Chapter 1: Units, Physical Quantities, and Vectors PDF

Chapter 1 © 2016 Pearson Education, Ltd.  Learning Goals for Chapter 1 Looking forward at … the four steps you can use to solve any physics problem. three fundamental quantities of physics and the units physicists use to measure them. how to work with units and significan...

Chapter 1 © 2016 Pearson Education, Ltd.  Learning Goals for Chapter 1 Looking forward at … the four steps you can use to solve any physics problem. three fundamental quantities of physics and the units physicists use to measure them. how to work with units and significant figures in your calculations. how to add and subtract vectors graphically, and using vector components. two ways to multiply vectors: the scalar (dot) product and the vector (cross) product. © 2016 Pearson Education, Ltd.  The nature of physics Physics is an experimental science in which physicists seek patterns that relate the phenomena of nature. The patterns are called physical theories. A very well established or widely used theory is called a physical law or principle. © 2016 Pearson Education, Ltd.  Solving problems in physics All of the Problem-Solving Strategies and Examples in this book will follow these four steps: Identify the relevant concepts, target variables, and known quantities, as stated or implied in the problem. Set Up the problem: Choose the equations that you’ll use to solve the problem, and draw a sketch of the situation. Execute the solution: This is where you “do the math.” Evaluate your answer: Compare your answer with your estimates, and reconsider things if there’s a discrepancy. © 2016 Pearson Education, Ltd.  Idealized models To simplify the analysis of (a) a baseball in flight, we use (b) an idealized model. © 2016 Pearson Education, Ltd.  Standards and units Length, time, and mass are three fundamental quantities of physics. 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. © 2016 Pearson Education, Ltd. © 2016 Pearson Education, Ltd. © 2016 Pearson Education, Ltd. © 2016 Pearson Education, Ltd.  Unit prefixes Prefixes can be used to create larger and smaller units for the fundamental quantities. Some examples are: 1 µm = 10−6 m (size of some bacteria and living cells) 1 km = 100 m (a 10-minute walk) 1 mg = 10−6 kg (mass of a grain of salt) 1 g = 10−3 kg (mass of a paper clip) 1 ns = 10−9 s (time for light to travel 0.3 m) © 2016 Pearson Education, Ltd.  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, by forming a ratio of the same physical quantity in two different units, and using it as a multiplier. For example, to find the number of seconds in 3 min, we write: © 2016 Pearson Education, Ltd.  Uncertainty and significant figures 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. As this train mishap illustrates, even a small percent error can have spectacular results! © 2016 Pearson Education, Ltd.  Vectors and scalars A scalar quantity can be described by a single number. A vector quantity has both a magnitude and a direction in space. In this book, a vector quantity is represented in boldface italic type with an arrow over it:. The magnitude of is written as A or | |. © 2016 Pearson Education, Ltd.  Drawing vectors 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. © 2016 Pearson Education, Ltd.  Adding two vectors graphically © 2016 Pearson Education, Ltd.  Adding two vectors graphically © 2016 Pearson Education, Ltd.  Adding two vectors graphically © 2016 Pearson Education, Ltd.  Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order. © 2016 Pearson Education, Ltd.  Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order. © 2016 Pearson Education, Ltd.  Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order. © 2016 Pearson Education, Ltd.  Subtracting vectors © 2016 Pearson Education, Ltd.  Multiplying a vector by a scalar If c is a scalar, the product c has magnitude |c|A. The figure illustrates multiplication of a vector by (a) a positive scalar and (b) a negative scalar. © 2016 Pearson Education, Ltd.  Addition of two vectors at right angles To add two vectors that are at right angles, first add the vectors graphically. Then use trigonometry to find the magnitude and direction of the sum. In the figure, a cross- country skier ends up 2.24 km from her starting point, in a direction of 63.4° east of north. © 2016 Pearson Education, Ltd.  Components of a vector Adding vectors graphically provides limited accuracy. Vector components provide a general method for adding vectors. Any vector can be represented by an x-component Ax and a y-component Ay. © 2016 Pearson Education, Ltd.  Positive and negative components The components of a vector may be positive or negative numbers, as shown in the figures. © 2016 Pearson Education, Ltd.  Finding components We can calculate the components of a vector from its magnitude and direction. © 2016 Pearson Education, Ltd.  Calculations using components We can use the components of a vector to find its magnitude and direction: We can use the components of a set of vectors to find the components of their sum: Refer to Problem-Solving Strategy 1.3. © 2016 Pearson Education, Ltd.  Example 1.7 © 2016 Pearson Education, Ltd.  Unit vectors A unit vector has a magnitude of 1 with no units. The unit vector 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 © 2016 Pearson Education, Ltd. © 2016 Pearson Education, Ltd.  The scalar product © 2016 Pearson Education, Ltd.  The scalar product The scalar product can be positive, negative, or zero, depending on the angle between and. © 2016 Pearson Education, Ltd.  Calculating a scalar product using components In terms of components: The scalar product of two vectors is the sum of the products of their respective components. © 2016 Pearson Education, Ltd.  Finding an angle using the scalar product Example 1.10 shows how to use components to find the angle between two vectors. © 2016 Pearson Education, Ltd.  The vector product If the vector product (“cross product”) of two vectors is then: The direction of the vector product can be found using the right- hand rule: © 2016 Pearson Education, Ltd.  The vector product is anticommutative © 2016 Pearson Education, Ltd.  Calculating the vector product Use ABsinϕ to find the magnitude and the right-hand rule to find the direction. Refer to Example 1.11. © 2016 Pearson Education, Ltd.