measurements and calculations chapter 2. units of measurement measurements involve number and unit...
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Measurements and Calculations
Chapter 2
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Units of Measurement
• Measurements involve NUMBER and UNIT
• Represent a quantity: has magnitude, size, or amount
• Gram = unit of measurement
• Mass = quantity
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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
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SI Base Units
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Units of Measurement
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Units of Measurement
• Metric Prefixes – make units easier to use
• Make the unit smaller or larger
• Unit = prefix + base unit
• Table pg. 35
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Mass
• Measures quantity of matter
• SI unit: kilogram, kg
• ______ kg = _____ g
• gram used for smaller masses
• Weight: measure of gravitational pull
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Length
• SI unit: meter, m
• Longer distances: kilometer, km
• _______ km = _______ m
• Shorter distances: centimeter, cm
• _______ m = ________ cm
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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
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Derived Units
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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
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Scientific Notation
• Review: Write in scientific notation32,700
0.0003412
3.901 x 10-6
4.755 x 108
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Significant Figures (sig figs)
• How many numbers mean anything?• When we measure, we can (and do)
always estimate between the smallest marks.
21 3 4 5
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Significant figures (sig figs)
• Better marks better estimate.• Last number measured actually an estimate
21 3 4 5
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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.
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Sig Figs.
• 405.0 g
• 4050 g
• 0.450 g
• 4050.05 g
• 0.0500060 g
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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
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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
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Rounding rules
• Round 454.62 to four sig figs– to three sig figs– to two sig figs– to one sig fig
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Calculations
1. 165.86 g + 4.091g - 140 g + 27.32 g
2. (35.6 L + 2.4 L) / 4.083 =
3. 2.524 x (16.408 m – 3.88 m) =
Answers: 57g 9.31 L 31.62 m
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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
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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…
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Density
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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
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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
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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
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Unit Conversions
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Unit Conversions
• Given information in one unit need to find the equivalent in another unit
1. Identify what’s given
2. Organize plan of attack
3. Carry out plan WITH UNITS!!
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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.
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Conversion factorsConversion factors
100 cm1 m =100 cm 100 cm
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Conversion factorsConversion factors
11 m =100 cm
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Conversion factorsConversion factors
11 m =100 cm
100 cm=1 m1 m 1 m
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Conversion factorsConversion factors
11 m =100 cm
100 cm=1 m
1
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Conversion Factors
• Unique way of writing the number 1.
• Does NOT change the VALUE, it changes the UNITS.
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Write the conversion factors for the following
• kilograms to grams
• feet to inches
• 1 L = 1 dm3 = 1000mL = 1000 cm3
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Let’s See How They Work
• We can multiply by a conversion factor creatively to change the units .
• 13 inches is how many yards?
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Let’s Try Some!
• 323 mm = _____ nm
• 3.2 miles = _____ in
• 250 gallons = _____ mL
• 15 days = _______ min
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More Unit Conversions
More Involved
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Derived Unit Conversions
• 54.3 cm3 = ______ m3
• 7.54 ft2 = _______ in2
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Derived Unit Conversions
• 125.3 m/s = ______ mi/hr
• 625 g/mL = ______ kg/m3
• 100 km/hr = ______ mi/hr
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Where do these measurements come from?
Recording Measurements
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Making Good Measurements
• We can do 2 things:
1. Repeat measurement many times
- reliable measurements get the same number over and over
- this is PRECISE
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Making Good Measurements
2. Test our measurement against a “standard”, or accepted value
- measurement close to accepted value is ACCURATE
Video - 46
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Measurements are Uncertain1. Measuring instruments are never
perfect
2. Skill of measurer
3. Measuring conditions
4. Measuring always involves estimation– Flickering # on balance– Between marks on instrument
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Estimating Measurements
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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
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Calculating Percent Error• Compares your measurement to
accepted value
• Negative if measurement is small
• Positive if measurement is big
experimental accepted
accepted
Value -ValuePercentage error = × 100
Value
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Calculating Percent Error
• What is the % error for a mass measurement of 17.7g, given that the correct value is 21.2g?
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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…
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Direct Proportions
• Solve for y: y = k
x
• Look familiar?
• Eqn for a straight line: y = mx + b
• Slope is the constant
kx
y
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Direct Proportion
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Inverse Proportions• Two quantities are inversely proportional if
their product is a constant
• “y is proportional to 1 divided by x”
• Gen eqn: xy = k
• Ex: speed and travel time
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Inverse ProportionGraph is called “hyperbola”
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Calculations
• Convert 3.23 x 104 kg to g. Give answer with correct sig. figs.
• How many miles are in 450,000 in?
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Calculations
• What is the mass of an object with a density of 25.98 g/mL and a volume of 4.2 mL?
• What is the density of a 430 g object that takes up 25.5 cm3?