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 cm1 cm

1 cm

Volume = length x width x height 1 cm3 = 1 cm x 1 cm x 1 cm

cc means cubic centimeter

1 milliliter

1.00 ml = 1.00 cc1000 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

PLACE THE NUMBERS IN THE SPOTS INDICATED BY THE UNIT LABELS

CANCEL UNITS TO LEAVE UNITS OF THE ANSWER

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 106 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 107 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 1As the decimal is moved to the left

The power of 10 increases onevalue for each decimal place moved

Any number to theZero power = 1

RULE 2As the decimal is moved to the right

The power of 10 decreases onevalue for each decimal place moved

Any number to theZero power = 1

RULE 3When scientific numbers are multiplied

The powers of 10 are added

RULE 4When scientific numbers are divided

The powers of 10 are subtracted

RULE 5When scientific numbers are raised to powers

The powers of 10 are multiplied

RULE 6Roots of scientific numbers are treated as fractional

powers. The powers of 10 are multiplied

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

Powers of 10 areDifferent. ValuesCannot be added !

Power are now theSame and values

Can be added.

Move the decimalAnd change the power

Of 10

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 0C)

• 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 room fullof air

A leadFishingsinker

THE AIR! WHY?BECAUSE THERE IS MUCH MORE OF IT !

LEAD HAS A GREATER DENSITY BUT NOTNECESSARILY 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.

1 2 3 4 5 6 7

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

3 4

With more calibration a more precise measurement is possible

The pencil is 3.64 cm long!

3.6 3.7

Now 3.640 cm !The calibration of the instrument

determines measurement precision

ACCURACY MEANS HOW CLOSE A MEASUREMENTIS TO THE TRUE VALUE

PRECISION REFERS TO THE DEGREE OFSUBBDIVISION OF THE MEASUREMENT

FOR EXAMPLE, IF A ROOM IS 10 FEET LONG ANDYOU MEASURE IT TO BE 15.9134 FT LONG, YOUR

MEASUREMENT IS VERY PRECISE BUT INACCURATE !

MEASUREMENTS SHOULD BE ACCURATE AND ASPRECISE 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!

No decimalpoint

Zeros are not significant!

2 sig figs

DecimalPoint

All digits includingzeros to the left ofThe decimal are

significant. 6 sig figs

Zeros betweenNon zeros are

significant

All figures areSignificant4 sig figs

Zero to theRight of theDecimal aresignificant

All figures areSignificant5 sig figs

Zeros to the right ofThe decimal with no

Non zero values Before the decimalAre not significant

3 sig figs

Zeros to the right of the decimalAnd to the right of non zero values

Are significant

5 sig figs

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 numbers inthese positions arenot zeros, they are

unknown

The sum of an unknown number

and a 6 is not valid.The same is true

For the 2The answer is rounded to theposition of least significance