Chapter One, Section Two

OBJECTIVES

  1. To become familiar with the SI system for stating the units of measurements.
  2. To learn how to work with significant figures.
  3. To learn how to use the factor-label method to convert between different units of measurements.

I. UNITS

We use units often in our daily lives and probably don't think a lot about it. We say, "It will take me 30 minutes to get to KSU, if the traffic is not a problem." The word "minutes" is a unit of time. We say that it is 27 miles from KSU to the Omni. The word "miles" is a unit of distance. We say that we want a gallon of milk. The word "gallon" is a unit of volume. We say that we weigh xxx pounds. The word "pounds" is a unit of weight. If we made these statements without using units, the sentences would be meaningless. Similarly in working problems related to science, any number without units is meaningless -- remember that! We do sometimes say that it is 95 degrees outside, and strictly speaking that is a number without units. The units are implied to be the units used in our country, Fahrenheit. We are just lazy and say the number without units. In science, we can't be lazy!

In science, we don't use the same units that we are used to using in our everyday life in the United States of America. We use the Systeme Internationale units or SI units. The name is French because the system was developed in France. So we have to learn a new system of units, which is used by most of the people in the world outside of the U.S.A.

The Fundamental SI Units are as listed below:
Physical QuantityName of UnitAbbreviation
Masskilogramkg
Lengthmeterm
Timesecondt
TemperatureKelvinK
Electric Currentamperea
Amount of substancemolemol
Luminous intensitycandelacd

Derived SI units come from combinations of the above such as the following:
areasquare meterm2
volumecubic meterm3
velocitymeter per secondm s-1
accelerationmeter per second squaredm s-2
densitykilogram per cubic meterkg m-3
molar masskilogram per molkg mol-1
molar volumecubic meter per mole m3 mol-1
molar concentrationmole per cubic metermol m-3

The prefixes used with these units are given below and in your book (Table 1-2) and you must memorize them today.

PrefixSymbolExponential Notation
exaE1018
pentaP1015
teraT1012
gigaG109
megaM106
kilok103
hectoh102
dekada101
decid10-1
centic10-1
millim10-3
micromc or µ10-6
nanon10-9
picop10-12
femtof10-15
attoa10-18

II. MEASUREMENT

When you have seen one measurement you have seen them all is certainly not true. There are three concepts involved:

If we want to measure this line: ______________________________

we could get several different answers depending upon the measuring device. One device might give 3.5 inches by one person with measuring device A and 3.56 inches by another person with measuring device B. However when they swap devices, they might not get the same results as recorded initially with A and B. So all three concepts are involved.

Accuracy and Precision

The images below neatly shows the difference between accuracy and precision.

precise imprecise accurate


Imagine that the above represents the results of a "ring toss" game. The first image is clearly very precise, but not accurate. The second has rings all over the place and hence is neither precise nor accurate. The third is "just right" -- precisely accurate. If both people measure with the same precision then when they swap measuring devices, their results should agree. Each person should get 3.5 inches with device A and each should get 3.56 inches with device B.

Measuring Device

We have all seen rulers with different accuracies and scales with different accuracies. Some rulers are marked to the nearest quarter inch and some to the nearest eighth inch. Some scales are marked to the nearest pound and some to the nearest ounce (to use U.S. units). Which are correct? They are both correct, just one is reporting more accuracy than the other. The results of a measurement then depend upon the preciseness of the measurement and the accuracy of the measuring device.

III. SIGNIFICANT FIGURES

Now suppose that person 1 measures the above line with device A and reports a length of 3.5 inches and person 2 measures the length of this line __________with device B and reports a length of 1.45 inches. What is the sum of the lengths of the two lines? First we must be certain that the accuracy of A is really only in the tenths of inches and the accuracy of B is really to the hundredths of inches. We also have to be certain that each person is measuring with the same precision. Assuming that both are true, then how do we determine the sum? Well now we are into a new topic: significant figures. The number of significant figures of a measurement depend upon the accuracy of the measuring device (we will assume that the people doing the measurements are precise in their measurement). When measurements are made with devices of different accuracy, we need to be very careful in combining the measurements to make sure that we consider the proper use of significant figures.

There are some rules to follow to allow you to get the computation done correctly considering significant figures. I summarize them as follows:

Going back to our original problem. One measurement was 3.5 inches and the other measurement was 1.45 inches. Adding the two we might report a distance of 4.95 inches, but this is three significant figures which is too accurate a number considering the two measuring devices. We can only report an answer to two significant figures, so we report a combined distance of 5.0 inches. Note that we rounded up and we reported our result with a zero after the decimal and that indicates that the zero is significant.

IV. DIMENSIONAL ANALYSIS or The FACTOR-LABEL METHOD

Often we have to convert from one system of units to another. For example, you may be wondering how much I really weigh since I told you earlier that I weigh 90.8 kgm. The method of unit conversion is called either the unit factor method or dimensional analysis. In order to make conversions we must know some conversion factors. Two of the more useful are that there are 2.54 cm in one inch and that there are 453.6 gms in one pound. Let's consider the gram/pound conversion.

We have the equality:

453.6 g = 1 pound

So divide both sides by 453.6 gms and we have

453.6 g/453.6 g = 1 = 1 pound/453.6g

So since 1 pound divided by 453.6g is equal to one, multiplying by that ratio will not change the value of the numerical value of the number -- just the unit. So let's give it a try.

90.8kg x 1000gm/kg = 90800 g where we used another definition ratio (1000g/1k). Now use the pound conversion:

90800g x 1 pound/453.6 g = 200.2 pounds. But what about sig figs? We only have 3 sig figs in the given number of 90.8 kg so we say that I weigh 200. pounds where I put a "." after the number 200 which indicates that all three digits are significant. Alternatively we could say that I weigh 2.00 x 102 pounds.

Another common conversion we Americans need to make is between Fahrenheit and Celsius. Here we can't use a factor-lable method directly because the different zero point for the two methods. Instead we have to use:

oF = oC x 9/5 + 32

and

oC = (oF - 32)*5/9

In learning this type of material it is crucial to practice, practice, practice.

ASSIGNMENTS

QUIZ TWO

After you have studied this material and practiced some problems, take quiz Two. If you score at least 80 on the test then you are ready to continue to the next section.


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Web Author: Dr. Leon L. Combs
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