A Mathematical Toolkit. Introduction: Chapter 2. Objectives

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1 A Mathematical Toolkit 1 About Science Mathematics The Language of Science When the ideas of science are epressed in mathematical terms, they are unambiguous. The equations of science provide compact epressions of relationships between concepts. The methods of mathematics and eperimentation have led to enormous successes in science. Introduction: Chapter 2 Objectives Define the SI standards of measurement. Use commen metric prefies. Estimate measurements and solutions to problems. Perform arithmetic operations using scientific notation. SI units for Physics The SI stands for "System International. There are 3 fundamental SI units for LENGTH, MASS, and TIME. They basically breakdown like this: SI Quantity Length Mass Time SI Unit Meter Kilogram Second Of course there are many other units to consider. Many times, however, we epress these units with prefies attached to the front. This will, of course, make the number either larger or smaller. The nice thing about the prefi is that you can write a couple of numbers down and have the unit signify something larger. Eample: 1 Kilometer The unit itself denotes that the number is actually larger than "1" considering fundamental units. The fundamental unit would be 1000 meters

2 Metric Measurement Most scientists use the metric system and SI units when collecting data and performing eperiments Metric System - decimal system of measurement based on certain physical standards and scaled on multiples of 10 (page 16) Metric Prefies These two measurements are different. 15 meters 15 kilometers How? Metric Prefies These two measurements are different. 15 meters 15 kilometers How? One has a prefi Metric Prefies You will need to convert between different metric units like the following 15 meters 15 kilometers How do you do this? Metric Prefies You will need to convert between different metric units like the following 15 meters 15 kilometers How do you do this? Know your prefies and how they relate to the base unit

3 Metric Prefies A prefi will make the unit bigger or smaller than the base unit We will use the base unit meter in this eample But remember: Prefies have the same number meaning when attached to other units like seconds, liters, etc. Bigger Metric Prefies Prefi: deca- Symbol: da Meaning: ten Metric Prefies 1 dam or decameter equals 10 meters Bigger Metric Prefies Prefi: hecto- Symbol: h Meaning: one hundred Metric Prefies 1 hm or hectometer equals 100 meters Bigger Metric Prefies Prefi: kilo- Symbol: k Meaning: one thousand Metric Prefies 1 km or kilometer equals 1000 meters Bigger Metric Prefies Prefi: mega- Symbol: M Meaning: one million Metric Prefies capital M Smaller Metric Prefies Prefi: deci- Symbol: d Meaning: one tenth Metric Prefies 1 Mm or Megameter equals meters 10 dm equals 1 m 1 dm

4 Smaller Metric Prefies Prefi: centi- Symbol: c Meaning: one hundredth 100 cm equals 1 m Metric Prefies Smaller Metric Prefies Prefi: milli- Symbol: m Meaning: one thousandth 1000 mm equals 1 m Metric Prefies 1 cm 1 mm Metric Prefies Metric Prefies Smaller Metric Prefies Prefi: micro- Symbol: µ Meaning: one millionth µm equals 1 m Too small to read on a meter stick SI PREFIX Megakilohectodeca- None decicentimillimicro- Unit Symbol M k h da ---- d c m µ Meaning in words million thousand hundred ten ---- tenth hundredth thousandth millionth Mathematical meaning 1,000,000 1, /10 1/100 1/1000 1/ Most commonly used prefies in Physics Prefi Factor Symbol Mega- ( mostly used for radio station frequencies) 10 6 M Kilo- ( used for just about anything, Europe uses the Kilometer instead of the mile on its roads) Centi- ( Used significantly to epress small distances in optics. This is the unit MOST people in AP forget to convert) Milli- ( Used sometimes to epress small distances) Micro- ( Used mostly in electronics to epress the value of a charge or capacitor) Nano ( Used to epress the distance between wave crests when dealing with light and the electromagnetic spectrum) K c m 10-6 µ 10-9 Tip: Use your constant sheet when you forget a prefi value n Conversions outside the metric system LENGTH 1 in = 2.54 cm 1 ft = m 1 mi = 5280 ft = km 1 m = ft 1 km = mi MASS 1 oz = g 1 kg = 2.2 lb TIME 1 hr = 60 min = 3600 s 1 day = 24 h = min = s 1 yr = 365 days = s SPEED 1 mi/h = km/h = ft/s = m/s 1 km/h = mi/h = m/s = ft/s

5 Estimates The ability to make rough estimates without actually solving a problem is a skill you will find useful all your life. Rough estimates have to start with reasonable lengths, times, weights, etc... Arrange these in order of largest to smallest measure of length dm dam mm m hm km cm Arrange these in order of largest to smallest measure of length dm dam mm m hm Place a 1 in front of the unit that is bigger Net write how many of the smaller unit fit in the bigger unit km cm Km hm dam m dm cm mm Scientific Notation

6 Doing a conversion Convert 65 m to km 1.Turn this into a question?km = 65m Doing a conversion Convert 65 m to km 1.Turn this into a question?km = 65m 1 km 1000 m 2. Draw the conversion line 3. Write the unit we are converting out of on the opposite side of the line (so it will cancel out later) 4. Write what you are converting into on the opposite side 5. Place a 1 net to the larger unit 6. Place the correct conversion factor net to the other

7 ?km = 65m Doing a conversion 1 km 1000 m 7. Cross out the units that have canceled out 8. Check to see if you are now at the desired unit 9. If yes.. Grab a calculator If no.. You will need another conversion line ( ) 65 1 ( 1000 ) Doing a conversion Using your calculator to get a successful result 1.Open parenthesis, multiply everything in the numerator, close parenthesis 2.Hit divide 3.Open parenthesis, multiply everything in the numerator, close parenthesis 4.solve cm to m?m = 32 cm Get some practice Get some practice minutes to seconds?s = 14 min cl to L?L = 507 cl Get some practice m to km?km = m Get some practice or

8 Get some practice cm 3 to ml?ml = 13 cm L to ml?ml= 3.004L (1cm 3 = 1 ml) kg to g?g = 86 kg mi to km?km = mi (0.62mi = 1km) ft to inches Net level of difficulty?in = 0.15ft km to mm?mm= 56 km Now were there Calculator time (1ft = 12in) Not there yet need another conversion line

9 2. 29 kg to cg km to µm?cg = 29 kg = cg or cg? µm = km = µm or µm Net level of problem The trick here is ignoring the part of the unit you are not working with 4. 69,000 dg to kg?kg = 69,000 dg km/min to m/s? m = 25 km s 1 min = m/s = 6.9 kg 1. Do eactly what we did before to convert 2. The numerator is now what you want, time to do the same with the denominator 3. To cancel out a denominator we need to start by placing the same unit up top km/h to mm/s km to ft? mm = km s 1 hr? ft = 20 km = ft (1 m = ft)

10 Dimensional Analysis Dimensional Analysis is simply a technique you can use to convert from one unit to another. The main thing you have to remember is that the GIVEN UNIT MUST CANCEL OUT. Suppose we want to convert 65 mph to ft/s or m/s. miles 1hour 1min 5280 ft ft 65 = 95 hour 60min 60sec 1mile s ft 95 s 1 meter 95 1 = ft m / s Good Accuracy - Poor Precision Good Accuracy - Good Precision Poor Accuracy - Good Precision 4. Dimensional Analysis a. No number without a unit, UNLESS IT IS A RATIO b. Dimension - qualitative nature of a physical item (1) E - length - the separation between two points (2) can use brackets to denote dimensions [ ]

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13 . ** ** What should do you do now? Read chapter 2. Study Chapter 2 Summary. Know the key terms listed in the summary. Complete the Reviewing concepts Smile...it will get a more fun... **