copyright Sautter 2003

Measurement

• All measurement is comparison to a standard. Most often that standard is an excepted standard such as a foot of length, liter of volume or gram of mass.

• Unusual standards may be used in obtaining measurements but this is rarely done since few people would be familiar with the standard used. For example, someone measuring a distance can pace off that distance but since the length of one’s step is variable and this would give a very unreliable measure.

• We generally work with two systems of measurement, English and metric. The metric system is used more frequently in science although the English system can be used.

Measurement

• Basic metric units are systematically subdivided using a series of prefixes. Each prefix multiplies the basic unit by a specific value. For example the prefix “centi” multiplies by 0.01 (one hundredth – 100 cents in a dollar), “deci” multiplies by 0.10 (one tenth - 10 dimes in a dollar) and so on.

• The prefix or multiplier may be applied to any basic measurement, grams, liters or meters and others yet to be discussed. The prefix may subdivide the unit or enlarge it. For example, “milli” divides the unit into a 1000 parts (0.001 or one thousandth) while “kilo” multiplies the unit by 1000 (a thousand times).

UNIT CONVERSIONS

• Quantities can be converted from one type of unit to another. This conversion may occur within the same system (metric or English) or between systems (metric to English or English to metric).

• Conversions cannot be made between measures of different properties, that is, mass units to length units for example.

• A method of unit conversion commonly used is called Dimensional Analysis or Unit Analysis. In this procedure, units are used to decide when to multiply or divide in order to obtain the correct answer.

1 cm 1 milliliter 1 cm 1 cm Volume = length x width x height 1 cm 3 = 1 cm x 1 cm x 1 cm cc means cubic centimeter

1.00 ml = 1.00 cc 1000 ml = 1000 cc = 1.0 liter

Unit Analysis

• Let’s apply Unit Analysis to a sample problem. In order to use this method we must have available a list of conversion factors from English to metric and vice versa. Some have been provided on the previous slides.

• To begin we will examine a metric to metric conversion problem.

FROM THE CONVERSION TABLE CANCEL UNITS TO LEAVE UNITS OF THE ANSWER PLACE THE NUMBERS IN THE SPOTS INDICATED BY THE UNIT LABELS

Unit Analysis – metric to metric

• Problem: How many millimeters are contained in 5.35 kilometers.

• Solution: First, we decide that units we are starting with (km) and the units we want to find (mm). Km.  mm.

• Next, we will examine the metric relationships that are available to be used for the conversion.

• Millimeter means 0.001 meters or 1000 mm = 1m • Kilometer means 1000 meter so 1000 m = 1km • Now, we will set up unit fractions so that all units will cancel out leaving only the unit for the answer (mm) • We are starting with km • Km x (m / km) x (mm / m) = mm (the units for our answer) • Km will cancel and m will cancel leaving just mm in our set up.

• Now place the numbers in the positions indicated by the units • • 5.35 x (1000/1) x (1000/1) =

5.35 x 10 6 mm Mental check: since mm are very small and km are large there should be a lot of mm in 5.35 km. 5.35 million is a lot!

Unit Analysis – English



metric

• Problem: How many milligrams are contained in 25 lbs?

• Solution: We are starting with pounds and want to find milligrams. Lbs  mg • We need an English – metric weight (mass) conversion. We will use 454 grams = 1.0 lbs. We will also use 1000 mg = 1.0 grams • Set up the units: lbs x (g / lb) x (mg / g) = mg • 25 x (454 / 1.0) x (1000 / 1.0) = 1.134 x 10 7 mg • Check: there are lots of grams in a pound and lots to milligrams in a gram, therefore expect a large number and 11.34 million is a large number!

In science, we often encounter very large and very small numbers. Using scientific numbers makes working with these numbers easier

Scientific numbers use powers of 10

RULE 1 As the decimal is moved to the left The power of 10 increases one value for each decimal place moved Any number to the Zero power = 1

RULE 2 As the decimal is moved to the right The power of 10 decreases one value for each decimal place moved Any number to the Zero power = 1

RULE 3 When scientific numbers are multiplied The powers of 10 are added

RULE 4 When scientific numbers are divided The powers of 10 are subtracted

RULE 5 When scientific numbers are raised to powers The powers of 10 are multiplied

RULE 6 Roots of scientific numbers are treated as fractional powers. The powers of 10 are multiplied

RULE 7 When scientific numbers are added or subtracted The powers of 10 must be the same for each term.

Powers of 10 are Different. Values Cannot be added !

Move the decimal And change the power Of 10

Power are now the Same and values Can be added.

DENSITY

• Density is a fundamental property of all matter. It measures the quantity of matter in a given volume of space. • For elements and compounds, density can be an identifying characteristic. For example, the density of gold is 19.5 grams per milliliter. Any substance with appearances similar to gold cannot be gold unless it has the density of 19.5 grams per milliliter.

• The density of solids is usually greater than that of liquids and the density of gases is always significantly lower than that of liquids or solids.A notable exception to the density relationships of solids and liquids is that for water. Water in the solid state (ice) has a lower density than liquid water (ice floats on water). Most substances do not have this inverted density relationship.

DENSITY

• Density units can be any mass unit divided by any volume unit. Usually grams / ml are used for liquids and solids. A volume measurement of cubic centimeters is also often used. One cubic centimeter (cc) equals one milliliter (ml). When measuring the density of gases grams per liter (g / l) are generally used.

• The term specific gravity is also used to measure density. It is a ratio of the density of substance divided by the density of water. The density of water is generally considered to be 1.0 grams per ml. (Actually, the density of water like all substances varies with temperature and is really 1.0 g/ml at 4 0 C) • Since the density of water is used as 1.0 g/ml dividing it into the density of any substance gives back the density of that substance however the units are divided out and therefore the specific gravity value has no units associated with it.

A lead Fishing sinker A room full of air THE AIR! WHY?

BECAUSE THERE IS MUCH MORE OF IT !

LEAD HAS A GREATER DENSITY BUT NOT NECESSARILY A GREATER MASS (WEIGHT) !

DENSITY

•

Problem

: A rectangular block to substance X measures 3.0 cm by 0.50 m by 20 mm and has a mass of 150 grams. Find its density.

• •

Solution

: Density = mass / volume

Volume = length x width x height

(rectangular solid) • Vol = 3.0 cm x 50. cm x 2.0 cm (all unit are converted to cm so that the volume is calculated in cubic centimeters) • Volume = 300 cc • Density = 150 grams / 300 cc =

0.50 g/cc

Significant Figures

• Significant figures are used to distinguish truly measured values from those simply resulting from calculation. Significant figures determine the precision of a measurement. Precision refers to the degree of subdivision of a measurement. • As an example, suppose we were to ask how much money you had and you replied “About one hundred dollars”. This would be written as $100 with no decimal point included.This is shown with one significant figure the “1”, the zeros don’t count and it tells us that you have about $100 but it could be $90 or even $110. If we continued to inquire you might say “ OK, ninety seven dollars. This would be written as $97. It contains two significant figures, the 9 and the 7. Now we know that you have somewhere between $96.50 and $97.49.

• If we continue to ask you may eventually say, “Ninety seven dollars and twenty cents”. This is written as $97.20 and in this case the zero is significant because it say that you have exactly 20 cents, not 19 or 21, in addition to the $97.The $97.20 contains four significant figures.

Measurements are always all measured values plus one approximated value. The pencil is 3.6 cm long.

1 2 3 4 5 6 7

3 4

With more calibration a more precise measurement is possible The pencil is 3.64 cm long!

3.6 3.7

The calibration of the instrument determines measurement precision Now 3.640 cm !

ACCURACY MEANS HOW CLOSE A MEASUREMENT IS TO THE TRUE VALUE PRECISION REFERS TO THE DEGREE OF SUBBDIVISION OF THE MEASUREMENT FOR EXAMPLE, IF A ROOM IS 10 FEET LONG AND YOU MEASURE IT TO BE 15.9134 FT LONG, YOUR MEASUREMENT IS VERY PRECISE BUT INACCURATE !

MEASUREMENTS SHOULD BE ACCURATE AND AS PRECISE AS THE MEASURING DEVICE ALLOWS

Significant Figures

• In working with significant figures, zeros are the most problematic. Non zero numbers are always significant. Zeros are sometimes significant and other times not as we saw in the previous frame. To work successfully with significant figures a set of rules are required. Here they are: • (1) Zeros to the right of non zeros and left of the decimal are significant. In 300 the zero are to the right of a non zero but not left of a decimal and are not significant. The number contains only one sig fig. Zeros to the right of the decimal and to the right of non zeros are significant. In 0.02300 the zeros are to the right of non zeros and the decimal and are significant. The zeros not preceded by non zeros are not significant. The number has four sig figs, the 2,3,0,0 while the 0.0 values are not significant.

Significant Figures

• (2) Zeros between non zeros are always significant. In the number 4009 the zeros are significant being in between the 4 and the 9. The number has four sig figs.

• (3) Zeros with no decimal to the right are not significant. In 4500 the zeros have no decimal to the right and are not significant. The number has two sig figs, the 4 and the 5.

• (4) Zeros with a decimal to the right are significant when preceded by non zeros. In 4500

.

the zeros are significant. The number has four sig figs, the 4, 5, and both zeros.

• (5) A significant zero means a value was measured and found to be zero in that position. A non significant zero (called a place holding zero) means that no measurement was taken in that position. The value is unknown!

2 sig figs No decimal point Zeros are not significant

!

Decimal Point All digits including zeros to the left of The decimal are significant. 6 sig figs

All figures are Significant 4 sig figs All figures are Significant 5 sig figs Zeros between Non zeros are significant Zero to the Right of the Decimal are significant

5 sig figs Zeros to the right of The decimal with no Non zero values Before the decimal Are not significant 3 sig figs Zeros to the right of the decimal And to the right of non zero values Are significant

Exact equivalences have an unlimited number of significant figures There are exactly 3 feet in exactly 1 yard. Therefore the 3 can be 3 or 3.0 or 3.00 or 3.000 etc.

and the 1can be 1 or 1.0 or 1.00 or 1.000 etc. !

The same is true for:

Mathematics and Significant Figures

• (5) Multiplying and dividing with significant figures.

The result of multiplication or division can have no more sig figs than the term with the least number. For example, 9 x 2 = 20 since the 9 has one sig fig and the 2 has one sig fig, the answer 20 must have only one and is written without a decimal to show that fact. By contrast, 9.0 x 2.0 = 18 each term has two sig figs and the answer must also have two.

• (6) Adding and subtracting with significant figures.

The position, not the number, of the significant figures is important in adding and subtracting. For example, 12.03 (the last sig fig is in hundredth place (0.01)) + 2.0205 (the last sig fig is in ten thousandth (0.0001)) 14.05 (the answer is rounded off to the least significant position hundredths place)

The answer is rounded to the position of least significance The numbers in these positions are not zeros, they are unknown The sum of an unknown number and a 6 is not valid.

The same is true For the 2