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: m 3

• A

derived unit

: combination of base units by multiplying or dividing • SI unit for

Area

:

l

x

w

= m x m =

m 2

• •

Volume

:

l

• Also: liters (L), mL, dm 3 and cm 3

1 L = 1 dm

x

3

w

x

h

= m x m x m =

m = 1000mL = 1000 cm 3 3

Derived Units

Scientific Notation • Put the numbers in the form a x 10 n •

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 notation 32,700 0.0003412

3.901 x 10 -6 4.755 x 10 8

Significant Figures (sig figs) • How many numbers mean anything?

• When we measure, we can (and do) always estimate between the smallest marks.

1 2 3 4 5

Significant figures (sig figs) • Better marks better estimate.

• Last number measured actually an estimate 1 2 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.

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

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 10 1 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/cm 3 or g/mL but

SI unit

is

kg/m 3

•

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 22 cm 3 = 3.0 g/cm 3

Density Examples • Aluminum has a density of 2.7 g/cm 3 . What volume of aluminum has a mass of 60 grams?

• D = M 20 cm 3 V

Density Examples • Gold has a density of 19.3 g/cm 3 . A block of metal has a mass of 80 g and a volume of 12 cm 3 . Could this block be a piece of gold?

• No, because this block has a density of 7 g/cm 3s

Unit Conversions

Unit Conversions • 1.

Given information in one unit  need to find the equivalent in another unit 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 factors 1 m 100 cm = 100 cm 100 cm

Conversion factors 1 m 100 cm = 1

Conversion factors 1 m 100 cm = 1 1 m 1 m = 100 cm 1 m

Conversion factors 1 m 100 cm = 1 1 = 100 cm 1 m

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 dm 3 = 1000mL = 1000 cm 3

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 cm 3 = ______ m 3 • 7.54 ft 2 = _______ in 2

Derived Unit Conversions • 125.3 m/s = ______ mi/hr • 625 g/mL = ______ kg/m 3 • 100 km/hr = ______ mi/hr

Where do these measurements come from?

Recording Measurements

Making Good Measurements • 1.

We can do 2 things: 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 Uncertain 1. 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

Percentage error = Value experimental -Value Value accepted accepted × 100

• •

Negative

if measurement is

small Positive

if measurement is

big

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 =

m

x + b • Slope is the constant

y



k x

Direct Proportion

Inverse Proportions • • Two quantities are

inversely proportional

their product is a constant if “

y

is proportional to 1 divided by x” • Gen eqn:

x

y

= k

• Ex: speed and travel time

Inverse Proportion Graph is called “hyperbola”

Calculations • Convert 3.23 x 10 4 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 cm 3 ?