To know standards and units and be able to do unit

Transcription

1 Chap. 1: Models, Measurements, Vectors Goals for Chapter 1 To know standards and units and be able to do unit conversions. To express measurements and calculated information with the correct number of significant figures. To be able to add vectors. To be able to break down vectors into x and y components. 1 Science ce-products-mysteries-of-science 2

2 Science Spin-Off World Wide Web (WWW) In 1989, an event at CERN changed the world forever. Tim Berners-Lee handed a document to his supervisor Tim Mike Berners-Lee Sendall entitled Information Management : a Proposal. Vague, but exciting is how Mike described it, and he approved it to go forward. The following year, the World Wide Web was born. We ve learned Earth s languages though the WWW. 3 Wrong units, $125M Loss! Units of Measurement (Metric mishap by NASA) Units are: Cultural cubit, span, foot, mile Changes with time and location Unified in 1889 by the General Conference on Weights and Measures Systéme International (SI) 4

3 Anatomy of Exam 1, Problem 1 [Science] Use a consistency in the number of significant figures [Science] Units are in SI. [Science] Full of vectors See the key words in the instruction: vector components 5 Exam 1, Problem 1 y x Trigonometry! 6

4 in Vectors? In the world of vectors, does not necessarily equal Graphically? parallelogram Vector Addition In the world of vectors, does not necessarily equal Graphically? 8

5 Or, decompose the vectors into components then solve. [Science] Use a consistency in the number of significant figures [Science] Units are in SI. [Science] Full of vectors See the key words in the instruction: vector components 9 Trigonometry Try example 1.6 to check yourself. 10

6 Which of the vectors A ~ E represents the vector sum of vectors 1 and 2? A. B. C. D. E. Quiz 11 Quiz Which of the vectors A ~ E represents the vector sum of vectors 1 and 2? A. B. C. D. E. 12

7 Q1.1 What are the x- and y-components of the vector E? A. E x = E cos, E y = E sin B. E x = E sin, E y = E cos C. E x = E cos, E y = E sin D. E x = E sin, E y = E cos E. E x = E cos, E y = E sin 13 A1.1 What are the x- and y-components of the vector E? A. E x = E cos, E y = E sin B. E x = E sin, E y = E cos C. E x = E cos, E y = E sin D. E x = E sin, E y = E cos E. E x = E cos, E y = E sin 14

8 Q1.6 Consider the vectors shown. What are the components of the vector E=A+D? A. E x = 8.00 m, E y = 2.00 m B. E x = 8.00 m, E y = m C. E x = 6.00 m, E y = 0 D cos(53.0) = 6.02 D sin(53.0) = 7.99 D. E x = 6.00 m, E y = m E. E x = 10.0 m, E y = 0 15 A1.6 Consider the vectors shown. What are the components of the vector E=A+D? A. E x = 8.00 m, E y = 2.00 m B. E x = 8.00 m, E y = m C. E x = 6.00 m, E y = 0 D. E x = 6.00 m, E y = m E. E x = 10.0 m, E y = 0 16

9 Problem 56 tunred into Exam 1. *) 15 min. *) a textbook figure was used. *) modified to have 3 parts. a) 228 km, 29.6 o b) 189 km, 10.5 o c) No 17 If one does good at science, it may be: Today s class SI units Dimensional analysis Scientific notation Errors Vectors Next class Chapter 2: Motion in 1D Example 2.10 and

10 If one does good at science, CBS comedy Big Bang Theory (Season 1 Episode 15) The poster was designed during lunch meetings at TAMU. 19 The Nature of Science Physics is an experimental science. Observe phenomena in nature. Make predictions. Observation Experimental Data Models, Hypotheses, Theories, Laws Application (e.g., Engineering) 20

11 The Nature of Science Observation Experimental Data E = m c 2 Nuclear Power 21 Three Fundamental S.I. Units. e.g., d = 10 7 m (Distance between North Pole and Equator) [Conversion: km = 1 mi ] 62,150 mi k mi x 10 4 mi Scientific notation 22

12 The Meter Now tied to Krypton discharge and counting a certain number of wavelengths. Exceptionally accurate, in fact redefining c, speed of light. New definition is the distance that light can travel in a vacuum in 1/299,792,458 s. Figure 1.2 So accurate that it loses only 1 second in 30 million years. 23 The Second Originally tied to the length of a day. Now, exceptionally accurate. Atomic clock 9,192,631,770 oscillations of a low-energy transition in Cs In the microwave region Figure 1.3a 24

13 The Kilogram The reference cylinder is kept in Sevres, France. A more modern, atomic reference is hoped for, waiting for a precise atomic technique. Figure 1.3b the reference kilogram 25 You can adjust the fundamental units. Imagine trying to measure the distance from San Francisco, CA to Charlotte, NC in meters. Instead, we attach prefixes to the units that adjust their size by powers of ten. In this case kilo (or x10 3 ) would likely be chosen. The distance would then be reported as 4,621 km or Mm or 4.621x10 6 m. o Getting good at converting units will be very important to us. o Use Table

14 Table The powers of ten are dramatic! Figure

15 Conversions Practice, practice, practice. Problems will come in many different styles of measurement but you will ultimately need to get back to (m), (kg), and (s) if your answers are going to fit other calculations. You ll need to overcome two hurdles: Derived units English S.I. Let s examine each. 29 Derived units The science will wait until we find it in a subsequent chapter. First, we should examine some units from those chapters. Imagine you need to work with energy. E = m c 2 The unit for energy is Joule (J) and it s built from other units. In this case [J] = [kg] [(m/s) 2 ]. Having mass in g (instead of kg) or distance in cm (instead of m) will destroy your answer. 30

16 Derived units (II) The science will wait until we find it in a subsequent chapter. First, we should examine some units from those chapters. Imagine you need to work with energy. E = m c 2 The unit for energy is Joule (J) and it s built from other units. In this case [J] = [kg] [(m/s) 2 ]. Having mass in g (instead of kg) or distance in cm (instead of m) will destroy your answer. [Quick Quiz] How about the units of force (Netwon or N)? F = m a [N] = [mass] [acceleration] 31 English S.I. In the United States, we often encounter measurements in miles, feet, pounds, quarts and gallons. It s useful to memorize one conversion for displacement, one for volume, and one for mass. Everything else could be reached by converting orders of magnitude. Mass is tricky. English units assume standard earth gravity and relate kg to lb even though kg is a mass and lb is a force. 32

17 English S.I. (II) Personal favorites for English to SI are listed below. Displacement: 2.54 centimeters = 1 inch. Mass: 454 grams = 1pound. Volume: 1liter = 1.06 quarts. You could certainly choose a different set. 33 Number of Significant Figures 15 1 m (1 digit in uncertainty, same 10 s as last digit) 1) m 2) m 3) m 34

18 Precision and Significant Figures Your measurement tools have limitations ( uncertainty ) and your reported results need to reflect those limitations. At the risk of making data feel unwanted, we dare not report everything our calculator tells us. Try this. Divide 424 (three SF) by 135 (three SF) and your calculator will tell you If you report this answer, the reader will believe you have measured carefully to billionths of the unit you are using. What can happen if 4241 and 1351? You expect the value could be between 3.11 and Precision and Significant Figures Your measurement tools have limitations ( uncertainty ) and your reported results need to reflect those limitations. At the risk of making data feel unwanted, we dare not report everything our calculator tells us. Try this. Divide 424 (three SF) by 135 (three SF) and your calculator will tell you If you report this answer, the reader will believe you have measured carefully to billionths of the unit you are using. What can happen if 4241 and 1351? You expect the value could be between 3.11 and

19 Scientific Notation and Error Propagation? 37 Scientific Notation and Error Propagation 38

20 Appendix: Vectors 39 Displacement (Vector) y x 40

21 Direction: North Magnitude: 2.6 km y x 41 Direction: East Magnitude: 4.0 km y x 42

22 Direction: 45 o North of East Magnitude: 3.1 km y x 43 Direction:? Magnitude:? D 2 D 3 D 1 D R D 1 D 2 D 3 x y D R 44

23 Direction:? Magnitude:? D 2 D 3 D 1 D R Adding vectors by components x y D 1 0 km 2.6 km D km km D 3 km km D R 45 Direction: 37.7 o North of East Magnitude: 7.83 km D 2 D 3 D R D 1 D 2 Rx tan 1 ( D Ry D / 2 Ry D Rx ) D R x y D 1 0 km 2.6 km D km km D 3 km km D R 6.19 km km 46

24 x1.5 longer, Same direction Vector Operations x2.0 longer, Opposite direction 47 [Quick Quiz 1] Find: (a)a+b (b)ab (c) A+B+C (d) A+BC (e) BC (f) AC 48

25 [Quick Quiz 1] Find: (a)a+b (b)ab (c) A+B+C (d) A+BC (e) BC (f) AC 49 Adding vectors using their components 50