LECSS Physics 11 Introduction to Physics and Math Methods 1 Physics 11 Introduction to Physics and Math Methods In this introduction, you will get a more in-depth overview of what Physics is, as well as an introduction to (or review of) the numerous math and problem-solving skills required in this course. I.1: What is Physics? If you Google the question: What is Physics?, you will be hard-pressed to get a good, concise definition. One of the most thorough definitions may be found at: http://biotsavart.tripod.com/physdef.htm Go to the site above and read the first two sections (Physics and Scope of Physics). Read the entire document if you wish. Let your teacher know if the link above is broken! The web-site above gives a pretty lengthy discussion of the history of Physics and many of its branches. However, it does not emphasize the extremely mathematical nature of Physics. Physics is the most mathematical of all the disciplines of Science. While the various branches of Physics encompass everything from the smallest sub-atomic particles to the universe in its entirety, all the branches of Physics have mathematics in common. Physics is about observation and measurement. From those things Physicists try to develop mathematical models and discover mathematical laws which describe and explain what has been observed and measured. I.2: Mathematics in Physics 11 In this section we will examine some mathematical concepts that are important for successful completion of this course. These concepts are: the metric system of measurement, scientific notation and metric prefixes, significant figures and rounding, graphing scientific data and the required steps for problem-solving. I.2.1: The Metric System of Measurement In this course we will only use metric (System International or SI) units of measure. Every system of measurement has established base units from which all other units are defined. While we will not encounter all the base units of the SI system in this course, you can find a list of them on the left side of page 548 of the Heath text. Page 549 lists many of the important quantities in the metric system, the accepted symbols used to represent those quantities and the standard SI units used to measure them. Please turn to page 548 and read through the list of SI base units. Also study the table on page 549 before continuing with the next section.
LECSS Physics 11 Introduction to Physics and Math Methods 2 1.2.2: Required Steps for Problem-Solving in Physics As you work through the assignments and write tests in this course, you will encounter both multiple-choice and written-response questions and problems. Some of these questions and problems will be conceptual (i.e., definitions or explanations), others will be computational. For written-response questions and problems, you are expected to communicate your knowledge and understanding of physics principles in a clear and logical manner. What follows is an outline of what is expected in your written-response answers to questions and problems. 1.2.2.1: Answering Conceptual Questions When asked to define terms or provide explanations, it is important to be clear and concise. While not mandatory, it is usually best to answer using point-form. For the most part, using point-form will help you stay on-topic. Here are some guidelines for answering conceptual questions: You do not have to re-state or re-iterate the question. Just get to the point! Use short sentences. Using long sentences will cause you to stray from the topic being addressed. Ensure that you have answered the question! For example, if a question asks you to state if some statement is true or false and then explain using the principles of Physics, start by stating true or false. If you start with the explanation you may forget to answer the question! Re-read your answer. Check for missing words and spelling errors (spelling counts if it is a scientific term!). Include a diagram if it helps with your explanation. If a diagram is asked for, make certain one is included. 1.2.2.2: Answering Computational Problems Showing your work when answering computational problems in Physics means much more than simply showing the arithmetic operation done to get the final answer. Your solutions to computational problems need to show that you understand what physical quantities are involved in the problem, what equation(s) relates those quantities and how the numerical values given in the problem are used to get to the answer. When you write an essay in English class, there is an expectation of how your essay is to be constructed. You start with an introduction. This is followed by the body of your essay which leads to the conclusion. When you answer a computational problem in Physics, there is an expectation of how to set up the problem, how to show the processes that lead to answer, and how to present your final answer.
LECSS Physics 11 Introduction to Physics and Math Methods 3 There are a number of suggested and required steps that must be included in your solutions to computational problems. Even if you are unable to reach the final answer to a problem, you may receive partial marks for setting up the problem as described below and writing the relevant equation(s). Listed below are a number of steps that you need to follow to maximize your success when solving computational problems in Physics. A. Read the problem carefully to extract the important given information. Write each numerical value given as being equal to its correct algebraic symbol. DO NOT make up your own notation for representing physical quantities (e.g., don t use D to represent distance; the proper symbol is d ). B. Write down the algebraic symbol of the quantity you are asked to calculate and set it equal to a question mark. It is hard to proceed to the final answer if you don t know what you are looking for. C. It is sometimes helpful to draw a picture of the situation. Sometimes an illustration is a REQUIRED part of your solution (I ll let you know when). Illustrations are often useful tools for organizing given information. D. Write down the equation(s) that you think you need to solve the problem as they appear on the equation sheet provided, WITHOUT the given numerical values substituted into the equation. NOTE: not all the relevant equations for solving problems in this course appear on the equation sheet for the course. You will need to memorize a small number of equations and be able to derive a few others using your understanding of the relevant concepts. E. If necessary, perform the required algebraic step(s) needed to solve the equation(s) in step D for the quantity you are looking for. Any algebraic steps shown must be clear and logical. F. Substitute the given numerical values into the algebraically-solved equation. NOTE: you do not have to include units with each numerical value in this step. Only your final answer needs to include units. For the most part, I tend to include units in this step. Steps E and F above describe the preferred method of problem-solving. If your algebra skills are weak, you are permitted to substitute the given numerical values into the relevant equation(s) before doing the necessary algebra. G. Perform the required calculation needed to get your final answer. Do this in a single calculation on your calculator (when possible), not as a series of calculations. H. Write down the unrounded calculated answer first. You don t need to write down all the digits shown in your calculator display, and then write down the rounded answer. Use the appropriate rules for significant figures to round your answer (these rules will be
LECSS Physics 11 Introduction to Physics and Math Methods 4 discussed later). Add appropriate units to your final answer. It is preferred that your final answer is underlined. As we work through this course, I will demonstrate the expectations for answering computational problems when working through example problems. I.2.3: Scientific Notation and Metric Prefixes In this course we will encounter some very small numbers, some very large numbers and many numbers in-between. An example of a very small number could be the wavelength of red light (0.000 000 62 m). The radius of the Earth measured in metres is a very large number (6 380 000 m). [NOTE: in the metric system we use spaces to separate digits into groups of 3 - we do not use commas. For example, one million is written as 1 000 000 not 1,000,000] Writing the zeros that are part of very small and very large numbers is inconvenient. Scientific notation (also called exponential notation is one method of writing small and large numbers that eliminates the need to write all or most of those zeros. Large and small multiples of 10 can be written using exponents. Some examples are shown in the table below. Multiple of 10 Written Using Exponents 0.000 001 10-6 0.000 01 10-5 0.000 1 10-4 0.001 10-3 0.01 10-2 0.1 10-1 1 10 0 10 10 1 100 10 2 1 000 10 3 10 000 10 4 100 000 10 5 1 000 000 10 6 The radius of the Earth measured in metres can be written as: Since 1 000 000 = 10 6 we can write this as: 6.38 x 1 000 000 m 6.38 x 10 6 m When written in scientific notation, a number consists of three parts: the coefficient, the base and the exponent. There are some rules for each of those three parts.
LECSS Physics 11 Introduction to Physics and Math Methods 5 6.38 x 10 6 m exponent coefficient base Coefficient: This is always a number greater or equal to one and less than 10. This means that there is ONLY ONE NON-ZERO IN FRONT OF THE DECIMAL POINT. Base: For numbers written in scientific notation, the base is always 10. Exponent: This is always a positive value for large numbers and a negative value for small numbers. The size of the exponent indicates the number of place-values that the decimal point is moved to get it behind the first non-zero digit of the number. For example: 6.380 000. 0.000 000 6.2 7 place-values Written in scientific notation, the wavelength of red light is 6.2 x 10-7 m. Notice: 6 place-values If the decimal point is moved to the left, the exponent is positive. When the decimal point is moved to the right, the exponent is negative. The units of measure come after the base and exponent (we would not write the radius of the Earth in metres as 6.38 m x 10 6 ) Additionally: A negative exponent does NOT mean that the entire number is also negative. For example: 2.6 x 10-2 = 0.026 NOT -0.026 Practice: Write the following numbers in scientific notation. a) 2500 b) 0.00035 c) 470 000 d) 0.0000567 e) -850 f) -0.0284 Using Scientific Notation on Your Calculator All scientific calculators have some built-in method of entering numbers in scientific notation. This will require using a button labelled EXP or EE. On a TI-83. 83+, 84 and 84+
LECSS Physics 11 Introduction to Physics and Math Methods 6 calculators, the EE function is a second function above the comma key (located above the number 7 key). To input 6.38 x 10 6 on one of the TI calculators mentioned above, type the following: 6. 3 8 2nd, 6 Your TI calculator will display 6.38E6 as shown in the first line of the screen capture at the left. The E stands for Exponential. Other calculators may display something different. IMPORTANT: AS PART OF YOUR SOLUTIONS TO WRITTEN PROBLEMS IN THIS COURSE, YOU MUST NEVER, EVER WRITE A NUMBER IN SCIENTIFIC NOTATION USING E IN PLACE OF THE x 10. When the exponent is negative, nearly all scientific calculators require that you use a sign change key, and NOT the subtraction key, to make the exponent negative. Changing From Scientific Notation Back to Standard or Common Notation At times it may be necessary to do the reverse of what was described above. Standard notation and common notation (the text uses this term) are common names given to the way that numbers are commonly written. Changing a value back to standard (common) notation requires that we do the reverse of what was described earlier. If a number written in scientific notation has a positive coefficient, we must move the decimal point to the right. The size of the exponent tells us the number of placevalues that the decimal point must be moved. Example: 2.38 x 10 2 = 238 2 place-values Depending on the size of the exponent, you may need to insert some number of zeros between the last digit in the given value and the decimal point. Example: 4.72 x 10 6 = 4720000 6 place-values
LECSS Physics 11 Introduction to Physics and Math Methods 7 If the exponent is negative, the decimal point must be moved to the left. It will always be necessary to insert some number of zeros in front of the coefficient. Examples: 9.2 x 10-3 = 0.0092 3 place-values 1.60 x 10-19 = 0.000000000000000000160 Mathematical Operations Using Scientific Notation 19 place-values When performing mathematical operations with numbers already written in scientific notation, there is rarely a need to convert the numbers back to regular form. The only time you may need to do so is when you are determining where to round off calculated values. This will be discussed later on. Converting numbers back to regular form adds unnecessary steps to your calculations and is a common source of errors in calculations. Example: Do the following calculation. (2.6 x 10 12 )(3.0 x 10-2 ) Despite instructions to the contrary, some students resist using the built-in scientific notation feature of their calculators. Instead, they choose to use the 10 x feature (located above the log button). While this can lead to the correct answer, it requires more keystrokes. This is shown on right. The screen capture to the left shows the calculation done using the scientific notation function of the TI calculator. Other calculators may display something different. As noted above, you would not write your answer as 7.8E10. The answer is written as 7.8 x 10 10. Using the 10 x feature instead of the scientific notation feature often causes order of operations (BEDMAS) errors when dividing.
LECSS Physics 11 Introduction to Physics and Math Methods 8 NOTE: The solutions to the Sample Problems on page 552 of the textbook show another method of doing calculations using scientific notation. That method is rather oldfashioned ; it is the method that was commonly used before scientific calculators were invented! Example: Do the following division. 1.4 10 2.0 10 3 3 The screen capture to the left shows the calculation done using the scientific notation function of the TI calculator. Other calculators may display something different. Note: in order to display the correct answer on a cheap scientific calculator, you may have to force it to display answers in scientific notation. If you don t know how to do that, ask for assistance. Consider the same calculation done using the 10 x feature (shown on the left). Why is the answer incorrect? The calculation at the left is actually: 1.4 10 10 3 2.0 In order to get the correct answer, the entire denominator would need to be in brackets. This will ensure that the correct order of operations is used. Practice: Perform the following calculations. Ensure that you use the scientific notation feature of your calculator. Express your answers in scientific notation. Don t worry about rounding your answers (the proper way to round calculated answers will be discussed later). a) (1.6 x 10 3 )(4.5 x 10 6 ) b) (9.6 x 10-5 )(1.2 x 10-2 ) c) 5.6 10 2.8 10 3 5 d) 7.5 10 3.0 10 2 8 CAUTION: DON T FORGET TO WRITE DOWN THE POWER OF 10 PORTIONS OF YOUR ANSWERS! THIS IS A COMMON ERROR FOR SOME STUDENTS.
LECSS Physics 11 Introduction to Physics and Math Methods 9 Extra Practice: If you want some extra practice writing and using scientific notation, go to the following website: http://janus.astro.umd.edu/cgi-bin/astro/scinote.pl Metric Prefixes Rather than use scientific notation to express large or small measured values, it is common to use the system of metric prefixes. The table below shows the most commonly used metric prefixes. This table is provided to you on the formula sheets for the course. A more extensive table can be found on page 548 of the textbook. Consider the radius of the Earth measured in metres (6.38 x 10 6 m). We can express this same measure using the metric prefix mega (M). The conversion from metres to megametres is shown below. 1Mm 6 Radius of Earth = 6.38 10 m 6.38 Mm 6 10 m Similarly, the wavelength of red light (6.2 x 10-7 m) can be expressed in micrometres. 1μm 7 Wavelength of red light = 6.2 10 m 0.62 μm 6 10 m You will find that many textbooks tend to use metric prefixes to express large and small values instead of scientific notation. I believe this is done to reduce the printing cost of the book (every exponent adds one-half of a line of text!). Keep this in mind when you work through practice
LECSS Physics 11 Introduction to Physics and Math Methods 10 questions from the textbook. If your answer and the numerical answer given in the text differ by several factors of 10, it may be because a metric prefix has been used to eliminate the use of scientific notation. Practice: Re-write each measured value below using the metric prefix indicated. Answers are on the last page of this handout. a) 0.028 m in cm b) 1250 J in kj c) 7.3 x 10-8 g in ng d) 0.275 A in ma 1.2.4: Rounding Calculated Values and the Rules of Significant Digits (a.k.a. Significant Figures) One mathematical aspect of problem-solving in the physical sciences that gives some students difficulty deals with the rounding of computed numerical values. Why we need to follow a set of rules for rounding off computed answers arises from the fact that every measured value contains an amount of uncertainty, meaning that no measured value is exact. When we do calculations with measured values, this uncertainty must be expressed in the final calculated value we obtain. There are a number of methods of expressing uncertainty in computed answers. The method that is most commonly used is the method of Significant Digits and Rounding. I am of the habit of using the term significant figures instead of significant digits. In every measured value, the significant figures include all the certain digits plus one digit which is uncertain. Identifying the digits in a measured value that are significant is the first step in learning to round-off computed values correctly. NOTE: THESE RULES APPLY ONLY TO MEASURED VALUES. NUMERIC VALUES THAT ARE PRESENT IN MATHEMATICAL EQUATIONS AND COUNTED VALUES ARE CONSIDERED TO BE EXACT AND THUS CONTAIN NO UNCERTAINTY. Identifying Significant Digits There are 5 rules for identifying significant digits in a measured value. Four of these rules involve zeros. Rule 1: All non-zero digits in a measured value are significant. Examples: 2.38 s 3 significant figures 42 m 2 significant figures Rule 2: Zeros between non-zero digits in a measured value are significant.
LECSS Physics 11 Introduction to Physics and Math Methods 11 Examples: 102 m 3 significant figures 4003 s 4 significant figures Rule 3: Zeros at the end of a measured value that contains a decimal portion are significant. Examples: 18.20 m 4 significant figures 1.00 s 3 significant figures Rule 4: Zeros at the front of a measured value are not significant. Such zeros are merely decimal place holders. Examples: 0.058 s 2 significant figures (the 5 and the 8) 0.00260 m 3 significant figures (the 2, the 6, and the last 0) If you find you are uncertain if such zeros are significant or not, rewrite the value in scientific notation. When you do so, the zeros at the front of the value disappear. If we rewrite the above examples in scientific notation, we get: 5.8 x 10-2 s 2.60 x 10-3 m Rule 5: Zeros at the end of a whole number measured value may or may not be significant. Example: 1400 m This could have 2, 3 or 4 significant figures How do we determine the actual number of significant figures in the above value? The only way to be absolutely certain how many of the digits are significant is to ask the person who made the measurement. If that person had rounded the measure to the nearest hundred metres, then only the 1 and the 4 are significant. If the measure was rounded to the nearest ten metres, then the 1, the 4, and the first 0 are significant. If the person rounded to the nearest one metre, then all four digits are significant. Because the significance of the two zeros in the above value is uncertain, such zeros are called ambiguous zeros. Sometimes they are significant, other times they are only decimal place holders. In a perfect world, one should not encounter ambiguous zeros. To indicate the significance of zeros at the end of a whole number measure, scientific notation should be used. To properly illustrate the number of significant digits in the above value, the person should have recorded the value in one of the following ways: 1.4 x 10 3 m if only 2 significant figures 1.40 x 10 3 m if 3 significant figures 1.400 x 10 3 m if 4 significant figures So what does one do if ambiguous zeros are encountered?
LECSS Physics 11 Introduction to Physics and Math Methods 12 If no statement to the contrary is made, you MAY assume that all ambiguous zeros are NOT significant. However, in this course, I prefer that you use Don s Rule for Ambiguous Zeros: If you encounter a measured value with one ambiguous zero, assume it IS significant. If you encounter a measured value with two or more ambiguous zeros, assume that the FIRST ONE IS SIGNIFICANT AND THE REST ARE NOT. Some textbooks will explicitly state something like: All zeros not clearly significant shall be taken as significant. Your textbook discusses significant digits in Appendix B. Their rule (which they often do not follow!) is to assume all ambiguous zeros are significant unless you know otherwise. We won t use that assumption it is ridiculous. In fact, the opposite is often assumed, especially when doing Chemistry! Now that the rules for identifying significant figures has been established, we can examine how these rules are applied to the rounding of calculated values. There are two rules that must be applied; one for addition and subtraction, the other for multiplication and division. The Precision Rule: The first rule to be examined applies to addition and subtraction. There are different ways of defining precision. One definition states that precision is a measure of how closely two or more measures of some quantity correspond. This definition is usually applied to experimentally determined measures of some value. A second definition (my own) is this: When comparing measured values, the value that has its last significant figure in the smallest numerical place value is the more precise measure. This definition of precision may only be applied to two measures of the same type of quantity. That is, we apply when comparing two measures of length, or two measures of mass, or two measures of volume, etc. Given two measures of time such as 21.0 s and 2.48 s, the latter measure is more precise as its final significant figures (the 8), falls in the hundredths place whereas the final significant figure in the other value (the 0) falls in the tenths place. The Precision Rule states that when we add or subtract two or more values, their sum or difference must be rounded to the precision of the least precise value involved. Examples: 1. 21.0 s + 2.48 s = 23.48 s = 23.5 s
LECSS Physics 11 Introduction to Physics and Math Methods 13 The sum of the two numbers had to be rounded to the tenths place because the less precise value (21.0 s) is precise only to the tenths place. The other value is precise to the hundredths place. 2. 985 m + 368 m 36.3 m = 1316.7 m = 1317 m The final answer to the above calculation had to be rounded to the ones place as the least precise values (985 m and 368) are precise only to the ones place. The other value (36.3) is precise to the tenths place. The Accuracy Rule: The second rule to be examined applies to multiplication and division. There are different ways of defining accuracy. One definition states that accuracy is a measure of how closely a value corresponds to an accepted given value. This definition is usually applied to experimentally determined values. For example, if we performed an experiment to determine the speed of light, the accuracy of the result would be determined by how closely that result matches the actual speed of which happens to be 3.00 x 10 8 m/s (to 3 significant digits). A second definition (my own) is this: When comparing measured values, the one possessing the greater number of significant figures is the more accurate measure. This definition may be used when comparing any measured values, not just measures of the same type of quantity. For example, if you are told that the distance to the Sun is 1.50 x 10 11 m and that it takes light 5.0 x 10 2 s for light from the Sun to reach us, the measure of the distance to the Sun is a more accurate measure as it has 3 significant figures while the given measure of time has only 2 significant figures. The Accuracy Rule tells us that when we multiply or divide two or more values, the final computed answer must be rounded to the accuracy of the least accurate value involved in the calculation. Examples: 1. 0.28 m m 0.0608695652 6.1 4.65 s s -2 x 10 m/s or 0.061 m/s The answer to the division above had to be rounded to 2 significant figures because the least accurate value (0.28 m) has only 2 significant figures. The more accurate value (4.65 s) has 3 significant figures.
LECSS Physics 11 Introduction to Physics and Math Methods 14 2. (0.0246 s)(142.5 m/s) = 3.5055 m = 3.51 m The answer to the multiplication above had to be rounded to 3 significant digits because the least accurate value (0.0246 s) has only 3 significant digits. The more accurate value (142.5 m/s) has 4 significant digits. Clear as mud? Good. Here are a couple more rules. When Do We Do the Rounding Off? It is VERY important that only final calculated values be rounded off. What this means is that when you do multi-step calculations, intermediate values are not rounded off before being used in subsequent calculation steps. If you need to write down an intermediate value for some reason, you can either write down ALL the digits in your calculator display, or write down a good number of them and store the value somewhere in your calculator s memory. If your calculator does not have more than one place to store calculated values, it is inferior and should be replaced with something that cost more than $8. What this also means is that you may have to go through and analyze each step in a multi-step calculation to determine where to round your final answer. Example: Later in this course we will study the concept of average speed (v av ). If an object moves a distance, d, in an interval of time, Δt, its average speed is given by: d vav t Knowing an object s average speed, we can calculate the distance it travels in an interval of time using: d = v t av a) If it takes me 0.73 h to drive 24.3 km to work, what is my average speed? b) At the average speed calculated in part a), how far would I drive in 1.36 h? Solution: a) Start by writing down the given information. d = 24.3 km Δt = 0.73 h v av =? v av d t 24.3 km 0.73 h 33.2876... 33 km/h The ellipses (the ) indicates that the answer to the above calculation has more digits. According to the accuracy rule tells us that the above answer must be rounded to 2 significant
LECSS Physics 11 Introduction to Physics and Math Methods 15 figures due to the fact that the measured value of time (0.73 h) has only 2 significant figures. The measurement of the distance (24.3 km) has 3 significant figures, but we round to the accuracy of the least precise value involved. b) Using the calculated value for average speed, we can determine the distance driven in 1.36 h. v av = 33.2876 km/h Δt = 1.36 h d =? d = vav t = (33.2876...km/h)(1.36 h) = 45.27...km = 45 km In the above calculation, the value of t has 3 significant figures. While the value used for the average speed is the unrounded value, this value has only two significant figures as explained above. Because the average speed has only two significant figures, we must round the calculated answer to two significant figures. In the above calculation, we needed to use the unrounded value of the average speed to avoid round-off error. Often, using rounded values in calculations will still lead to a correct answer when rounded, but be aware that on occasion an incorrect answer will result. This is why we must always use unrounded values in calculations. The Odd/Even Rule for Rounding Values Ending in a 5: An obscure and seldom necessary rule to remember involves rounding calculated values ending exactly with the number 5. For example, if we performed the following calculation: 1.23 m Average speed, v av = = 0.615 m/s 2.0 s According to the accuracy rule, the answer to the above calculation must be rounded to 2 significant figures. The final rounded answer is, therefore, 0.62 m/s. If the distance were longer, say, 12.5 m the calculation is: 1.25 m Average speed, v av = = 0.625 m/s 2.0 s Again, the accuracy rule tells us to round this answer to 2 significant figures. Is the answer 0.63 m/s? NOT ALWAYS!!!! Reason: the odd/even rule of 5 s. Some textbooks (including yours), employ this rule which usually applies only to Pure Chemists (as far as I have seen, anyways). Here it is:
LECSS Physics 11 Introduction to Physics and Math Methods 16 For calculated values ending with a 5, we round up if the digit in front of the 5 is odd. If the digit in front of the 5 is even, the 5 is dropped. NOTE: this rule applies only to calculated values ending exactly with a 5 and nothing after the 5 AND when the place value where rounding occurs is immediately in front of that 5. Thus, if we wish to round the value 6.253 to 2 significant digits, it becomes 6.3. It is incorrect to apply the odd/even rule in the following way: Since the number after the 2 is a 5, and 2 is an even number, we drop the 5 This reasoning is incorrect because the number does not end exactly with a 5. If we wish to round the value 6.1453 to 2 significant figures, it becomes 6.1. It is incorrect to extend this rule in the following way: Since the number ends with a value after the 5, the 4 rounds up to give us 6.15. Since the number in front of the 5 we now have is odd, it rounds up and we get 6.2 This reasoning is incorrect because the place value where the rounding occurs (the tenths place) is not immediately in front of the 5 in the calculated value. NOTE: WE WILL NOT USE THE ODD/EVEN RULE OF 5 S IN THIS COURSE. INSTEAD WE WILL USE WHAT I CALL. I PREFER TO USE THE ENGINEER S RULE. This rule states that ALL 5 s are rounded up. This rule comes from the following ideas: 1. If we drop the 5, we may not have enough. 2. If it is slightly too long, we can shave some off. If we drop the 5, it may not be long enough and we can t glue some more back on. Rules for Other Mathematical Operations Calculations in this course are not limited to addition, subtraction, multiplication and division. Sometimes we will square root values, apply exponents or calculate trigonometric ratios for angles. For all other mathematical operations other than addition and subtraction, the accuracy rule is used. Many of the calculations that you will do require application of BOTH the precision rule and the accuracy rule. This is especially true in Unit 2: Kinematics and Dynamics. In such cases, it is important to remember that ONLY FINAL CALCULATED ANSWERS ARE ROUNDED. The following calculation shows how one might calculate the speed, v, of an object fired vertically upward with a speed of 16.0 m/s, 1.30 after it is launched. This calculation requires that we do a multiplication and an addition.
LECSS Physics 11 Introduction to Physics and Math Methods 17 v = 16.0 m/s + ( - 9.80 m/s 2 )(1.30 s) = 16.0 m/s + - 12.74 m/s = 3.26 m/s = 3.3 m/s Notice that all the numerical values used have 3 significant figures, yet the final rounded answer has only 2 significant figures! This can be explained by applying the appropriate rule to each step of the calculation. The order of operations (BEDMAS) tells us that we must do the multiplication before the addition. When we multiply - 9.80 m/s 2 and 1.30 s, we get - 12.74 m/s. Because of the accuracy rule, there are only 3 significant figures in the value - 12.74 m/s. This means that only the 1, the 2 and the 7 are significant digits. We do not drop the insignificant 4 at this point because only final answers are rounded. When we add 16.0 m/s and - 12.74 m/s, we must apply the precision rule. Both values are precise only to the first decimal place. Both values are equally precise. The precision rule tells us that the sum of 16.0 m/s and - 12.74 m/s must be rounded to the first decimal place. This is why the sum (3.26 m/s) is rounded to 3.3 m/s. Required Practice: Complete the following questions from the textbook: Page 550 Practice (a) through (g). Page 551 (left side of page) Sample Problems (a) through (h). Pages 552 and 553 Review 1, 2, 4, 5(i) through 5(q). --------------------------------------------------------------------------------------------------------------------- It is now time to begin your study of Unit 1 Part 1: General Wave Properties. --------------------------------------------------------------------------------------------------------------------