Chapter 1 Units, Physical Quantities, and Vectors 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 this lecture 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 2 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. 3 Copyright © 2012 Pearson Education Inc.  Solving problems in physics A problem-solving strategy offers techniques for setting up and solving problems efficiently and accurately. 4 Copyright © 2012 Pearson Education Inc.  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. 5 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! 7 Copyright © 2012 Pearson Education Inc.  Significant Figures – Rules 1) ALL non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant. 2) ALL zeroes between non-zero numbers are ALWAYS significant. 3) ALL zeroes which are SIMULTANEOUSLY to the right of the decimal point AND at the end of the number are ALWAYS significant. 4) ALL zeroes which are to the left of a written decimal point and are in a number >= 10 are ALWAYS significant. A helpful way to check rules 3 and 4 is to write the number in scientific notation. If you can/must get rid of the zeroes, then they are NOT significant. Copyright © 2012 Pearson Education Inc.  Significant Figures - examples Number Number of significant figures a. 8.0100 five b. 0.00326 three c. 8.01 x 10-5 three d. 1.200 x 108 four e. 27300 three f. 27301 five Copyright © 2012 Pearson Education Inc.  Math with Significant Figures - rules 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 (the least precise quantity). 10 Copyright © 2012 Pearson Education Inc.  Math with Significant Figures - examples Operation Calculator displays Number rounds to a. 22.101 - 0.9307 21.1703 21.170 b. 23.1 + 4.77 + 125.39 + 3.581 156.841 156.8 c. (3.4617 x 107) ÷ (5.61 x 10-4) 6.1706 x 1010 6.17 x 1010 d. [(561.0) (34908) (23.0)] ÷ 2280.3972 2280 = (2.28 x 103) [(21.888) (75.2) (120.00)] Copyright © 2012 Pearson Education Inc.  Unit prefixes Table 1.1 shows some larger and smaller units for the fundamental quantities. 13 Copyright © 2012 Pearson Education Inc.  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 the Textbook, 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.  Vectors and scalars a. 5 m Scalar (distance) b. 30 m/s, East Vector (velocity) c. 5 mi., North Vector (displacement) d. 20 degrees Celsius Scalar (temperature) e. 400 Joules Scalar (energy) f. 5 kg Scalar (mass) g. 100 m/s Scalar (speed) 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.  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. Commutative 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. Associative Copyright © 2012 Pearson Education Inc.  Subtracting vectors Figure 1.14 shows how to subtract vectors. Copyright © 2012 Pearson Education Inc.  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.  Trigonometric functions, Pythagorean theorem Sine, cosine and tangent in the four quadrants When Sin, Cos and Tan are positive? Add Sugar To Coffee Pythagorean theorem h2 = a2+b2 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. 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. 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.  Calculations using components We can use the components of a vector to find its magnitude Ay and direction: A = Ax + Ay and tan = 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 +Cy + Refer to Problem-Solving Strategy 1.3. Copyright © 2012 Pearson Education Inc.  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, jj points in the +y- direction, and kk points in the +z-direction. Any vector can be expressed in terms of its components as → A =Axî+ Ay jj + Az kk. Copyright © 2012 Pearson Education Inc.  Unit vectors Using unit vectors, → we can express the vector sum R → R =(Ax + Bx)î+ (Ay + By) jj Rx Ry That also works in three dimensions! → R =(Ax + Bx)î+ (Ay + By) jj + (Az + Bz) kk Follow Example 1.9. Copyright © 2012 Pearson Education Inc.  The scalar product—Figures 1.25–1.26 The scalar product (also called the “dot product”) of two vectors is A B = AB cos. Figures 1.25 and 1.26 illustrate the scalar product. Follow Example 1.10. Copyright © 2012 Pearson Education Inc.  The vector product—Figures 1.29–1.30 The vector product (“cross product”) of two vectors has magnitude | A B | = ABsin and the right- hand rule gives its direction. See Figures 1.29 and 1.30. Copyright © 2012 Pearson Education Inc.  Calculating the vector product In terms of components C= Can be expressed in determinant form Copyright © 2012 Pearson Education Inc.  Lecture Summary Physical quantities and units: Three fundamental physical quantities are mass, length, and time. The corresponding basic SI units are the kilogram, the meter, and the second. Derived units for other physical quantities are products or quotients of the basic units. Equations must be dimensionally consistent; two terms can be added only when they have the same units. (See Examples 1.1 and 1.2.) Significant figures: The accuracy of a measurement can be indicated by the number of significant figures or by a stated uncertainty. The result of a calculation usually has no more significant figures than the input data. When only crude estimates are available for input data, we can often make useful order-of-magnitude estimates. (See Examples 1.3 and 1.4.) Copyright © 2012 Pearson Education Inc.  Lecture Summary A scalar quantity can be described by a single number. A vector quantity has both a magnitude and a direction in space. Unit vectors Can be used with components to describe vectors The scalar (i.e. dot) product of vectors is a scalar The vector (i.e. cross) product of vectors is a vector Copyright © 2012 Pearson Education Inc.