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# Measurements & Calculations

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1. Measurements & Calculations Chapter 2

2. Section 2.1 Objective: show how very large or very small numbers can be expressed as the product of a number between 1 and 10 AND a power of 10 Scientific Notation

3. Scientific Numbers In science, we often encounter very large and very small numbers. Using scientific numbers makes working with these numbers easier 5,010,000,000,000,000,000,000 a very large number (the number of atoms in a drop of water) 0.000000000000000000000327 a very small number (mass of a gold atom in grams)

4. Scientific Numbers RULE 1 As the decimal is moved to the left The power of 10 increases one value for each decimal place moved Any number to the Zero power = 1 0 450,000,000 = 450,000,000. x 10 8 2 3 0 1 450,000,000 = 450,000,000. x 10 8 4.5 x 10

5. Scientific Numbers RULE 2 As the decimal is moved to the right The power of 10 decreases one value for each decimal place moved Any number to the Zero power = 1 0 0.0000072 = 0.0000072 x 10 -6 -2 -3 0 -1 0.0000072 = 0.0000072 x 10 -6 7.2 x 10

6. Objective: to learn english, metric and SI systems of measurement Section 2.2 and 2.3 UNITS and MEASUREMENTS OF LENGTH, VOLUME & MASS

7. Section 2.3 Objective: understand metric system for measuring length, volume and mass Measurements of length, volume & mass

8. FundamentalQuantities Property English Unit Metric Unit masssluggram 1.0 slug14,590 g lengthfootmeter 1.0ft0.305 m volumequartliter 1.06qt1.0L

9. Property Metric Unit English Unit time second second temperature Kelvin Fahrenheit

10. Section 2.5 Objective: to learn how to determine the number of sig figs Significant Figures

11. Significant Figures • All number other then zero are significant • Ex. 23 = 2 sig figs • Leading zeros- zeros that are at the beginning of a number are NEVER significant • Ex 034 = 2 sig figs and .0578 = 3 sig figs • Trapped zeros – zeros that are trapped between two other significant figures are ALWAYS significant • Ex 304 = 3 sig figs and 8.0091 = 5 sig figs • Trailing zeros – zeros that are at the end of a number – depends on if there is a decimal point expressed in that number • If there is a decimal point showing in the number then the zeros are significant • Ex 60 = 1 si fig but 60. = 2 sig figs and 60.0 = 3 sig figs • Ex .05 = 1 sig figs • If there is NOT a decimal point showing in the number then the zeros are NOT sinificant

12. Example: 120000 120000.

13. Significant Figures 120000 No decimal point 2 sig figs Zeros are not significant! 120000. Decimal Point All digits including zeros to the left of The decimal are significant. 6 sig figs

14. Significant Figures 1005 All figures are Significant 4 sig figs Zeros between Non zeros are significant 123.00 All figures are Significant 5 sig figs Zero to the Right of the Decimal are significant

15. Significant Figures 0.00523 3 sig figs Zeros to the right of The decimal with no Non zero values Before the decimal Are not significant 0.0052300 5 sig figs Zeros to the right of the decimal And to the right of non zero values Are significant

16. SignificantFigures Exact equivalences have an unlimited number of significant figures 1 in = 2.54cm Therefore in the statement 1 in = 2.54 cm, Neither the 1 nor the 2.54 limits the number of Sig figs when used in a calculation The same is true for: 4 quarts = 1 gallon 100 centimeters = 1 meter 1000 grams = 1 kilograms and so on !

17. Exact numbers (numbers that were not obtained using measuring devices, but determined by counting) also have an unlimited number of sig figs Examples: 3 apples 8 molecules 32 students

18. Section 2.4 Uncertainty in Measurement Objective: to understand how uncertainty in measurement arises Difference between accuracy and precision

19. Significant Figures • Significant figures are used to distinguish truly measured values from those simply resulting from calculation. Significant figures determine the precision of a measurement. Precision refers to the degree of subdivision of a measurement. • As an example, suppose we were to ask you to measure how tall the school is, you replied “About one hundred meters”. This would be written as 100 with no decimal point included. This is shown with one significant figure the “1”, the zeros don’t count and it tells us that the building is about 100 meters but it could be 95 m or even 104 m. If we continued to inquire and ask you to be more precise, you might re-measure and say “ OK, ninety seven meters. This would be written as 97m. It contains two significant figures, the 9 and the 7. Now we know that you have somewhere between 96.5m and 97.4m. If we continue to ask you to measure even more precise with more precision, may eventually say, “97.2 m”. • THE PRECISION OF YOUR MEASUREMENT IS DICTATED BY THE INSTRUMENT YOU ARE USING TO MEASURE!!!!

20. Precision = Accuracy ACCURACY MEANS HOW CLOSE A MEASUREMENT IS TO THE TRUE VALUE PRECISION REFERS TO THE DEGREE OF SUBBDIVISION OF THE MEASUREMENT FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR MEASUREMENT IS VERY PRECISE BUT INACCURATE ! MEASUREMENTS SHOULD BE ACCURATE AND AS PRECISE AS THE MEASURING DEVICE ALLOWS

21. Illustration of accurate vs percise

22. You tell me. What is it?

23. Precision & Measurement Measurements are always all measured values plus one approximated value. The pencil is 3.6 cm long. 1 2 3 4 5 6 7 With more calibration a more precise measurement is possible The pencil is 3.64 cm long! • 4 3.6 3.7 The calibration of the instrument determines measurement precision Now 3.640 cm !

24. Scales and sig figs • In our class • Write down what the scale says • Most scales are taken to the hundreths place

25. Graduated Cylinders & Thermometers First – figure out scale Then – take measurement out to one guess past certainity

26. Section 2.5 continued Math and Sig Figs

27. Round each number to one sig fig: If the digit is to be removed: Is less than 5, the preceding digit stays the same EX. 1.49 rounds to ???? _____________ is equal to or greater then 5, the preceding digit is increased EX. 1.509 rounds to ???? _____________ In calculations: carry the extra digits through to the final result AND THEN round off Rules for rounding off

28. Addition/Subtraction with Sig Figs Adding and subtracting with significant figures. The position, not the number, of the significant figures is important in adding and subtracting. For example, 12.03 (the last sig fig is in hundredth place (0.01)) + 2.0205 (the last sig fig is in ten thousandth (0.0001)) 14.0505 14.05 (the answer is rounded off to the least significant position hundredths place)

29. Adding & Subtracting Sig Figs The numbers in these positions are not zeros, they are unknown 123.6 + 42.326 _ 165.946 Don’t even look at The 6 to determine Rounding. Only Look at the 4 165.9 The answer is rounded to the position of least significance

30. The result of multiplication or division can have no more sig figs than the term with the least number. *ex. 9 x 2 = 20 since the 9 has one sig fig and the 2 has one sig fig, the answer 20 must have only one and is written without a decimal to show that fact. * By contrast, 9.0 x 2.0 = 18 each term has two sig figs and the answer must also have two. *4.56 x 1.4 = 6.384 How many sig figs can this answer have? 6.4 (2 sig figs) Multiplying/Dividing with Significant Figures

31. Section 2.6 Objective: learn how to apply dimensional analysis to solve problems Dimensional analysis

32. NO KING HENRY • You must use dimensional analysis to convert from metric to metric • You must use your brain and logic to do this K H D b d c m

33. Some Common Metric Prefixes Prefix Multiplier Example ----------------------------------------- milli 0.001 milliliter centi 0.010 centimeter deci 0.10 decigram kilo 1000 kilometer micro 10-6 microgram 6 Mega 10 megabyte 9 Giga 10 gigabyte

34. From the last slide we learned the meaning of some of the common prefixes, BUT we are going to learn to dimensional analysis using the root prefixes and deciding bigger/smaller.

35. Some Common Metric Prefixes Prefix Multiplier Prefix ----------------------------------------- milli 1000 milli centi 100 centi deci 10 deci kilo 1000 kilo micro 10 6 micro 6 Mega 10 mega 9 Giga 10 giga

36. Conversions YOU Need to Memorize • Length • 1in = 2.54 cm • 39.37 in = 1 meter • 1 mile = 5280 feet • Mass • 1kg = 2.2 lbs • 1lb = 454 grams • Volume • 1 liter = 1.06qts • 1 gallon = 3.79 liters

37. Dimensional Analysis Rules • 1.37days = ? minutes • Always start with the known value over the number 1 • Always write one number over the other • Always, Always, Always, Always, Always write/include the unit with the number • 1.37 days 1

38. Single step examples Equivalence statements Length 1in = 2.54 cm 39.37 in = 1 meter 1 mile = 5280 feet Mass 1kg = 2.2 lbs 1lb = 454 grams Volume 1 liter = 1.06qts 1 gallon = 3.79 liters • 3.6 m = ? ft • 6.07 lb = ?kg • 4.2 L = ?qt • 35.92 cm = ? in

39. Double step Exampls Equivalence Statements Length 1m = 1.094 yd 2.54 cm = 1 in 1mi = 1760 yd Mass 1kg = 2.205 lb 453.6 g = 1lb Volume 1 L = 1.06qt • 56,345 s = ? yrs • 98.3 in = ?m • 3.2 mi = ?km

40. Section 2.7 Objective: to learn three temperature scales to convert from one scale to another Temperature Conversions

41. Temperature – the average kinetic energy in a substance • Boiling points • Fahrenheit 212 F • Celsius 100 C • Kelvin 373 K • Freezing points • Fahrenheit 32 F • Celsius 0 C • Kelvin - 273 K *O Kelvin or Absolute zero: point at which molecular motion stops

42. Temperature Conversion Formulas Celsius to Kelvin TK = TC + 273 Kelvin to Celsius TC = TK – 273 Celsius to TF = 1.80TC + 32 Fahrenheit Fahrenheit to TC = TF - 32 Celsius 1.80

43. Objective: to define density and its units Section 2.8 Density

44. Density: the amount of matter present in a given volume of a substance Formula Units Density = g/ml OR g/cm3 Mass = g (grams) Volume = ml OR cm3 Liquids OR solids • Density = mass/volume DENSITY of a substance never changes Ex gold is ALWAYS 19.3g/cm3 Less dense objects “FLOAT” in more dense objects

45. Example calculation Mercury has a density of 13.6g/ml. What volume of mercury must be taken to obtain 225 grams of the metal?

46. Example calculation: ANSWER • Mercury has a density of 13.6g/ml. What volume of mercury must be taken to obtain 225 grams of the metal? • 16.5 mL

47. THE END 