Unit One Math and Measurments Don t worry about your difficulties in mathematics, I can assure you that mine are still greater.

Transcription

1 Galena High School Unit One Math and Measurments Don t worry about your difficulties in mathematics, I can assure you that mine are still greater. Albert Einstein 1

2 Unit 1 Math and Measurement Supplements to Text Readings from Holt Physics by Serway and Faughn Chapter 1 Topic Page # 1. Unit 1 Objectives and Assignments.3 2. Are you ready for Physics math quiz Scientific Notation.9 4. Significant Figures Accuracy and Precision Number Systems SI vs the Stupid System Dimensional Analysis (number conversions) How to Draw and Interpret Graphs The Zen of estimating and using your brain as a calculator Using a Scientific Calculator some things you should know Unit Practice quiz (coming soon) Links for help on the web (coming soon).22 2

3 1) Physics Math Assessment Unit 1 Math and Measurement Text : Holt Physics by Serway and Faughn Chapter 1 and Appendixes A, B, C, and D a. Students should take the Physics Math Assessment quiz and evaluate areas of weakness and strengths. i. After evaluating the quiz results, students should use available text books and review aids to reinforce week areas. Group and individual help sessions will be available outside of class time. 2) Scientific Notation a. Students should understand how to write complex numbers in scientific notation when given standard notation. b. Students should understand how to write complex numbers in standard format when given in the scientific notation format. c. Students should be able to add and subtract numbers in the scientific format. d. Students should be able to multiple and divide numbers in the scientific format. e. Students should be able to do mixed calculations when given numbers in scientific notation format. 3) Significant Numbers a. Students should understand the importance of significant figures and demonstrate a full understanding of the rules for writing and doing mathematical operations with numbers in regards to significant figures. 4) Accuracy and Precision a. Students should understand and be ready to delineate the differences between accuracy and precision. 5) Number Systems-- SI vs the Stupid System a. Students will understand the reason for common and simple number systems. b. Students will learn the basic prefixes and suffixes of the metric system and how they relate. c. Students will be able to convert between numbers in the metric format with different prefixes and suffixes. 6) Drawing and Interpreting Graphs a. Students will learn and understand the importance of graphical representation of data and information. b. Students will understand how to construct a graph given specific data. c. Students will understand how to read a graph of data and interpret the correlations involved. d. Students will understand how to find the slope of a graph and interpret its meaning to the data. e. Students will be able to recognize different graphical relationships based on formulas relating the data. 3

4 7) Zen of Estimating and the Human Calculator a. Students will be able to do simple to medium calculations in their head. b. Students will learn the art of estimation or guessing with some reliability. 8) Using a Scientific Calculator a. Students will become familiar with the operations of a scientific calculator including adding, subtracting, multiplying, dividing, exponents, scientific notation, square roots, trigonometric functions, log functions and solving polynomials. b. Students will become familiar with the graphing function of their calculators or those supplied by the instructor. i. Student should be able to graph simple relationships using a graphing calculator. 4

5 Is Your Math Ready for Physics Many students take Physics after having completed a rigorous series of math courses including Algebra, Geometry, Advanced Algebra, and in some cases Trigonometry and even Calculus. Even with a good background of doing lots of math problems, it is easy to forget some of the mathematical operations that you have learned or maybe assumed you learned by taking a quiz. In this section of the unit, I would like to give you a little quiz of math problems you should be familiar with and help you understand where you might need to brush up on some basic skills. Don t feel bad or intimidated if you have difficulty with some of the problems, you may not have had the information yet or it just did not sink in when you did have it. If you are taking Trigonometry this year, you may not understand the Trig related problems at all. Following the quiz and the solutions, I will include a small section of review to help bring you up to speed on some of the problems. If you are still having a little difficulty, please check out the appendix in your book and spend some time reviewing old math books to reinforce you week areas. Mr R Is Your Math Ready Quiz Calculators are OK! Please show your work even on the simple problems, it is difficult to have students turn in their calculators x x = 63 2 y 7. If one inch equals 2.54 centimeters, how many inches equal 100 centimeters? = 9. (10 2 ) -1 = 10. (4-2 ) -2 = kl = ml 12. 3,800,000,000 meters in scientific notation =? 13. Express in scientific notation =? 5

6 14. 1/2 x = 5-2x x =? b = 4(2b-3) = 16 b =? x x? 17. Find y if y = mx = b, m = 3/4, x = 8, and b = 2/3 If g = 0.30, h = 1.2, and k= 0.25 evaluate the expressions in 18 and k(-g h) g(h + gk) 20. k - 7 = 1 find k k Find the slope of a line passing through P 1 and P 2 P 1 = { -2, -3 } P 2 = { 4, 9 } 22. Find the equations of the line determined by the given information, express your answer in the form ax + by = c A line passing through ( 2, 3 ) with a slope of 1/2 23. Evaluate the radical Solve y 2-25/36 = Solve t 1/2-1/2 = Solve by factoring -3a 2-7a + 6 = Solve by factoring -6m m + 4 = Solve using the quadratic equation d 2 = 8d Sketch a rough graph of y = -4x 2 6

7 30. Graph the following y = e 2x 31. Solve 3 2y-3 = Express in exponential form log 3 81 = Express in logarithmic form 2 5 = Find the logarithm log Find x log 4 x = Graph the equation y = log 3 x 36. If tan θ = 3/2, what is the value of θ? 37. Simplify sin x / cos x 38. Cos of 180 =? 39. Use Pythagorean theorem The diagnal of a square is 4 2 inches, find the length of a side. 40. A radius vector whose length is 8 inches makes an angle of 45 degrees with the x axis. Determine the components of this vector. 7

8 Solutions to the Ready Quiz 1. 83/ / / in 8. 1/ / ,400,000ml x x x / x-2y = ± 5/ / , 2/ /6, (double root) 29. inverted u shape with the apex at 0, = / / (graph) º 37. Tan x inches , 4 2 \ 8

9 Scientific Notation Introduction: Imagine trying to measure the distance to the Sun in inches or the diameter of an atom in inches. The numbers would include a lot of zeros and would be very hard to work with. For that reason, scientists (and everyone for that matter) use a system of abbreviations for numbers called Scientific Notation. Scientific notation is easy to do and easy to do mathematical calculations with once you learn the rules. Lets look at the rules and some examples to check our understanding. Scientific Notation is based on powers of 10 like those listed below 10 0 = = = 10 x 10 = = 10 x 10 x 10 = = 10 x 10 x 10 x10 = 10,000.and so on, you get the idea. The number of zeros determines the power to which 10 is raised, or the exponent of 10. For example a million dollars written as $1,000,000 can be written as $1 x 10 6 For numbers less than one, we can write the following 10-1 = 1/10 = = 1/10 x 10 = = 1/10 x 10 x 10 = and so on The value of the negative exponent equals the number of places the decimal point must be moved to the right of the first nonzero digit (in these cases 1 ) Numbers that are expressed as a number between 1 and 10 multiplied by a power of 10 are said to be in scientific notation. For example 5,943,000,000 is x 10 9 and is 8.3 x 10-5 when expressed in scientific notation. Easy stuff RIGHT! Calculations Involving Scientific Notation Addition and Subtraction is simple, just change the exponents of the numbers until they are all the same exponent, and then add or subtract. 2.5 x x 10 3 = 250 x x 10 3 = x 10 3 Multiplication and Division are just a little harder, but not much When multiplying, just multiply the whole numbers and add the exponents. (2.5 x 10 5 ) x (5.0 x 10 3 ) = 2.5 x 5.0 x = 12.5 x 10 8 (But we like to have no more than one digit to the left of decimal point so the answer would be 1.25 x

10 It looks like this 10 n x 10 m = 10 n + m When dividing, just divide the whole numbers and then subtract the exponents. It looks like this 10 n / 10 m = 10 n-m Hey, this stuff really is easy once you know the rules and understand the process. Scientific Notation Practice Express in scientific notation ,000,000,000 (3.45 x ) (4.52 x 10-8 ) 3. The sum of (4.67 x 10 2 ) + (2.99 x 10-3 ) - (3.88 x 10 2 ) ( x 10 1 ) 4. (3.0 x 10 8) x (4.21 x 10 4 ) / (2.25 x 10-6 ) = (5.61 x ) SPECIAL NOTE: If you have and know how to use a scientific calculator, you will find most of these problems rather easy except when we consider how precise a number can really be. For instance on my graphing calculator when I divide 7.1 by 3 I get an answer of , which is kind of ridiculous considering the numbers I entered in the first place. I still find it interesting that students will actually report the answer to 8 or nine places when doing work or quizzes. I hope most of you recognize that part of the answer is just a guess and should be ignored. This brings us to the subject of how to report numbers and our next topic, SIGNIFICANT NUMBERS (OR FIGURES). Significant Figures After teaching Physics and Chemistry for a number of years, I have come the conclusion that every teacher and most textbooks have their own twist on how to report significant figures. Even when the textbook lays out a specific set of rules the teacher and even the students often massage the rules to fit their own needs and qualifications. For that reason, I will just repeat the information from our current textbook (Holt Physics) and we will discuss my requirements in class. I will tell you that in my AP classes, it is acceptable to be within 2 significant figures of the actual answer and still get full credit on a AP exam. That is close enough for me. I will ask you to do some specific reporting of data and calculations using the correct number of significant figures, and then be generally relaxed about it the rest of the year. 10

11 Please don't give ridiculous answers like the one above in class though or I will throw things at you. It is important to record the precision of your measurements so that other people can understand and interpret your results. A common convention used in science to indicate precision is known as significant figures. In the case of the measurement of a pencil that shows a little more than 18.2cm, the measurement has three significant figures. The significant figures of a measurement include all the digits that are actually measured (18cm in this case), plus one estimated digit. Note that the number of significant figures is determined by the precision of the markings on the measuring scale in this case. The last digit is reported as a 0.2 (because the estimated distance beyond the 18cm mark looks like about 0.2cm and its my pencil, so I will do the guessing). Because this digit is an estimate, the true value for the measurement is actually somewhere between 18.15cm and 18.25cm. When the last digit in a recorded. When the last digit in a recorded measurement is a zero, it is difficult to tell whether the zero is there as a place holder or as a significant digit. for example, if a 11

12 length is recorded as 230 mm, it is impossible to tell whether this number has two or three significant digits. In other words, it can be difficult to know whether the measurement of 230mm means the measurement is known to be between 225mm and 235mm or is know more precisely to be between 229.5mm or 230.5mm. One way to solve such problems is to report all values using scientific notation. In scientific notation, the measurement is recorded to a power of 10, and all of the figures given are significant. For example, if the length of 230cm has 2 significant figures, it would be recorded as 2.30 x 10 2 cm. Scientific notation is also helpful when the zero in a recorded measurement such as cm should be expressed in scientific notation as 1.5 x 10-4 cm if it has two significant figures. The three zeros between the decimal point and the digit 1 are not counted as significant figures because they are present only to locate the decimal point and to indicate the order of magnitude. The rules for determining how many significant figures are in measurement that includes zeros are show below. Significant figures in calculations In calculations, the number of significant figures in your result depends on the number of significant figures in each measurement. For example, if someone reports that the height of a mountaintop, like Mt Rose is 1710 m, that implies that its actual height is between 1705 and 1715 m. If another person builds a pile of rocks 0.20 m high on top of Mt Rose, that would not suddenly make the mountain s new height known accurately enough to be measured as m. The final answer cannot be 12

13 more precise than the least precise measurement used to find the answer. Therefore, the answer should be rounded off to 1710 m even if the pile of rocks is included. Similar rules apply to multiplication. Suppose that you calculate the area of a room by multiplying the width and length. If the room s dimensions are 4.6 m by 6.7 m, the product of these values would be m 2. However, this answer contains four significant figures, which implies that it is more precise than the measurement of the length and width. Because the room could be as small as 4.55 m by 6.65 m or as large as 4.65 m by 6.75 m, the area of the room is known only to two significant figures because each measurement has only two. So it must be rounded off to 31 m 2. Table 1-5 summarizes the two basic rules for determining significant figures when you are performing calculations. Calculators and Significant Figures Most of the problems you will do this year will involve using a calculator. Calculators do not pay attention to significant figures so you will need some guidelines on what to do with the numbers and answers you come up with. Table 1-6 give you those rules. 13

14 Accuracy and Precision While discussing significant figures, I used the term precision several times. Some mention of the difference between accuracy and precision should be mentioned. Accuracy is the extent to which a reported measurement approaches the true value of the quantity measured. Precision is the degree of exactness or refinement of a measurement. If you were taking data numbers down during an experiment, you might collect a few numbers like 10.1, 10.0, 9.99, and If the number you should have gotten was 9.81, then your numbers would not be very accurate. But, if you look at your data, they are really close together so we say they are very precise. We will discuss this a little more when working on classroom labs and collecting data. 14

15 Number Systems - Metrics vs The Stupid System A small note to explain the title of this section. After teaching the metric or SI numbering system to science students for many years, I have come to refer to the common American system as the stupid system. This is not out of disrespect or lack of confidence with our system, just a way of pointing out how difficult it is to learn and remember. A simple example might be to ask How many quarts in a bushel or if there are 16 ounces in a pint, does that mean that a pint weighs a pound because there is 16 ounces in a pound. Another example may relate to how many of the current American units were derived. For instance, an inch was the width of someone s thumb (who s?) or a mile is a 1000 paces of a Roman soldier (which one?). You get the idea. I just point out how much easier the metric or SI system is. For scientists (and everyone else for that matter) to communicate with each other consistently, they need a common measurement system and have adopted the metric system worldwide. The formal name of the metric system is the System International or SI system. The metric system is very easy to learn and use because it is based on a 10 based numbering system (like our money and years etc..) and only requires learning how to use a few prefixes and the basic units for mass, length, volume, and time. The tables below reflect the minimum information you need to know to be successful in Physics. Prefix What it means Decimal Equivalent Kilo 1000 times x Milli 1/ Deci 1/ Centi 1/ Mega ,000,000 Micro 10-6 Note: there are a lot more you can research and learn, but these will take car of most situations. Type of Unit Mass Length Volume Time Common Metric Units Kilograms, grams Meters, centimeters, millimeters Liters, milliliters Seconds I would also include that the units for force such as weight is important to know. In addition, we will combine many types of units to describe Physics phenomena. KNOW THESE TWO CHARTS!! ENOUGH SAID, I WON T HELP YOU ON A QUIZ OR TEST! (MEAN GUY HUH?) 15

16 Dimensional Analysis or Unit Conversions in Physics When taking Chemistry, one of the first mathematical skills you needed to learn quickly was the ability to do unit conversions. The first unit conversions where just simply changing units like millimeters to kilometers. Then you started to do more complex conversions while solving problems like density, volume, and others. Finally, most the math in Chemistry required a combination of general problem solving and unit conversions. Physics is a continuation of applying abstract ideas, problem solving and converting and mixed units. Physics is how the physical world works and the study of physics is not complete without learning the language of physics and how it relates to the physical world. In addition, almost all physics concepts require combinations of units in their explanations. With this in mind, no study physics can be complete without learning the math of Physics and that math requires that one understand unit conversions and dimensional analysis. Dimensional Analysis and unit conversion are exactly the same for all practical purposes and we will start out with some simple examples and work towards some of the more complex problems will encounter. It really is simple once you understand the rules or procedures and follow them each time. It is based on the concept that you can multiple anything by the number "1" and not change that thing except for its units. Concepts to understand Units help define what a number really means. It can be something simple like inches or miles or hours or can be more complex like miles per hour or feet per second. It can even become quite complicated like meters per second squared times kilograms of mass. Let s start off looking at conversions of simple single units. Let s say we want to know how many inches are in 35 feet. We start off by writing our given units. 35 feet then multiply that number and units by a "unit conversion" that connects feet and inches but is equal to "one". "one" 1ft = 12 inches or 12 inches = 1 ft either one in a fractional form is equal to 12 inches 1 foot are both equal to "one" and we 1 foot 12 inches can multiple anything by one! so lets try the conversion 35 feet x 12 inches = 420 inches 1 foot 16

17 Notice that the feet (or foot) cancels out and you are left with your answer in "inches" which is want you want! And all we did was multiple by one! We can even multiple by one multiple times to get the desired units. For Example, how many inches in a mile would like 1 mile x 5280 feet x 12 inches = inches 1 mile 1 foot notice that the miles cancel and feet cancel to leave us with the desired units of inches and we just multiplied by one a couple of times. Notice also that while the units cancelled to give us the desired ones, we had to do a little math to get the actual number. Now let s get a little more complicated. Speed or velocity is usually reported in distance per an amount of time, like 60 miles per hour. Now if we want to change both the distance and time units we use the same process, but keep going until we have multiplied conversion factors to both the numerator and denominator. Change 60 miles / hour to inches per minute 60 miles 5280 feet 12 inches 1 hour = inches hour 1 mile 1 foot 60 min 1 min Pretty neat stuff huh? Most chemistry classes don't do conversions that require this (except mine - smile) however, it is quite common in physics. Even more, they can involve numbers that have units with exponents like squared or cubed. Change 10 meter/second 2 to centimeters / minute 2 10 meters 100 cm 60 sec 60 sec = cm 1 sec 2 1 meter 1 min 1 min 1 min 2 Notice how we have to put in a conversion factor for each power the exponent. In this case. the exponent is 2 so we put in the conversion 2 times. Cool right? Before we go on to solving some practice problems, I want to mention some expectations about how I look at and correct you work concerning dimensional analysis problems. First, you problems need to be set up as a SINGLE problems and not done in steps. Later I will accept work differently once I know you can do the problem correctly. Second, you must include and show unit cancellation to get the desired final units. Lastly, you cannot turn in you calculator, so set the problem up correctly on your paper, and only do the math at the end. 17

18 How to Draw and Interpret Graphs Graphs are used to show how one quantity depends on another quantity (e.g. distance traveled depends on time of travel). Since discovering and understanding relationships between measured quantities is one of the main activities of Physics, making and using graphs is important in many laboratory activities. A graph should be thought of as a capsule summary of the data for a laboratory activity, and the graphs should be intelligible without any auxiliary description. 1. Plan the Axes Usually it is clear in a laboratory activity that one of the quantities being measured depends on the other. For instance, when a ball rolls down an incline its speed at any instant depends on the time since it was released. In this case, the speed is considered the dependent variable, and the time is the independent variable. It is customary to plot the dependent variable (in this case the speed) on the vertical or Y-axis, and the independent variable (in this case time) on the horizontal or X-axis. A graph should be easy to read. Thus, a single square of your graph paper can stand for one, two, five, ten, one-half, two tenth, or one-tenth of a unit, but never for a difficult-to-read value like seven or one-third. Don t try to number every line along the axis. You need to provide enough divisions on the axis to make the graph easy to read without being cluttered. It s not necessary to use the same scale on both axes: after all, they often represent different quantities. Try to choose scales so that the whole graph will fill the page. Label each axis with the name of the quantity being plotted, and the units in which it is expressed (e.g. speed in [meters/seconds]). A title (e.g. Speed versus Time for Falling Object ) should be placed at the top of the graph paper. 2. Plot the Points Each point on a graph represents a pair of measured values. Locate each point on the graph as exactly as you can, and mark it using a small but clear dot. Make a circle around each dot so it can easily be spotted. The size of the circle should represent the approximate uncertainty of the measurements. Some people use little + s for the points. In more advanced work it is customary to show the probable uncertainty in each quantity by the size of the arms of a cross drawn through the data point. 3. Draw the Line For many laboratory activities the graphed points will suggest a straight line or a smooth curve, often beginning at the origin. Due to uncertainty of measurement, there is bound to be some random scattering of the points. If the actual relationship is not apparent, additional measurements may be needed. 18

19 50 40 O speed (cm/s) 30 O 20 O 10 O time (seconds) If you think the data represents a straight line graph, use a ruler to draw a straight line, locating it with about as many points scattered above the ruled line as below it. The circles drawn around the data points will prevent the data points from being lost when you draw the graph. If a single point seems to be quite far from the line suggested by the other points, try to return to the laboratory to check the values. If you cannot check it, leave it on the graph but ignore it when drawing the line. If the points indicate a curve, draw it smoothly, passing close to as many points as possible. Don t draw a series of broken point-to-point lines. In elementary physics laboratory activities, the curved graphs are generally simple sweeps which are concave upwards or downwards. 4. Translating graphs into English and Algebra What can the shape of a graph tell you about the possible relationship between the dependent variable (we ll call it Y) and the independent variable (call it X)? Here are some examples you will encounter in your physics course. Acc (m/s 2 ) Straight line parallel to the X-axis 10 English: the Y value is constant for all values of X. Algebra: Y = k, where k is a constant Example: a g = 9.8 m/s 2. the acceleration due to gravity 5 for a body falling in a given location has a constant 19

20 value. (Experimental values will vary randomly about an average, as shown at the left) time (secs) distance (m) Sloping straight line through the origin 80 English: The quantity plotted on the Y axis is directly 60 proportional to quantity plotted on the X axis. Note that 40 in this relationship whenever X doubles, Y also doubles; 20 and when X triples, so dies Y Algebra: Y/X = k, or Y = kx time (secs) Example: d = vt. For an object moving with a constant speed v, the distance it travels is directly proportional to the time the object has been traveling. In this case k is represented by v. At times you can extend a graph back to the origin without making a measurement. In the speed Versus time example, one can argue: Since the ball was released from rest, when time is zero the speed is zero Sloping straight line not through the origin English: the slope of the line is constant (as for case b), Therefore the change in Y is directly proportional to the Corresponding change in X. Algebra: 20

21 Zen and the Art of Using your Brain as a Calculator I will be adding a couple of pages hear explaining how to do some simple and even a little complex math in your head. Until then, please use your brain to do simple math problems rather that always picking up the MBWB (that is magic box with batteries). I would expect easy times tables, simple division and even some simple roots should not need a calculator. What You Should Be Able to Do With Your Calculator 1. Any multiplication, division, addition, subtraction, squares and other roots etc. 2. Solve polynomials in a quadratic format for multiple answers. 3. Graph simple and complex formulas including polynomials. 4. Program in formulas and constants. 5. Save and plot lab data using a interconnecting device. I am currently typing up a workbook just for these tasks, but will cover the last 2 in class. Read your manual. 21