Appendix A from ast 1022 Lab Manual

Appendix A from ast 1022 Lab Manual - !"‘\ ....

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Unformatted text preview: !"‘\ . ,x‘ APPENDIX A MATH AND SCIENCE BASICS INTRODUCTION Students enter this course from a wide variety of fields of study. Some are mathematically sophisticated, while others use little math. These notes are intended to assist the latter in handling the measurements and simple calculations used in this course. Some instructors may wish to cover at least part of the material in class, while others may assign it for self-study. The notes include the uso of an inexpensive pocket calculator to perform certain operations that may be difficult to do by hand, but are childishly simple on the calculator. These sections are marked with a calculator icon. If you do not yet use such a calculator, you are depriving yourself of a wonderful aid in everyday life! If you have done little or no math since 7th grade algebra, please do not be frightened by this material. None of the calculations used in this course can by any stretch of the imagination be called “advanced!” If at any point you hate difficulties, remember that you always have at hand an instructor who will be happy to give you one-on-one assistance. We hope that an important by-product of this course will be that in the future you will be empowered to read newspaper and magazine articles on science and technology with increased understanding (and perhaps even to detect the all-too-frequent errors in such articles!). SCIENTIFIC NOTATION In astronomy, we often have to deal with, huge numbers. For instance, the age of the Universe, the distance to a galaxy or the mass of- a star, represent times, lengths and masses, respectively, well bewnd our everyday experience. Astronomers have two ways to cope with this inconvenience. The first is to use an intelligent way” to represent numbers, calied “scientific notation”. The second is to use different units of time, length, era, that are more appropriate for astronomy—related problems (see UNITS Section). The scientific notation is a way to express very large (and very small) numbers, as powers of ten: ' 10" = 1.0 _ 101 = 10.0 10" = 0.1 102 = 100.0 10'2 = 0.01 103 = 1,000.0 10'3 = 0.001 104 = 10,0000 _ 104 = 0.0001 105 = 100,0000 10-5 = 0.00001 10‘5 = 1,000,0000 10*5 = 0.000001 etc. etc. A quick way of changing a number from a decimal representation to a Scientific notation representation is simply to count the number of position to the left or right of the decimal point. The number of positions corresponds to the power of ten. In most cases, scientific notation expresses numbers as the product of a number and a power of A.1 ten. For example, the mean distance from the Earth to the Sun is 149,600,0000 km. In _ scientific notation, we would express this as 1.496 x 108 km. The decimal point was moved 8 positions to the left. In order to enter Scientific notation in your calculators, you should use the key usually labeled EXP or E. For instance, if your calculator has the key EXP, to type in 1.496 x 103, you must press 1.496 EXP s. ? Problems 1. Express the following numbers in scientific notation: i. 394,000,000 ii. 0.0007985 52 iv. 0.65 MEASURING ANGLES About five thousand years ago, ancient Babylonians divided the circle into 360 degrees (or 360°), each degree into 60 are minutes (or 60’) and each arc minute into 60 are seconds (or 60"). Although this system may have been consistent with philosophical beliefs of the time, it sounds rather arbitrary today. - We will use the Babylonian system throughout this course: 1 degree = 60 arcminutes (1° = 60’) 1 arcminute = 603rcseconds (1’ =60”) ? Problems 2. This question is intended to get you used to manipulating angles: ii. How many arcseconds are there in a degree? 7 How many degrees are there in an arcsecond? UNITS In the section about scientific notation, we saw how to write numbers in scientific notation in order to conveniently express very big or very small numbers. A second way out is to use appropriate units, so that we do not have to deal with cumbersome numbers at all. For example, using miles to measure the distance to the Sun is much like using inches to measure the distance from New York to Los Angeles. So, when we deal with distances of objects within our solar system, we use a “yardstick” called the astrOnornical unit (or just AU). One astronomical unit is equal to the average distance of the Sun from the Earth; ' A2 1 AU: 1.496x 108km=9.3>< 10" miles However, distances between the stars of our galaxy are huge compared to the distance of the Earth'from the Sun, so the astronomical unit is not a convenient unit any more. Therefore, we have to further stretch our Wardstick” and use as a unit of length the distance a light beam travels in a year, which we call light year (or 1y). Thus, although “year” is a unit of time, a “light year” is a unit of length! 11y:6.324x10‘AU=9.46x10”km There is yet another unit to measure interstellar distances within galaxies. It is called parsec (pc): 1 pc = 3.26 ly One kiloparsec (kpc) is equal to one thousand parsecs: 1 kpc =‘ 103 pc One megaparsec (Mpc) is equal to one million parsecs: 1 Mpc = 10‘ pc Although the awe-inspiring size of the Universe makes necessary the use of such long yardsticks, the measurement of distances is not the only case where we have to introduce new units. For instance, masses of stars and galaxies "are so huge that instead of using kilograms (kg), we use our Sun’s mass itself” as-a unit of mass. It is called a solar mass (Me): 1 M9: 1.992 x103" kg Other common and useful units are: 1 kilogram (or kg) = 1000 grams (or g) 1 meter (or m) = 100 centimeters (or em) = 1000 millimeters (or m) I Angster (or A) = 10‘“ meters (or m) 1 nanometer (or run) = 10" m = 10 A 1 inch (or in) = 2.54 centimeters (or cm) 1 foot (or ft) = 0.305 In 1 mile (or mi) = 1.609 kilometers (or km) A3 ? Problems 3. These questions are designed to give you a feeling for metric and astronomical units as compared to English measures. Express: 1 cm in inches iv. 1 kilometer in miles v. 8.9 x 109 kilometers in AU SIGNIFICANT FIGURES The theory of significant digits is concerned with the useful and n‘ustwonhy digits in a number. Suppose you measure a child's height to the nearest centimeter as 63 cm. This means that you are sure of the 6 and have determined that the 3 is better than 2 or 4; therefore, both the 6 and the 3 are significant digits. A significant digit represents a number whose value is supported by a reliable measurement. Exact numbers may be defined or counted and knoWn to be exactly correct- For example, you can count the number of pages in this appendix”. If there are 10 pages, there is no doubt that this is exact and not 9.7 or 10.] pages. Defined numbers are exact as well. A kilometer is defined to: be exactly 1000 meters. Exact numbers have an infinite number of significant figures. ‘ Inexact numbers are what is yielded by most measurements, whose accuracy is limited by the measuring device used. For example, the smallest unit on most metric rulers is miliimeters. The first estimated digit in length obtained from such a ruler would be 01 millimeters. One can still measure the diameter of a penny to be 18.9 millimeters using this” ruler. When measuring the penny you see that its size lies somewhere between 18 and 19 millimeters and is most likely 18.9 millimeters. The last is- considered significant because it is the best estimate of the person who measured the penny. If only the results of a measurement are reported, the following conventions are used to tell which digits are significant: 1. All non—zero numbers are significant. 2. Zeros used only to position the decimal point are not significant. Three statements cover all cases involving zeros: a . Leading zeros (those to the left of the first nonzero digit) are never significant. b . Imbedded-zeros (those betvveen nonzero digits) are always significant. c. Trailing zeros (those to the right of nonzero digits) are significant only if the decimal point is shown. For example: (a) 0.00334 three sig figs (b) 1.0058 five sig figs (c) 34.00 four sig figs (d) 0.678 three sig figs (e) 0.00004 one sig fig A.4 r_,‘m_\ . (f) 1.38 x 105 three Sig figs (g) 1.3800x105 five sig figs The rules used in this lab for determining the number of significant digits in a calculated final answer are given below: (1.). When adding or subtracting numbers, the result should not be carried beyond the first column having a doubtful figure. For example: , 8.16 + 74 = 82, not 82.16. (2). When multiplying or dividing, the result should have the same number of digits as the least significant number used in the calculation. For example: 8-3 x 1045 = 8700, not 8673.5. (3). To show the results of calculations to the proper number of significant . figures, rounding must be used. In all labs be sure that all the measured and calculated numbers that you record have the intended number of significant digits. ? Problems 4. Find the answers to the following with the correct number of sig figs: (iv) 4.256 + 0.008 (v) so -:— 5.876 (vi) (4.6789 x 1024) - (9.267 x 1073) Be sure to use the correct number of sig figs for all the following problems! UNIT TRANSFORMATIONS With so many units around, it is not surprising that we often have to make unit transformations. For instance, we may want to express a result in certain units, but we are given data in different units. We will have to deal with unit transformations many times during this courSe, so it is imperative that you feel comfortable with them. There is a very simple and general way to perform unit transformations, which will be illustrated with the following two examples. Example 1: You measure the length of your desk to be 48.0 inches long. What is the length of the desk in cm? Use the fact that l in = 2.54 cm. If we are given: 1 in = 2.54 cm A.5 We can divide both sides of the equation by 1 inch to get: 254 I = . (A) 1 m 7 Notice that equation (A) equals 1. Multiplying any side of an equaIiOn by i is allowed in mathematics, so we have: 43.0311 = (48.0 in) x (1) Which is the same as: 48.0in = (48049) x (23m) = 122cm (B) 1-in- Now notice that by doing this, the unit of inches has canceled out, leaving just cm as the final unit for (B). When multiplying an equation by anothet equation that is equal to onethe whalepaint istoputinthedenomitwtortheunitwe wanttogetridafi so that it eventually cancels out. If we knew that the length of the desk is 122 cm and wanted to find it in inches: 122cm=(12263§)x[ )=48.0in 254655 Example 2: A car is traveling at a speed of 60.0 miles per hour. How many meters per second is this? 1 mile: 1.6km, 1 hr: 3,600 sec. We now have to perform several times what we did in the previous example: . (60-01%)x 1.6km x 1,000m 600 _/.hr _ 603m: _ 1-mi- 1km __ 96,000111 __ 267 I ‘m‘ "' 1hr _ (1_ 3,6005ec ‘3,6005ec" ‘mm X —-.— he 1%?- ? Problems 5. The purpose of the question is to see if you can keep track of units and compute answers to simple formulae. Calculate the mass of the Earth by using the formula; M: 21!: 9 ER 3 A6 , ’"W-w where M is the mass of the Earth (what we are trying to find); p = 5.5 glam3 is the average density of the Earth; and Si = 6.4 x 108 cm is the radius of the Earth. Express your results in scientific notation (i) in grams, (ii) in kilograms. 6. Suppose your car was modified so you could travel through space. At a velocity of 65 mi/hr, and driving non-stop, how long would it take for you to get to: (i) Pluto (usually the outermost planet in our solar system), at a distance of 5.85 x 109 km? (ii) Proxima Centauri (the nearest star to the Sun), at a distance of 1.33 pc? ERRORS When we perform an experiment, we usually want to take measurements of physical quantities, such as lengths, times, weights, etc. We may then substitute these numbers into formulae and seek relations among them. In this process, there are two sources of error involved: Human errors These include: Reading errors, eg. misreading a ruler when measuring Calculation errors, eg. punching the wrong number in the calculator Bias, Le. errors induced into data collection by the experimenter when helshe knows the outcome of the experiment ' These errors can be avoided by being more careful, repeating the experiment, asking a friend to verify our numerical results, and so on. Errors due to limitations imposed by nature For example: Measuring error, 3.3. when we measure the length of a line with a ruler, our reading cannot by any better than the ruler’s smallest subdivision. We can of course use a better ruler or a different measurement tool, but this will only decrease the error; it will not-eliminate it altogether. Unlike human errors, these errors are unavoidable. ' In the course of this lab, you will have to take many measurements. In most labs . you will only he asked to explain what are some possible sources of error in your results. It should be understood that you are asked to mention errors that occur due to the second reason, i.e., due to limitations imposed by the laws of nature, and not due to human error. Occasionally, you will be asked to find a numerical answer to describe the errors of the experiment. The easiest way to do this is mathematically to compare your observed result with the known result value in the following way: ' A.7 Resultm — Resth ‘7 ERROR = a Re sultm x 100.0 This is called percentage error. ? Problems '7. Through an experiment you conducted you found that the Moon’s diameter is 3.550 x 103 Inn. The accepted value for the Moon’s diameter is 3.476 x 103 km- What is the percentage error of your result? SCALE FACTORS We will need to use scale factors when examining photographs and pictures- These pictures are representative of real life in every way except one. The distances between objects in the pictures are obviously not the actual distances between actual objects. They are sealed so as to fit on the photograph. The best way to eXpla‘in the concept is with an example. 7 Suppose you have an aerial photograph of the University of Florida and you want to determine the size of the campus. On closer" inspection of the photograph, you find The Swamp is clearly visible. You know that the length of a regulation football field is 120 yards (including the endzones). Knowing this, you can develop a scale for the photograph. Suppose you measure the length of thetfieid in the photograph to be S centimeters. Then we say that: 120 ds scale = mu??— 2 24yardslcm 5cm Now you can determine the true length of campus along University Avenue. Suppose you measure the length of University Avenue on the photograph, and you find it to be 40 centimeters. Now multiply by the previously determined scale: real length = (scale) x (ruler measurement) = [24 x (40cm) 2 960 yards In other words, to find the scale of a picture, we find a feature whom length we somehow" know. Then we measure its length on the picture with a ruler. The ratio of these two lengths is the scale, and we can use it to determine the sizes of other features on the picture. All ? Problems 8. You measure the distance from Gainesville to Miami on a map. The measured distance is 14 cm. The map gives a scale of 40 km per cm. What is the actual distance in kilometers? 9. In the picture above the distance between the cities Que Que and Harare is 100 miles. ,. (i) What is the scale of this picture in miles/mm? (ii) How long is Lake Kariba in miles? COMMON MATH FUNCTIONS -' This section is designed to refresh your memory on basic math functions you will see in this course, as well as to acquaint you with how to use the special function keys of your calculator. -._' To perform logarithms with your calculator, locate the key marked LOG or log on your calculator. If you want to calculate log 5 and your calculator is a graphing calculator, you most likely have to press LOG then 5. Sometimes you have to press = to get the answer.__ Otherwise, you first press 5 then LOG. Try it! logs = 0.698970004 Ifyou have the equation: x = log y and you have to solve for y, you must put both sides in the exponent of 10 (in this course, logarithms are base 10 unless otherwise noted). It will look like this: 10 " = 10 “g Y But a basic identity of logarithms is 101°“ 2 y. So we get: 10"=y A.9 BE ‘ To perform 10 7'9 on your calculator (don’t confuse this with scientific notation), first locate the button labeled 10 x or 10 3' on yOur calculator. Type 7.9 and then the 10 ‘ (or 10 y). Sometimes you must press = to get the answer. Try it! 107'9 = 7943282147 There is yet another important exponentiai key to know about. What if the base of the exponential isn’t 10? Occasionally, you must perform calculations like 5 3 or 3.2 7'89. To perform a calculation such as 5.6 9, first find the button marked x’ or y’. Most of the time you get the result by typing 5.6 then x’ (or y‘) then 9 then = . Try it! 5.69 = 54l6169.448 Often we have an answer already displayed on our calculator that we would like to divide into 1. Rather than writing the number dowu or storing it, clearing the calculator and doing the operation, many calculators are equipped with a special button just for this task. I: E To perform an operation like 3%; first locate the button labeled 11x. Type in 2.867 then press 11x. Try it! 3.3;? = 0348795651 ? Problems . 10. When solving these problems use the calculator techninIes above: ' 6.67 10” (i) Lugs“) =? (ii) You have the following equation: 3 = log: I). Let s = 32.4 and solve for t. c. max. 7 d. Now find 5.4’. M96 Dee. 7997 A.10 _. mm} ...
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Appendix A from ast 1022 Lab Manual - !&amp;quot;‘\ ....

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