PHYSICS 221 Fall 2017

1. PHYSICS 221Fall 2017 Instructor: Tracy Furutani

2. How to Succeed • Work together. Form a study group. Social/communication skills are important for engineers and scientists. • Join the North Seattle College Rocketry Club, the Engineering Club or the RST Academy: www.seattlecolleges.edu/rst/north • Apply for summer technical jobs/research internships. • Study a little each day. Don’t wait until the last minute. • Keep your work in a notebook. Save it for years. It may help with future classes and help you avoid transfer problems. • Don’t be afraid to be wrong. You will learn faster if you speak up and ask questions. • Come to class! Come to office hours!

3. Units, Physical Quantities, and Vectors Or why the answer is never simply “6”.

4. 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.

5. The nature of physics Physics is an experimental science in which physicists seek patterns that relate the phenomena of nature. The patterns are called physicaltheories. A very well established or widely used theory is called a physicallaw or principle.

6. The steps of the scientific method are something like: • Careful observations are made of some phenomenon. • A speculative hypothesis or prediction is made to explain the phenomena. • The speculation is tested with many independent experiments. • If the speculation correctly predicts the results of the experiments, and if it is general enough to make new predictions that also prove to be correct, it may be accepted as a theory. • Note that for a scientist, for something to be called a theory, it has to be well established and backed up by many independent experiments.

7. What is the difference between a law and a theory? • Laws are generalizations, principles or patterns seen in nature. • Theories are the explanations of those generalizations, substantiated by repeated experiments and testing. • Theories do not “mature” and end up as laws! • Example: Newton’s Laws of motion and Law of gravity can be used to plan space flights with great precision. But there currently is no well accepted theory of gravity (gravity waves may end up providing the explanation, but there is little evidence supporting this). That is, no one knows exactly how or why gravity acts like it does.

8. Solving problems in physics All of the Problem-SolvingStrategies 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. SetUp 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.

9. Idealized models To simplify the analysis of (a) a baseball in flight, we use (b) an idealized model.

10. Standards and units T = tera =1012 G = giga = 109 M = mega = 106 k = kilo = 103 c = centi = 10-2 m = milli = 10-3  = micro = 10-6 n = nano = 10-9 p = pico = 10-12 Length, time, and mass are three fundamental quantities of physics. The InternationalSystem (SI for SystèmeInternational) is the most widely used system of units. In SI units, length is measured in meters, time in seconds, and mass in kilograms.

11. Unit prefixes T = tera =1012 G = giga = 109 M = mega = 106 k = kilo = 103 c = centi = 10-2 m = milli = 10-3  = micro = 10-6 n = nano = 10-9 p = pico = 10-12 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 = 103 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)

12. Unit consistency and conversions An equation must be dimensionallyconsistent. 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:

13. Unit Conversions 1 m = 3.28 ft 2.54 cm = 1 in 1 mi = 1.609 km More on inside cover of your book. You will need to remember conversion factors inside a unit system: 12 in = 1 ft 3 ft = 1 yd 5280 ft = 1 mi 60 s = 1 min 60 min = 1 hr 24 hr = 1 day 1000 m = 1 km 100 cm = 1 m 1000 mm = 1 m 1000 g = 1 kg You can use google.com to do conversions:

14. Unit Conversions Write out your unit conversions by hand—don't try them in your head. Write out the conversion so the units you don't want cancel out and the units you want are left. Example: 104 m = ? cm = 10400 cm Example: Car speedometer reads 35 mi/hr. How many m/s is that? = 15.6464 m/s Is this correct? How accurate should you report the answer? 16 m/s – the same number of significant figures as the original number.

15. More Conversions Example: 1208 square feet = ? m2 = 112.28435 m2 = 112 m2 Why square the conversion factor? 1 m = 3.28 ft Area = 1m2 = (3.28 ft)2 = 10.8 ft2 1 m =3.28 ft

16. Accuracy vs Precision Precision: How consistent the measurements are. Accuracy: How close the measurements are to the “actual” value.

17. Significant Figures No measurement is ever perfect, no matter how good the instrument is or how accurate the measurer is. There are always uncertainties. You should indicate to others the uncertainty of your numbers are that are based on measurements. The number of digits that are meaningful are called significant figures. • Ideally, numbers are reported something like this: 1.25 ± 0.01 m. All three digits are significant, but the last has a little uncertainty in it. Note that the precision (number of places past the decimal point) of the uncertainty matches the precision of the measurement. Figures such as 1.25 ± 0.1 m and 1.25 ± 0.0001 m do not report a meaningful uncertainty.

18. Significant Figures People often leave off the “±” making it hard to figure out what the uncertainty is. For example, how many significant figures are there in 120 m? Without knowing how the measurement was made you should assume it is only significant to the last non-zero number, so 2 sig figs and this measurement is assumed to be ±10 m.

19. How many sig figs? Usually we assume ± 1 for the last significant digit. Note that all measurements should have the uncertainty ± stated. 100.02 m 8.2 x 104 m 0.000709 m 2500 m 100. m

20. Math with sig figs Addition and subtraction: The result’s precision is the same as the least precise input. For example, 12.034 +4.2 16.2 The least precise number limits the precision of the result.

21. Multiplication and Division: The result has the same number of sig figs (not places past the decimal point) as the number with the fewest number of sig figs. For example: 12.34 x 2.1 26 Your calculator says the answer is 25.914. Round this to 2 sig figs. Your first lab will discuss how to figure out the uncertainty (the ±) of a measured quantity. Please read it before coming to lab.

22. Vectors and scalars A scalarquantity can be described by a singlenumber. A vectorquantity 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 | |.

23. 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.

24. Adding two vectors graphically

25. Adding two vectors graphically

26. Adding two vectors graphically

27. Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order.

28. Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order.

29. Adding more than two vectors graphically To add several vectors, use the head-to-tail method. The vectors can be added in any order.

30. Subtracting vectors

31. 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.

32. 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.

33. 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.

34. Positive and negative components The components of a vector may be positive or negative numbers, as shown in the figures.

35. Finding components We can calculate the components of a vector from its magnitude and direction.

36. 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-SolvingStrategy1.3.

37. Unit vectors A unitvector 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

38. The dot (or scalar) product

39. The dot (or scalar) product The dot product can be positive, negative, or zero, depending on the angle between and .

40. Calculating a dot product using components • In terms of components: • The dot product of two vectors is the sum of the products of their respective components. • Incidentally, some books will simply bold-face vectors: A • Those same books will use non-bold-face to indicate scalar quantities: A

41. Finding an angle using the dot product Example 1.10 shows how to use components to find the angle between two vectors.

42. The cross (or vector) product If the vector product (“cross product”) of two vectors is then: The direction of the cross product can be found using the right-hand rule:

43. The cross product is anticommutative

44. Calculating the cross product Use ABsinϕ to find the magnitude and the right-hand rule to find the direction.

45. Calculating the cross product (a different way) If A = Axi + Ayj + Azk and B= Bxi+ Byj+ Bzk , then the cross productof A and B is the vector A ´ B = (AyBz- AzBy)i + (AzBx – AxBz)j + (AxBy – AyBx)k Wow, why so complicated?

46. Matrix notation (plural of matrix is matrices) • The definition of a cross-product is derived from matrix notation. • A matrix is any array of numbers with n columns and m rows; such a matrix is defined as an “n by m matrix”. • For instance, a 3 by 3 matrix will have the form: where each of the letters can be a number or a function

47. Determinants • The determinant of a square (n by n) matrix is a single value assigned to a matrix. For a 2 by 2 matrix, the determinant is calculated as follows: Note the square bracket notation denotes the calculation of a determinant (and not the matrix itself, which is set off by parentheses).

48. Determinant • An example of determinant calculation:

49. Determinants • So a 3 by 3 matrix can have its determinant calculated in the following way: Each term on the right side of the equation involves a number aiin the first row of the determinant

50. Determinants • So a 3 by 3 matrix can have its determinant calculated in the following way: Each term on the right side of the equation involves a number aiin the first row of the determinant • This is multiplied by the second-order determinant obtained from the left side by deleting the row and column in which itappears.