CHAPTER 3 SCIENTIFIC MEASUREMENT

OBJECTIVES:a. Compare and contrast accuracy and precisionb. Place values in proper scientific notation with significant digitsc. Determine percent errord. Measure in the SI and convert unit quantitiese. State and give example of fundamental and derived unitsf. Demonstrate dimensional analysis

The accuracy of an instrument reflects how close the reading is to the 'true' value measured.

Accuracy indicates how close a measurement is to the accepted value.  For example, we'd expect a balance to read 100 grams if we placed a standard 100 g weight on the balance.  If it does not, then the balance is inaccurate

The precision of an instrument reflects the number of significant digits in a reading;

Precision indicates how close together or how repeatable the results are.  A precise measuring instrument will give very nearly the same result each time it is used.

Precise and Accurate

Accurate, Not Precise

Neither Precise Nor Accurate

Precise, Not Accurate

ESTIMATED UNCERTAINTY

The measurement of a board might be written as 8.8 + 0.1 cm. The +0.1 represents the estimated uncertainty in the measurement, so that the actual width most likely lies between 8.7 and 8.9cm.

PERCENT ERROR:Is the absolute value of the error divided by the accepted value, multiplied by 100

Percent error = (actual – accepted/accepted ) x100

Significant Figures Is a measurement that includes all the known digits plus a last digit that is estimated

Significant figures are the number of reliably known digits used to locate a decimal point reported in a measurement. Proper use of significant figures ensures that you correctly represent the uncertainty of your measurements. For example, scientists immediately realize that a reported measurement of 1.2345 m is much more accurate than a reported length of 1.2 m.

When a measurement is properly stated in scientific notation all of the digits will be significant. For example: 0.0035 has 2 significant figures which can be easily seen when written in scientific notation as 3.5 x 10-3. Fortunately, there are a few general guidelines that are used to determine significant figures:

Guidelines for Significant Figures:

1. Whole Numbers

2. Integers and Defined Quantities

3. Multiplication and Division

4. Addition and Subtraction

Whole Numbers: The following numbers are all represented by three significant digits. Note that

zeros are often place holders and are not significant.

0.00123 0.123 1.23 12.3 123

12300 (The zeros here often cause confusion. As written here, the zeros are not significant. If they were, in fact, significant, then the use of scientific

notation would remove all ambiguity and the number would be written 1.2300 x 104.)

The following numbers are all represented by one significant digit.

0.005 0.5 5

500 5,000,000

The following numbers are all represented by four significant figures.

0.004001 0.004000

40.01 40.00 4321 432.1

43,210,000

Multiplication and Division: When multiplying or dividing numbers, the result should have only as many significant figures as the quantity with the smallest number of significant figures being used in the calculation. For example, with your calculator multiply 4.7 and 5.93. The calculator returns 27.871 as the answer. A common mistake students make is to record what comes out of the calculator as the correct answer. However, since 4.7 has only 2 significant figures, the result must be truncated to 2 significant figures as well. Taking all this into account and remembering to round appropriately, the result should be reported as 28.

Addition and Subtraction:

RULE: When adding or subtracting your answer can only show as many decimal places as the measurement having the fewest number of decimal places.

3.76 g + 14.83 g + 2.1 g = 20.69 g = 20.7g

When adding or subtracting numbers in scientific notation, their powers of 10 must be equal. If the powers are not equal, then you must first convert the numbers so that they all have the same power of 10.

(6.7 x 109) + (4.2 x 109) = (6.7 + 4.2) x 109 = 10.9 x 109 = 1.09 x 1010

Addition and Subtraction

Multiplication and Division

It is very easy to multiply or divide just by rearranging so that the powers of 10 are multiplied together

(6 x 102) x (4 x 10-5) = (6 x 4) x (102 x 10-5) = 24 x 102-5 = 24 x 10-3 = 2.4 x 10-2.

ADDITIONAL PRINCIPLES

PRINCIPLE NUMBER ONEIf you are using exact constants, such as thirty-two ounces per quart or one thousand milliliters per liter, they do not affect the number of significant figures in you answer. For example, you might need to calculate how many feet equal 26.1 yards. The conversion factor you would need to use, 3 ft/yard, is an exact constant and does not affect the number of significant figures in your answer. Therefore, 26.1 yards multiplied by 3 feet per yard equals 78.3 feet which has 3 significant figures.

PRINCIPLE NUMBER TWOIf you are using constants which are not exact (such as pi = 3.14 or 3.142 or 3.14159) select the constant that has at least one or more significant figures than the smallest number of significant figures in your original data. This way the number of significant figures in the constant will not affect the number of significant figures in your answer. For example, if you multiply 4.136 ft which has four significant figures times pi, you should use 3.1416 which has 5 significant figures for pi and your answer will have 4 significant figures.

PRINCIPLE NUMBER THREEWhen you are doing several calculations, carry out all of the calculations to at least one more significant figure than you need. When you get the final result, round off. For example, you would like to know how many meters per second equals 55 miles per hour. The conversion factors you would use are: 1 mile equals 1.61 x 103 meter and 1 hour equals 3600 seconds. Your answer should have two significant figures. Your result would be 88.55 divided by 3600 which equals 24.59 m/sec. This rounds off to 25 m/sec. By carrying this calculation out to at least one extra significant figure, we were able to round off and give the correct answer of 25 m/sec rather than 24 m/sec.

SI Base Units

Name Symbol Unit of

meter m length

kilogram kg mass

second s time

ampere A electric current

Kelvin K thermodynamic temperature

mole mol amount of substance

candela cd luminous intensity

SI Derived Units

Name Symbol Unit ofEquivalent in

Base UnitsOther Equivalents

becquerel Bq activity (of a radionuclide) 1/s -

coulomb C quantity of electricity, electric charge A·s F·V = J/V

degree Celsius

°C Celsius temperature K K – 273.15

farad F capacitance A²·s4/kg·m² C/V=A·s/V

gray Gyabsorbed dose, specific energy imparted, kerma

m²/s² J/kg

henry H inductance kg·m²/A²· s² Wb/A = V·s/A

hertz Hz frequency 1/s -

joule J energy, work, quantity of heat kg·m²/s² N·m = W·s = Pa·m³

katal kat catalytic activity mol/s -

lumen lm luminous flux cd cd·(4·π sr) = lx·m²

newton N force kg·m/s²J/m = W·s/m =

Pa·m²

ohm electric resistance kg·m²/A²·s³ V/A = 1/S

pascal Pa pressure, stress kg/m·s² N/m² = J/m³

radian rad plane angle 11/(2·) of a

circle

siemens S electric conductance A²·s³/kg·m² A/V = 1/

sievert Sv dose equivalent m²/s² J/kg

steradian

sr solid angle 11/(4·) of a

sphere

tesla T magnetic flux density kg/A·s²Wb/m² = N/A·m

volt Velectric potential difference, electromotive force

kg·m²/A·s³W/A = J/C =

Wb/s

watt W power, radiant flux kg·m²/s³J/s = V·A =

N·m/s

weber Wb magnetic flux kg·m²/A·s²V·s = H·A = T·m² = J/A

Name

Symbol Unit ofEquivalent inBase Units

OtherEquivalents

TEMPERATURE CONVERSIONS

STEP ONE: Write the value (and its unit) from the problem, then in order write: 1) a multiplication sign, 2) a fraction bar, 3) an equals sign, and 4) the unit in the answer. Put a gap between 3 and 4. All that looks like this:

The fraction bar will have the conversion factor. There will be a number and a unit in the numerator and the denominator

DIMENSIONAL ANALYSIS:

STEP TWO: Write the unit from the problem in the denominator of the conversion factor, like this:

STEP THREE: Write the unit expected in the answer in the numerator of the conversion factor.

STEP FOUR: Examine the two prefixes in the conversion factor. In front of the LARGER one, put a one.

STEP FIVE: Determine the absolute distance between the two prefixes in the conversion unit. Write it as a positive exponent in front of the other prefix

Now, multiply and put into proper scientific notation format. Don't forget to write the new unit.

Here are all five steps for the second example, put into one image:

Why a one in front of the larger unit? I believe it is easier to visualize how many small parts make up one bigger part, like 1000 m make up one km

Two Comments 1) If you do the conversion correctly, the numerical part and the unit will go in opposite directions. If the unit goes from smaller (mm) to larger (km), then the numerical part goes from larger to smaller. There will never be a correct case where number and unit both go larger or both go smaller.2) A common mistake is to put the one in front of the SMALLER unit. This results in a wrong answer. Put the one in front of the LARGER unit.

DENSITY:Is the ratio of the mass of an object to its volume.

D = M/V units g/cm3

Density is an intensive property that depends only on the composition of a substance, not on the size of the sample.

The density of a substance generally decreases as its temperature increases. Do you know any special exceptions?