Measurements and Calculations

Chapter 2

Units of Measurement

• Measurements involve NUMBER and UNIT

• Represent a quantity: has magnitude, size, or amount

• Gram = unit of measurement

• Mass = quantity

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

SI Base Units

Units of Measurement

Units of Measurement

• Metric Prefixes – make units easier to use

• Make the unit smaller or larger

• Unit = prefix + base unit

• Table pg. 35

Mass

• Measures quantity of matter

• SI unit: kilogram, kg

• ______ kg = _____ g

• gram used for smaller masses

• Weight: measure of gravitational pull

Length

• SI unit: meter, m

• Longer distances: kilometer, km

• _______ km = _______ m

• Shorter distances: centimeter, cm

• _______ m = ________ cm

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

Derived Units

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

Scientific Notation

• Review: Write in scientific notation32,700

0.0003412

3.901 x 10-6

4.755 x 108

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

Significant figures (sig figs)

• Better marks better estimate.• Last number measured actually an estimate

21 3 4 5

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.

Sig Figs.

• 405.0 g

• 4050 g

• 0.450 g

• 4050.05 g

• 0.0500060 g

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

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

Rounding rules

• Round 454.62 to four sig figs– to three sig figs– to two sig figs– to one sig fig

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

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

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…

Density

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

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

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

Unit Conversions

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!!

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.

Conversion factorsConversion factors

100 cm1 m =100 cm 100 cm

Conversion factorsConversion factors

11 m =100 cm

Conversion factorsConversion factors

11 m =100 cm

100 cm=1 m1 m 1 m

Conversion factorsConversion factors

11 m =100 cm

100 cm=1 m

1

Conversion Factors

• Unique way of writing the number 1.

• Does NOT change the VALUE, it changes the UNITS.

Write the conversion factors for the following

• kilograms to grams

• feet to inches

• 1 L = 1 dm3 = 1000mL = 1000 cm3

Let’s See How They Work

• We can multiply by a conversion factor creatively to change the units .

• 13 inches is how many yards?

Let’s Try Some!

• 323 mm = _____ nm

• 3.2 miles = _____ in

• 250 gallons = _____ mL

• 15 days = _______ min

More Unit Conversions

More Involved

Derived Unit Conversions

• 54.3 cm3 = ______ m3

• 7.54 ft2 = _______ in2

Derived Unit Conversions

• 125.3 m/s = ______ mi/hr

• 625 g/mL = ______ kg/m3

• 100 km/hr = ______ mi/hr

Where do these measurements come from?

Recording Measurements

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

Making Good Measurements

2. Test our measurement against a “standard”, or accepted value

- measurement close to accepted value is ACCURATE

Video - 46

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

Estimating Measurements

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

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

Calculating Percent Error

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

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…

Direct Proportions

• Solve for y: y = k

x

• Look familiar?

• Eqn for a straight line: y = mx + b

• Slope is the constant

kx

y

Direct Proportion

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

Inverse ProportionGraph is called “hyperbola”

Calculations

• Convert 3.23 x 104 kg to g. Give answer with correct sig. figs.

• How many miles are in 450,000 in?

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?