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Measurements and Calculations

Measurements and Calculations

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Measurements and Calculations

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  1. Measurements and Calculations Chapter 2

  2. Units of Measurement • Measurements involve NUMBER and UNIT • Represent a quantity: has magnitude, size, or amount • Gram = unit of measurement • Mass = quantity

  3. Units of Measurement • Scientists around the world agree on one system… • International System of Units (le Systeme International d’Unites) • SI units • Built from seven base units

  4. SI Base Units

  5. Units of Measurement

  6. Units of Measurement • Metric Prefixes – make units easier to use • Make the unit smaller or larger • Unit = prefix + base unit • Table pg. 35

  7. Mass • Measures quantity of matter • SI unit: kilogram, kg • ______ kg = _____ g • gram used for smaller masses • Weight: measure of gravitational pull

  8. Length • SI unit: meter, m • Longer distances: kilometer, km • _______ km = _______ m • Shorter distances: centimeter, cm • _______ m = ________ cm

  9. Volume • SI unit: m3 • A derived unit: combination of base units by multiplying or dividing • SI unit for Area: l x w = m x m = m2 • Volume: l x w x h = m x m x m = m3 • Also: liters (L), mL, dm3 and cm3 • 1 L = 1 dm3 = 1000mL = 1000 cm3

  10. Derived Units

  11. Scientific Notation • Put the numbers in the form a x 10n • a has one # to left of decimal • If # is bigger than 1  + exponent • If # is less than 1  - exponent

  12. Scientific Notation • Review: Write in scientific notation 32,700 0.0003412 3.901 x 10-6 4.755 x 108

  13. 1 2 3 4 5 Significant Figures (sig figs) • How many numbers mean anything? • When we measure, we can (and do) always estimate between the smallest marks.

  14. Significant figures (sig figs) • Better marks better estimate. • Last number measured actually an estimate 1 2 3 4 5

  15. Sig Figs • What is the smallest mark on the ruler that measures 142.15 cm? • 142 cm? • 140 cm? • Does the zero mean anything? (Is it significant?) • They needed a set of rules to decide which zeroes count.

  16. Sig Figs. • 405.0 g • 4050 g • 0.450 g • 4050.05 g • 0.0500060 g

  17. Sig Figs • Only measurements have sig figs. • Counted numbers are exact – infinite sig figs • A dozen is exactly 12 • Conversion factors: 100 cm = 1 m

  18. Problems • 50 has only 1 significant figure • if it really has two, how can I write it? • Scientific notation • 5.0 x 101 2 sig figs • Scientific Notation shows ALL sig figs

  19. Rounding rules • Round 454.62 to four sig figs • to three sig figs • to two sig figs • to one sig fig

  20. Calculations • 165.86 g + 4.091g - 140 g + 27.32 g • (35.6 L + 2.4 L) / 4.083 = • 2.524 x (16.408 m – 3.88 m) = Answers: 57g 9.31 L 31.62 m

  21. Sig figs. • How many sig figs in the following measurements? • 458 g • 4085 g • 4850 g • 0.0485 g • 0.004085 g • 40.004085 g

  22. Density • Density = mass D = m volume V • Units: g/cm3 or g/mL but SI unit is kg/m3 • derived unit • Used to identify substances • Varies with temperature • As temp. increases density…

  23. Density

  24. Density Examples • If a metal block has a mass of 65.0 grams and a volume of 22 cubic centimeters, what is the density of the block? • D = m V • D = 65.0 g = 3.0 g/cm3 22 cm3

  25. Density Examples • Aluminum has a density of 2.7 g/cm3. What volume of aluminum has a mass of 60 grams? • D = M V 20 cm3

  26. Density Examples • Gold has a density of 19.3 g/cm3. A block of metal has a mass of 80 g and a volume of 12 cm3. Could this block be a piece of gold? • No, because this block has a density of 7 g/cm3s

  27. Unit Conversions

  28. Unit Conversions • Given information in one unit  need to find the equivalent in another unit • Identify what’s given • Organize plan of attack • Carry out plan WITH UNITS!!

  29. Conversion factors • “A ratio of equivalent measurements.” • Start with two things that are the same. 1 m = 100 cm • Can divide by each side to come up with two ways of writing the number 1.

  30. 1 m 100 cm 100 cm 100 cm Conversion factors =

  31. Conversion factors 1 m = 1 100 cm

  32. 1 m = 100 cm 1 m 1 m Conversion factors 1 m = 1 100 cm

  33. Conversion factors 1 m = 1 100 cm = 100 cm 1 1 m

  34. Conversion Factors • Unique way of writing the number 1. • Does NOT change the VALUE, it changes the UNITS.

  35. Write the conversion factors for the following • kilograms to grams • feet to inches • 1 L = 1 dm3 = 1000mL = 1000 cm3

  36. Let’s See How They Work • We can multiply by a conversion factor creatively to change the units . • 13 inches is how many yards?

  37. Let’s Try Some! • 323 mm = _____ nm • 3.2 miles = _____ in • 250 gallons = _____ mL • 15 days = _______ min

  38. More Unit Conversions More Involved

  39. Derived Unit Conversions • 54.3 cm3 = ______ m3 • 7.54 ft2 = _______ in2

  40. Derived Unit Conversions • 125.3 m/s = ______ mi/hr • 625 g/mL = ______ kg/m3 • 100 km/hr = ______ mi/hr

  41. Where do these measurements come from? Recording Measurements

  42. Making Good Measurements • We can do 2 things: • Repeat measurement many times - reliable measurements get the same number over and over - this is PRECISE

  43. Making Good Measurements 2. Test our measurement against a “standard”, or accepted value - measurement close to accepted value is ACCURATE Video - 46

  44. Measurements are Uncertain • Measuring instruments are never perfect • Skill of measurer • Measuring conditions • Measuring always involves estimation • Flickering # on balance • Between marks on instrument

  45. Estimating Measurements

  46. Error • Probably not EXACTLY 6.35 cm • Within .01 cm of actual value. • 6.35 cm ± .01 cm • 6.34 cm to 6.36 cm

  47. Calculating Percent Error • Compares your measurement to accepted value • Negative if measurement is small • Positive if measurement is big

  48. Calculating Percent Error • What is the % error for a mass measurement of 17.7g, given that the correct value is 21.2g?

  49. Direct Proportions • Two quantities are directly proportional if dividing one by the other gives a constant • yx “y is proportional to x” • Gen. Eqn: y = k x • Ex: mass and volume… constant is…

  50. Direct Proportions • Solve for y: y = k x • Look familiar? • Eqn for a straight line: y = mx + b • Slope is the constant