2. Vectors in Mechanics....
More informationELEMENTS OF VECTOR ALGEBRA
ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions
More information2. List the fundamental measures and the units of measurement for each. 3. Given the appropriate information, calculate force, work and power.
Lab Activity 1 The WarmUp Interpretations Student Learning Objectives After Completing this lab, you should be able to: 1. Define, explain and correctly use the key terms. 2. List the fundamental measures
More informationVector Algebra II: Scalar and Vector Products
Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define
More informationv w is orthogonal to both v and w. the three vectors v, w and v w form a righthanded set of vectors.
3. Cross product Definition 3.1. Let v and w be two vectors in R 3. The cross product of v and w, denoted v w, is the vector defined as follows: the length of v w is the area of the parallelogram with
More informationUnit 3.3 Metric System (System International)
Unit 3.3 Metric System (System International) How long is a yard? It depends on whom you ask and when you asked the question. Today we have a standard definition of the yard, which you can see marked on
More informationReview of Scientific Notation and Significant Figures
II1 Scientific Notation Review of Scientific Notation and Significant Figures Frequently numbers that occur in physics and other sciences are either very large or very small. For example, the speed of
More informationLecture 6. Weight. Tension. Normal Force. Static Friction. Cutnell+Johnson: 4.84.12, second half of section 4.7
Lecture 6 Weight Tension Normal Force Static Friction Cutnell+Johnson: 4.84.12, second half of section 4.7 In this lecture, I m going to discuss four different kinds of forces: weight, tension, the normal
More informationPhysics 170 Week 9, Lecture 3
Physics 170 Week 9, Lecture http://www.phas.ubc.ca/ gordonws/170 Physics 170 20 Week 9, Lecture 1 Textbook Chapter 1: Section 1.6 Physics 170 20 Week 9, Lecture 2 Learning Goals: After this lecture, you
More informationMaths Pack. For the University Certificates in Astronomy and Cosmology
Maths Pack Distance Learning Mathematics Support Pack For the University Certificates in Astronomy and Cosmology These certificate courses are for your enjoyment. However, a proper study of astronomy or
More informationHow do I do that unit conversion?
Name Date Period Furrey How do I do that unit conversion? There are two fundamental types of unit conversions in dimensional analysis. One is converting old units to new units with in the same system of
More informationDimensional Analysis I: Length, Mass, Volume and Area Conversions
EXPERIMENT Dimensional Analysis I: Length, Mass, Volume and Area Conversions Student will use the Dimensional Analysis Map to introduce / reinforce the process of basic unit conversions in chemistry. OBJECTIVE
More informationReview A: Vector Analysis
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Review A: Vector Analysis A... A0 A.1 Vectors A2 A.1.1 Introduction A2 A.1.2 Properties of a Vector A2 A.1.3 Application of Vectors
More informationMechanics 1: Vectors
Mechanics 1: Vectors roadly speaking, mechanical systems will be described by a combination of scalar and vector quantities. scalar is just a (real) number. For example, mass or weight is characterized
More informationU Step by Step: Conversions. UMultipliersU UExample with meters:u (**Same for grams, seconds & liters and any other units!
U Step by Step: Conversions SI Units (International System of Units) Length = meters Temperature: o C = 5/9 ( o F 32) Mass = grams (really kilograms) K = o C + 273 Time = seconds Volume = liters UMultipliersU
More informationAS Physics INDUCTION WORK XKCD. Student. Class 12 A / B / C / D Form
AS Physics 201415 INDUCTION WORK XKCD Student Class 12 A / B / C / D Form MCC 2014 1. Physical Quantities Maths and Physics have an important but overlooked distinction by students. Numbers in Physics
More informationExample SECTION 131. XAXIS  the horizontal number line. YAXIS  the vertical number line ORIGIN  the point where the xaxis and yaxis cross
CHAPTER 13 SECTION 131 Geometry and Algebra The Distance Formula COORDINATE PLANE consists of two perpendicular number lines, dividing the plane into four regions called quadrants XAXIS  the horizontal
More informationPosition, Velocity, and Acceleration
Chapter 2 Position, Velocity, and Acceleration In this section, we will define the first set of quantities (position, velocity, and acceleration) that will form a foundation for most of what we do in this
More informationA. The Metric System (this section can be completed before or during lab)
Lab I: INTRODUCTION TO MEASURING TECHNIQUES The basis for obtaining scientific knowledge is systematic observation and experimentation. A key component of most scientific investigations is measurement.
More informationPH2213 : Examples from Chapter 4 : Newton s Laws of Motion. Key Concepts
PH2213 : Examples from Chapter 4 : Newton s Laws of Motion Key Concepts Newton s First and Second Laws (basically Σ F = m a ) allow us to relate the forces acting on an object (lefthand side) to the motion
More informationWEEK 6: FORCE, MASS, AND ACCELERATION
Name Date Partners WEEK 6: FORCE, MASS, AND ACCELERATION OBJECTIVES To develop a definition of mass in terms of an object s acceleration under the influence of a force. To find a mathematical relationship
More informationDay 18 ENERGY CONSERVATION. 1 Introduction: More Kinds of Energy
Day 18 ENERGY CONSERVATION 1 Introduction: More Kinds of Energy Suppose I move an object between two points in space. Also suppose that a force acts on the object as it moves. If the work done by the force
More informationSCIENTIFIC MEASUREMENT
3 SCIENTIFIC MEASUREMENT Conceptual Curriculum Concrete concepts More abstract concepts or math/problemsolving Standard Curriculum Core content Extension topics Honors Curriculum Core honors content Options
More informationUnits of Measurement and Dimensional Analysis
POGIL ACTIVITY.2 POGIL ACTIVITY 2 Units of Measurement and Dimensional Analysis A. Units of Measurement The SI System and Metric System T here are myriad units for measurement. For example, length is
More informationVector Lab Teacher s Guide
Vector Lab Teacher s Guide Objectives: 1. Use s to show addition of force vectors. 2. Vector addition using tiptotail method and trigonometry. Materials: Each group must have: 2 ring stands, 2 s, 4 washers,
More information1 of 6 9/5/2009 6:13 PM
1 of 6 9/5/2009 6:13 PM Chapter 1 Homework Due: 9:00am on Tuesday, September 1, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]
More informationDimensional Analysis is a simple method for changing from one unit of measure to another. How many yards are in 49 ft?
HFCC Math Lab NAT 05 Dimensional Analysis Dimensional Analysis is a simple method for changing from one unit of measure to another. Can you answer these questions? How many feet are in 3.5 yards? Revised
More informationCHAPTER 4 DIMENSIONAL ANALYSIS
CHAPTER 4 DIMENSIONAL ANALYSIS 1. DIMENSIONAL ANALYSIS Dimensional analysis, which is also known as the factor label method or unit conversion method, is an extremely important tool in the field of chemistry.
More informationEDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1  LOADING SYSTEMS
EDEXCEL NATIONAL CERTIFICATE/DIPLOMA MECHANICAL PRINCIPLES AND APPLICATIONS NQF LEVEL 3 OUTCOME 1  LOADING SYSTEMS TUTORIAL 1 NONCONCURRENT COPLANAR FORCE SYSTEMS 1. Be able to determine the effects
More informationThe car is pulled up a long hill. 2. Does the roller coaster ever get higher than the first hill? No.
Roller Coaster Physics Answer Key Vocabulary: friction, gravitational potential energy, kinetic energy, momentum, velocity Prior Knowledge Questions (Do these BEFORE using the Gizmo.) [Note: The purpose
More informationI look forward to seeing you next year, Mr. Koch
AP Physics 1 Summer Assignment Welcome to AP Physics 1! I am very glad you are interested in physics and I look forward to investigating energy, matter and well the universe with you next year. We will
More information1 GRAM = HOW MANY MILLIGRAMS?
1 GRAM = HOW MANY MILLIGRAMS? (1) Take the 1gram expression and place the decimal point in the proper place. 1 gram is the same as 1.0 gram (decimal point in place) (2) Move that decimal point three places
More information1.2 ERRORS AND UNCERTAINTIES Notes
1.2 ERRORS AND UNCERTAINTIES Notes I. UNCERTAINTY AND ERROR IN MEASUREMENT A. PRECISION AND ACCURACY B. RANDOM AND SYSTEMATIC ERRORS C. REPORTING A SINGLE MEASUREMENT D. REPORTING YOUR BEST ESTIMATE OF
More informationIn order to describe motion you need to describe the following properties.
Chapter 2 One Dimensional Kinematics How would you describe the following motion? Ex: random 1D path speeding up and slowing down In order to describe motion you need to describe the following properties.
More informationPhysics of Rocket Flight
Physics of Rocket Flight In order to understand the behaviour of rockets it is necessary to have a basic grounding in physics, in particular some of the principles of statics and dynamics. This section
More informationExperiment 1 The Metric System and Measurement Tools
Experiment 1 The Metric System and Measurement Tools Preparation Prepare for this week's quiz by reading about the metric system in your textbook. Read the introduction to the manual and the first experiment
More information10 g 5 g? 10 g 5 g. 10 g 5 g. scale
The International System of Units, or the SI Units Vs. Honors Chem 1 LENGTH In the SI, the base unit of length is the Meter. Prefixes identify additional units of length, based on the meter. Smaller than
More informationLecture L3  Vectors, Matrices and Coordinate Transformations
S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3  Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between
More information2 Using the definitions of acceleration and velocity
Physics I [P161] Spring 2008 Review for Quiz # 3 1 Main Ideas Two main ideas were introduced since the last quiz. 1. Using the definitions of acceleration and velocity to obtain equations of motion (chapter
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationSample problems: A. What s the concentration of a solution containing 0.1 moles of NaCl in 0.25 liters of water?
EEWWWW. CHEMISTRY!! Or, dumb math problems Dr. Moss says I gotta know how to do for lab. I m not going to collect these assignments and grade them. However, there WILL be at least one math question on
More information6. Vectors. 1 20092016 Scott Surgent (surgent@asu.edu)
6. Vectors For purposes of applications in calculus and physics, a vector has both a direction and a magnitude (length), and is usually represented as an arrow. The start of the arrow is the vector s foot,
More informationThe following is Dirac's talk on projective geometry in physics at Boston University on October 30, 1972.
Audio recording made by John B. Hart, Boston University, October 30, 1972 Comments by Roger Penrose [JOHN HART] The following is Dirac's talk on projective geometry in physics at Boston University on October
More informationMetric System. The tables above lists the most common metric prefixes and their relationship to the central unit that has no prefix.
Metric System Many properties of matter are quantitative; that is, they are associated with numbers. When a number represents a measured quantity, the unit of that quantity must always be specified. To
More informationBASIC ELECTRONICS PROF. T.S. NATARAJAN DEPT OF PHYSICS IIT MADRAS LECTURE4 SOME USEFUL LAWS IN BASIC ELECTRONICS
BASIC ELECTRONICS PROF. T.S. NATARAJAN DEPT OF PHYSICS IIT MADRAS LECTURE4 SOME USEFUL LAWS IN BASIC ELECTRONICS Hello everybody! In a series of lecture on basic electronics, learning by doing, we now
More informationGravitational Potential Energy
Gravitational Potential Energy Consider a ball falling from a height of y 0 =h to the floor at height y=0. A net force of gravity has been acting on the ball as it drops. So the total work done on the
More informationScientific Notation, Engineering Notation
Scientific, Engineering Scientific Scientific or Standard Form is a way of writing numbers in a compact form. A number written in Scientific is expressed as a number from 1 to less than multiplied by a
More information6 EXTENDING ALGEBRA. 6.0 Introduction. 6.1 The cubic equation. Objectives
6 EXTENDING ALGEBRA Chapter 6 Extending Algebra Objectives After studying this chapter you should understand techniques whereby equations of cubic degree and higher can be solved; be able to factorise
More informationPhysics 2101, First Exam, Fall 2007
Physics 2101, First Exam, Fall 2007 September 4, 2007 Please turn OFF your cell phone and MP3 player! Write down your name and section number in the scantron form. Make sure to mark your answers in the
More information