Physics1_CH1_Section10.pdf

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 Copyright © 2012 Pearson Education Inc.  G...

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 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.  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. Copyright © 2012 Pearson Education Inc.  Solving problems in physics A problem-solving strategy offers techniques for setting up and solving problems efficiently and accurately. Copyright © 2012 Pearson Education Inc. Ex convert 230 am to inch 1 inch 2.54 cm given 2304mA I q a men 90.55inch 230am Ex convert 120 given 1 mile 1609 M I hour 3600 S IF 1 9 120 53.63  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. Copyright © 2012 Pearson Education Inc.  Unit prefixes Table 1.1 shows some larger and smaller units for the fundamental quantities. kilo eared kilo By Kilo mF Gegaf kilo Copyright © 2012 Pearson Education Inc. grand  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.  given I mil 1609 m 1 h 36005 763 341.01 763 141 3 0  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.  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.  0 29 73108160 4130 29.7 29 73 30 Rules nonzero digit is significant any digits are always significant 29 Zeros between non zero significant 19419.71.5 are always NOT 4 trailing zeros at theright a if there is a decimal point they are significant b if there is no decimal point it may or may not be significant 4 0 7.30 3 9 4 7.31 4 1  ex 30.7 a 30 971 If b 0 156 8688 157 1 202 0.6 7 5 0.7 755 432 432.8 d 430.47 2.3 7,1 e 12 17.3 29.3 29 f 86.3 3 2.3 88.72 88.7243  6 2.99792758 18 2 E m 1531 18 9 1 212758 15 J 828 5967 8 8.19 10 145  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.  Vectors and scalars 5 W'die A scalar quantity can be described by a single number. 85190 5.14 5 A vector quantity has both a magnitude and a direction in space. IUD In this book, a vector quantity is represented → in boldface italic type with an arrow over it: A. → → The magnitude of A is written as A or |A|. 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. head tail 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. start end A B TAB AT B  eat w̅ x̅ start  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.  Subtracting start F end B A end B At start B A A B B  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.  Addition of two vectors at right angles First add the vectors graphically. Then use trigonometry to find the magnitude and direction of the sum. Follow Example 1.5. Copyright © 2012 Pearson Education Inc.  Iersan We E 2km end s I 1km 0 2.24 sat ff.de 0.89 0 630 T since 2 4 OR 0.446 6 630 Case OR i 0 63 tano sino 0 89 630 skiff 0.89  Components of a vector—Figure 1.17 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. Use trigonometry to find the components of a vector: Ax = Acos θ and Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis. adjacent Copyright © 2012 Pearson Education Inc.  it 155 155in 35 8.8 1 35 i 1505351 12.29 17 61 12.291 8.6 156535 15 250 No 1554350  i 15 Ex write the components F 2560555 F 20.47 14.33J 14.33 25 255in 55 20.47 - 70 COS 400 - 53.6 4 40 70 N - 70 sin 40 - 45 N - 53.6 - 45 J  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. 0 30545 2.12 370 4.5 4 4.5512 3 35h45 2.12 Copyright © 2012 Pearson Education Inc.  solved 8 155 85 IT 17 1162 158 17 tano E 0 620 3600 298  tan 280 12 9 9 12 tan tan XEII.FI  Calculations using components We can use the components of a vector to find its magnitude Ay and direction: A = Ax + Ay and tanθ = 2 2 A x We can use the components of a set of vectors to find the components of their sum: Rx = Ax + Bx + Cx +, Ry = Ay + By + C y + Refer to Problem-Solving Strategy 1.3. 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. a I É 101 25 7k b 353 181 9 j 3k c 2É 8 105 16k d 253 E 25 121 65 2k  Unit vectors—Figures 1.23–1.24 A unit vector has a magnitude of 1 with no units. The unit vector î points in the +x-direction, j points in the +y- direction, and k points in the +z-direction. Any vector can be expressed in terms of its components as → A =Axî+ Ay j + Az k. Follow Example 1.9. Copyright © 2012 Pearson Education Inc.  vector vector scalar dot product f scalar product vector multiplication vectornector vector cross product vector product  Dot product I B 181181 cos a 0 angle between A and BE i i coso I JJ I R.pl I i f o I 1 A 6590 i k o j k o EI 15 8 27 34  13053 77 5 4 F Ñ I I Blase 4 5 COS 77 4.499 4.5 Sx 551440 4 553 3 21 2.4 13 8 53 3.21 3.838  2.41 3.2J a.IE I 7.704 12.256 4.552  The scalar product—Figures 1.25–1.26 The scalar product (also called the “dot product”) of two vectors   is AB = AB cosφ. Figures 1.25 and 1.26 illustrate the scalar product. Copyright © 2012 Pearson Education Inc.  Calculating a scalar product   In terms of components, AB = Ax Bx + Ay By + Az Bz. Example 1.10 shows how to calculate a scalar product in two ways. [Insert figure 1.27 here] 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.  Cross product vector vector Vector A A 1181 since direction 7 9 sink my 7 21.5 A ATE 1 It BX A R F  6 B 130 A 6 ÑxÑ 4 sinzo k 12  The vector product—Figures 1.29–1.30 The vector product (“cross product”) of two vectors has magnitude   | A× B | = AB sinφ 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 ABsinφ to find the magnitude and the right-hand rule to find the direction. Refer to Example 1.12. Copyright © 2012 Pearson Education Inc. EI 7 ii M B find a w̅ ALIBI 656 9 7 65120 31.5 b AT 7 Asin 2 k 54.51 k 54.56 c B A 54.56