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**Unit Outline--Topics What is Physics? Branches of Science**

Science Terms Scientific models Measuring and Units Powers of Ten and conversions Graphing Experimental Design Science vs. Technology Analyzing in Physics

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**Main Topics Identifying and using significant figures**

Using scientific notation Converting

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Significant Figures Significant figures are the method used to indicate the precision of your measurements. Significant figures are those digits that are known with certainty plus the first digit that is uncertain. If you know the distance from your home to school is between 12.0 and 13.0 miles, you might say the distance is 12.5 miles. The first two digits (1 and 2) are certain and the last digit (5) is uncertain.

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**Significant Figures Chapter 1**

Section 2 Measurements in Experiments Chapter 1 Significant Figures 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. Significant figures are those digits in a measurement that are known with certainty plus the first digit that is uncertain.

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**Significant Figures, continued**

Section 2 Measurements in Experiments Chapter 1 Significant Figures, continued Even though this ruler is marked in only centimeters and half-centimeters, if you estimate, you can use it to report measurements to a precision of a millimeter.

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**Rules for Determining Significant Zeros**

Section 2 Measurements in Experiments Chapter 1 Rules for Determining Significant Zeros

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**Counting Significant Figures**

Examples 50.3 m s 0.892 kg ms 57.00 g kg 1000 m 20 m Scientific notation simplifies counting significant figures. Have students read the rule and then apply it to the two measurements to the right. (Each rule has two examples.) Give them some time to read the rules and try on their own to apply the rules. Then go over the answers with them (below) so they can see where they made mistakes. Answers: three, five, three, one, four, seven, one, one This activity also provides an opportunity to show them that converting the above numbers to different units does not affect the number of significant figures. For example, converting kg into 892 g still yields three significant figures. Also show them that converting g into kg yields kg and not kg because the number must still have 4 significant figures. The zeros at the end are measured values and cannot be ignored.

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**Rules for Rounding in Calculations**

Section 2 Measurements in Experiments Chapter 1 Rules for Rounding in Calculations

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**Rounding Round to 3 figures: 30.24 32.25 32.65000 22.49 54.7511 54.75**

Have students practice rounding answers to the correct number of significant figures (three in this case). Answers: 30.2, 32.3, 32.6, 22.5, 54.8, 54.8, 79.4

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**Rules for Calculating with Significant Figures**

Section 2 Measurements in Experiments Chapter 1 Rules for Calculating with Significant Figures

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**Calculating with Significant Figures**

123 x 5.35 Have students apply the rule to the sample. (Answers are below.) Remind them often that “calculators do not keep track of significant figures.” Some students simply write down every digit they see on their calculator or round off in some arbitrary way. Students are very reluctant to take an answer such as 2540 m and round it off to 2500 m in order to get just 2 significant figures that the problem might require. However, many will simply take a number like m and call it 0.08 m even though the problem might require three significant figures in the answer. Answers: 103.2, 658

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**Identifying and using significant figures**

Using scientific notation Converting

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**SCIENTIFIC NOTATION Left is a positive exponent 1200 m (1.2 x 103 m)**

Used by scientists and engineers to express very large and very small numbers. Changes by powers of ten Count decimal places either to the right or left Left is a positive exponent 1200 m (1.2 x 103 m) Right is a negative exponent m (1.2 x 10-3 m)

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**What is a power of ten? A power of ten represents a decimal place.**

One power of ten can mean ten times less or ten times greater. Examples 10 m and 1 m differ by one decimal place or one power of ten. 0.001 m and m differ by two decimal places or two powers of ten.

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**3.1 x 108 m SCIENTIFIC NOTATION**

The very large measurement 310,000,000 m can be rewritten: number 3.1 x 108 m 10 multiplied by itself 8 times

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**7.1 x 10-7 SCIENTIFIC NOTATION 1 107**

The very small measurement can be rewritten: 7.1 x 10-7 number 1 divided by 10 multiplied by itself 7 times 1 107

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**SCIENTIFIC NOTATION AND YOUR CALCULATOR**

It is possible to compute using numbers written in scientific notation. Here’s how it’s done: For 3 x 108 x 85 Enter the number ‘3’ Press 2nd and then the ‘EE’ key. Some calculators (Casio) use the ‘EXP’ key Enter ‘8’ for exponent (press the -/+ key if exponent is negative) Press multiplication key Enter ‘85’ Press = to solve the problem Answer is x 1010

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**Identifying and using significant figures**

Using scientific notation Converting

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Prefixes It might be a good time to let students know which prefixes are more commonly seen, such as micro through mega. They will need to know these in order to convert units.

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**Prefixes represent different powers of ten**

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Converting Units Build a conversion factor from the previous table. Set it up so that units cancel properly. Example - Convert 2.5 kg into g. Build the conversion factor: This conversion factor is equivalent to 1. 103 g is equal to 1 kg Multiply by the conversion factor. The units of kg cancel and the answer is 2500 g. Try converting .025 g into mg .22 km into cm When presenting the example, you may wish to ask students for the conversion factor before you show it. Be sure students know that they must set up the conversion factor such that the units cancel properly. To discuss this issue, ask students how we know that the conversion factor is 103 g /1 kg rather than 1 kg /103 g. Answers: 0.25 g = 250 mg, 0.22 km = 22,000 cm. Note that the second problem requires two conversions (if using the table), first km into m, and then m into cm.

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**Classroom Practice Problem**

If a woman has a mass of mg, what is her mass in grams and in kilograms? Answer: g or 60 kg Show students how to get the conversion factor using the table (1 g / 1000 mg). The reverse (1000 mg / 1 g) will not work because the mg will not cancel out. Similarly, they need to find the conversion from g into kg. In order to make the grams cancel, the conversion factor is 1 kg / 1000 g.

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Dimensional Analysis Dimensions can be treated as algebraic quantities. They must be the same on each side of the equality. Using the equation y = (4.9)t2 , what dimensions must the 4.9 have in order to be consistent? Answer: length/time2 (because y is a length and t is a time) In SI units, it would be 4.9 m/s2 . Always use and check units for consistency. It is important to stress the use of units at all times. Discuss the example in the text showing the calculation of the time required when the speed is given in km/h and the distance traveled is given in km. You could also discuss quantities such as density in g/cm3 or tire pressure in lb/in2 to show them that you can often deduce the equation from the units. (For instance, the equation for density is D = m/V, because g is a measure of m and cm3 is a measure of V.) However, be sure to show students that unit analysis does not always yield the correct equation. Tell them that the units for acceleration are m/s2 and ask them to write an equation. They will probably come up with (a = d/t 2) and this is not a correct equation. Try rewriting the units of m/s2 as (m/s)/s and see what they come up with for an equation.

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**How do I interpret the prefixes?**

1 meter is 100 power 10 meters are 101 power milli- is 10-3 power or m (three powers of ten less than 1 meter or three decimal places less) kilo- is 103 power or 1000 m (three powers of ten more than 1 meter or three decimal places greater) giga- is 109 power or 1,000,000,000 m (nine powers of ten more than one meter or nine decimal places greater)

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Why Convert? To compare the results from measurements using different units, one unit must be converted into the other unit. Two basic types System conversions English to metric example: inches to centimeters Power of ten conversions Change in prefix reflects powers of ten example: meters to centimeters

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How do you convert? Use the factor-label method (also called dimensional analysis) 1. decide what must be converted 2. select conversion factor 3. set up factoring equation 4. perform math and solve

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**Meters in a kilometer? 103 m = 1 km**

Meters in a millimeter? m = 1 mm 0.001 m = 1 mm

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**Sample Problem Chapter 1**

Section 2 Measurements in Experiments Chapter 1 Sample Problem A typical bacterium has a mass of about 2.0 fg. Express this measurement in terms of grams and kilograms. Given: mass = 2.0 fg Unknown: mass = ? g mass = ? kg

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**Sample Problem, continued**

Section 2 Measurements in Experiments Chapter 1 Sample Problem, continued Build conversion factors from the relationships given in Table 3 of the textbook. Two possibilities are: Only the first one will cancel the units of femtograms to give units of grams.

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