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## Measurements and Calculations

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### Measurements and Calculations

### More Unit Conversions

### Where do these measurements come from?

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

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

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

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.

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.

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

- 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

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 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

- 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!!

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

- 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 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

Recording Measurements

Making Good Measurements

- 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

- Measuring instruments are never perfect
- Skill of measurer
- Measuring conditions
- Measuring always involves estimation
- Flickering # on balance
- Between marks on instrument

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

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
- yx “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

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 Proportion

Graph 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?

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