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Unit 2:SCIENTIFIC MEASUREMENT

OBJECTIVES (Don’t Copy!)

- Convert Between Standard Notation to Scientific Notation
- Identify Significant Figures & Uncertainty in Measurements
- Perform Operations with Significant Figures
- Addition & Subtraction
- Multiplication & Division

What is Scientific Notation?

- a way of expressing really big numbers or really small numbers.
- For some numbers, scientific notation is more concise.

Scientific notation consists of two parts:

- A number between 1 and 10 (the “coefficient”)

- A power of 10
- Ex:

7.01

x 108

Converting from StandardScientific Notation

EX: Convert 289,800,000 to scientific notation.

_______ x 10___

STEP 1: Moving the decimal, convert this number so that it falls between 1 and 10.

STEP 2: Count the number of places you moved the decimal. This is the exponent.

- If the number was large to start with, exponent is positive
- If the number was small to start with, exponent is negative

8

2.898

Examples

- Given: 0.000567

NOTE: This is a very small number, so the exponent is SMALL (negative)

5.67

-4

x 10 ___

___________

STEP 1

STEP 2

Scientific Notation & Your Calculator

- Video Instructions- Using Calculator

Plugging Scientific Notation into My Calculator

- Find the button on your calculator that is used to enter SCIENTIFIC NOTATION.

OR

- Note: If you find one of the symbols ABOVE a key, rather than ON a key, you must push

OR

EE

EXP

2nd

EE

2nd

EXP

Plugging Scientific Notation Into My Calculator

- Ex: Plug this number into your calculator: 8.93 x 1o-13
- Write the steps you used below
- Type 8.93
- Push ______ button.
- Type_____13_____
- Push ____OR____ button
- What I see on the screen is _________

NOTE: Use your calculator’s keys if they differ from what is written here

EE

NOTE: Sometimes these 2 steps can be reversed

(-)

+/-

8.93 -13

Scientific Notation:Adding & Subtracting

Ex: 3 x 104 + 2.5 x 105

USE

CALCULATOR:

NOTE: Use your calculator’s keys if they differ from what is written in table

NOTE: Answers must be in scientific notation!

Calculator says: 280000

Correct Answer: 2.8 x 105

Scientific Notation:Multiplying&Dividing

Ex: (6.1 x 10-3) (7.2 x 109)

USE

CALCULATOR:

NOTE: Use your calculator’s keys if they differ from what is written in table

NOTE: Answers must be in scientific notation!

Calculator says: 43920000

Correct Answer: 4.392 x 107

Stating a Measurement

In every measurement there is a

- Number followed by a
- Unit from a measuring device

The number should also be as precise as the measuring device.

Ex: Reading a Meterstick

. l2. . . . I . . . . I3 . . . .I . . . . I4. . cm

First digit (known) = 2 2.?? cm

Second digit (known) = 0.7 2.7? cm

Third digit (estimated) between 0.05- 0.07

Length reported =2.75 cm

or 2.74 cm

or 2.76 cm

Significant Figures

- The numbers reported in a measurement are limited by the measuring tool
- Significant figures in a measurement include the known digits plus one estimated digit

Shortcuts to Sig Figs

The Atlantic-Pacific Rule says:

"If a decimal point is Present, ignore zeros on the Pacific (left) side.

If the decimal point is Absent, ignore zeros on the Atlantic (right) side.

Everything else is significant."

Counting Significant Figures:Unlimited Sig Figs

2 instances in which there are an unlimited # of sig figs.

- Counting. Ex: 23 people in our classroom.
- Exactly defined quantities. Ex: 1hr = 60 min.
- Both are exact values. There is no uncertainty.
- Neither of these types of values affect the process of rounding an answer.

Learning Check

A. Which answers contain 3 significant figures?

1) 0.4760 2) 0.00476 3) 4760

B. All the zeros are significant in

1) 0.00307 2) 25.300 3) 2.050 x 103

C. 534,675 rounded to 3 significant figures is

1) 535 2) 535,000 3) 5.35 x 105

Learning Check

In which set(s) do both numbers contain the samenumber of significant figures?

1) 22.0 and 22.00

2) 400.0 and 40

3) 0.000015 and 150,000

Rounding With Sig Figs

- When rounding an answer, determine which is the last significant figure. This is where you will round your number.
- If the digit immediately to the right of the last sig fig is less than 5, the value of the last sig fig remains the same.
- 34, 231 rounded to 3 sig figs
- If it is 5 or greater, round up.
- Ex: 0.09246 rounded to 3 sig figs

34,200

0.0925

Practice Rounding (p 69)

- Round off each measurement to the number of sig figs shown in parentheses.
- 314.721 meters (four)
- 0.001775 meter (two)
- 8792 meters (two)
- 25,599 (four)

= 314.7 meters

=0.0018 meter

= 8800 meters

= 2.560 x 10 4

NOTE: Sometimes the only way to show sig figs properly is to use scientific notation!

Significant Numbers in Calculations

- A calculated answer cannot be more precise than the measuring tool.
- A calculated answer must match the least precise measurement.
- Significant figures are needed for final answers from

1) adding or subtracting

2) multiplying or dividing

- If you must round to obtain the right # of sig figs, do so after all calcs are complete

Adding and Subtracting

The answer has the same number of decimal places as the measurement with the fewest decimal places.

25.2one decimal place

+ 1.34two decimal places

26.54

answer 26.5one decimal place

Learning Check

In each calculation, round the answer to the correct number of significant figures.

A. 235.05 + 19.6 + 2.1 =

1) 256.75 2) 256.8 3) 257

B. 58.925 - 18.2 =

1) 40.725 2) 40.73 3) 40.7

Multiplying and Dividing

Round (or add zeros) to the calculated answer until you have the same number of significant figures as the measurement with the fewest significant figures.

Learning Check

A. 2.19 X 4.2 =

1) 9 2) 9.2 3) 9.198

B. 4.311 ÷ 0.07 =

1)61.582) 62 3) 60

C. 2.54 X 0.0028 =

0.0105 X 0.060

1) 11.3 2) 11 3) 0.041

Volume

Cover the variable you are solving for and perform the operation with the given amounts

M

D = _M_

V

÷

Mass

D

V

X

Density

Volume

Common Units for Density Probs

- Volume = L (mL), cm3
- NOTE: 1 mL = 1 cm3
- Mass = g (kg, mg, etc.), lbs

Density Practice Problem #1 (p 91)

QUESTION: A copper penny has a mass of 3.1 g and a volume of 0.35 cm3. What is the density of copper?

SOLUTION:

Givens: Unknown:

- mass = 3.1 g density = ?
- Volume = 0.35 cm3
- Find equation and solve D = m/v
- D = 3.1 g /0.35 cm3
- D = 8.8571 g/cm3

Density Practice Problem #2 (p 92)

What is the volume of a pure silver coin that has a mass of 14 g? The density of silver (Ag) is 10.5 g/cm3.

- Givens: Unknown:
- Mass = 14 g volume = ?
- Density = 10.5 g/cm3
- Find formula & derive it: D = M/V V = M/D
- Substitute values & solve V = 14 g

10.5 g/cm3

UNITS OF MEASUREMENT

Use SI units — based on the metric system

Length

Mass

Volume

Time

Temperature

Meter, m

Kilogram, kg

Liter, L

Seconds, s

Celsius degrees, ˚C

kelvins, K

Conversion Factors

Fractions in which the numerator and denominator are EQUAL quantities expressed in different units

Example: 1 km = 103 m

Factors: 1 km and 103 m

103 m 1 km

They don’t change the value of the measurement, just the units in which it is expressed.

How to set up conversion factors in the metric system

- If you always select the larger of the 2 units as your “1” unit,
- Then the multiplier (on prefixes table) will have a positive exponent.

Ex 1: Compare Megagrams & grams. Which is larger?

Megagrams

This is our “1” unit

1 Mg = 106 grams

This is the multiplier from our table

Ex: Compare meters & millimeters

- Which is the larger of the 2 units?

meters

This is your “1” unit

NOTE: when you look on your prefixes chart, the “milli-” multiplier is 10-3. When we use the method of making the larger unit the “1” unit, we always have a positive exponent for our multiplier.

1 meter =

103 mm

How many millimeters in 1205 meters?

4. Place units of the unknown in the numerator of the conversion factor. (if you can find a relationship between the 2 units. We can! 1m = 103 mm)

1. Identify your given. Place it far left.

103

mm

1205 m

= _____mm

1.205 x106

m

1

2. Identify your unknown. Place it far right.

The given units are in the NUMERATOR. Your goal is to get rid of the units of your given. How?

5. Cancel units & do the math! (Sig figs!)

3. Place the units of the given in the denominator of the conversion factor!

Conversion Factors

Fractions in which the numerator and denominator are EQUAL quantities expressed in different units

Example: 1 hr. = 60 min

Factors: 1 hr. and 60 min

60 min 1 hr.

They don’t change the value of the measurement, just the units in which it is expressed.

Ex: Convert your weight from pounds (lbs) to kg.

4. Place units of the unknown in the numerator of the conversion factor. (if you can find a relationship between the 2 units. We can! 1kg = 2.2lb)

1. Identify your given. Place it far left.

1

kg

150 lbs

= _____kg

68

lbs

2.2

2. Identify your unknown. Place it far right.

The given units are in the NUMERATOR. Your goal is to get rid of the units of your given. How?

5. Cancel units & do the math! (Sig figs!)

3. Place the units of the given in the denominator of the conversion factor!

Ex: How many minutes are in 2.5 hours?

Conversion factor

2.5 hr x 60 min = 150 min

1 hr

cancel

By using dimensional analysis, the UNITS ensure that you have the conversion right side up, and the UNITS are calculated as well as the numbers!

Learning Check

How many seconds are in 1.4 days?

Unit plan: days hr min seconds

1.4 days x 24 hr x _60min x 60 s =___s

1 day 1 hr 1 min

ANSWER: 120,960 s.

FINAL ANSWER (in sig figs) = 120,000 s

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