Objectives:

-To become familiar with the SI system, scientific notation, and interconversion of units.

-To conduct simple measurements to test physical laws.

Units are the arbitrary standards in which measurable quantities are expressed.

A common problem when working with environmental data, especially when looking at global patterns, or using data collected by different people or organizations, is that the units in which the variable was measured are often different. For example, temperatures can be measured in degrees Fahrenheit (°F) or degrees Celsius (°C); wind speed in miles per hour (m.p.h.) or meters per second (ms^-1); precipitation in inches or in millimeters, etc. All measurements have to be in the same units before they can be compared and analyzed.

In 1790 the French Assembly ordered the French Academy to develop a standardized system of measurements. Until that time, every country and commonly each province and city had been using a different system of measurements. The result was the system of SI units (Systeme International) or metric units, which was adopted by France on November 2, 1801. It has been adopted by most countries since and is the standard system of measurement for science. Although not used on an everyday basis in the U.S. it is the system used by scientists in the U.S., and it is the system of measurement that will be used throughout this course (and is used in your G107 textbook).

SI units involve a number of "base" units from which other "derived" units can be obtained. The base unit for length is the meter (m), for mass the gram (g), and for volume the liter (l). Units of a quantity are the ratio of its magnitude to the magnitude of the "base" unit. Derived units can be related to "base" units by multiplication or division using powers of ten as the only multiplying factor.

NOTE: The magnitude is a numerical quantitative measure expressed usually as a multiple of a standard unit.

The SI system uses prefixes, which are used to describe multiples of 10.

NOTE: It is important whether the letter is upper case or lower case (e.g. Mega is represented by M, while milli by m).

EXAMPLE:

kilo: kilometer = 1000 m = 10³ m = 1 km

kilogram = 1000 g = 10³ g = 1 kg

If we measure the length of an object, the "base" unit in the SI system will be the meter (m). The following table gives you the magnitude of derived units.

II. SCIENTIFIC NOTATION: Scientific notation is a convenient shorthand for representing very large or very small numbers without the use of many zeros. We use scientific notation because longhand numbers are clumsy and impratical. In general the notation:

aⁿ

means a is multiplied by itself n times "a to the n^th power" (n is called the exponent)

EXAMPLE:

2³ = 2 x 2 x 2 = 8

3² = 3 x 3 = 9

A negative exponent is written:

It is the reciprocal of aⁿ. For example the reciprocal of 2 is 1/2, which in scientific notation is written 2^-1. The solution for a slightly more complicated example 2^-3 is shown below:

NOTE: This works for units too. For example, ^-1 means 'per'. So m s^-1 means meters per second; ms^-2 means meters per second squared.

Any large or small number can be expressed in two terms:

10³ = 1000

10² = 100

10¹ = 10

10⁰ = 1

10^-1 = 0.1

10^-2 = 0.01

10^-3 = 0.001

NOTE: Think about moving the decimal place manually: to convert 1.5 to 1,500,000; you would have to move it 6 times to the right. For scientific notation, you have to leave only one non-zero digit to the left of the decimal point.

1. Multiplication of numbers with exponents

In general: aⁿ x a^k = a^n+k add exponents with the same base

EXAMPLE:

2. Division of numbers with exponents

In general: aⁿ / a^k = a^n-k subtract exponents with the same base

EXAMPLE:

To multiply or divide complete numbers in scientific notation it is necessary to operate on both parts of the number independently.

EXAMPLE:

NOTE: When adding or subtracting numbers using scientific notation all must be expressed in the same power of ten.

III. CONVERSIONS: In order to use the SI system of measurement often observations have to be converted from another system of measurement (e.g. imperial units of miles, pounds, °F) to SI units.

We need to do calculations in the same system of units. This lab will give you experience with this. In all future labs you should report your answers in SI units. Details concerning conversion factors are reported in the table at the end of the lab.

Except for temperature, a zero value in one unit system is a zero value in the other systems. Temperature conversion involves subtraction and addition because the zero point differs between the SI and Imperial units systems.

To convert a quantity of X old units to its equivalent in new units, we proceed as follow:

Where: Y is a conversion factor.

Convert 23.9 feet to meters:

Convert 80°F to °C:

The only trick in such operations is in knowing when to use a conversion factor directly and when to use its inverse. If we write out the problem as is done above and algebraically cancel units, the correct choice is clear.

NOTE: Conversion of temperature differences does not involve addition or subtraction because we are dealing with distances on the temperature scales.

EXAMPLE:

IV. ACCURACY, PRECISION AND SIGNIFICANT FIGURES

Accuracy: how close a measured value is to the actual value.

Precision: number of digits reported with the value.

EXAMPLE:

If you measure the air temperature with a digital thermometer that gives you a reading of 65.2° F, this means that the thermometer’s precision is to tenths of a degree Fahrenheit.

But if this reading seems too high, due to a problem with calibration, the accuracy of the thermometer might only be within 5° F.

Significant figures: number of digits in a value that have real meaning and reflect the accuracy of the value or measurement.

EXAMPLE:

The length of this manual is reported as 28.2 cm.There are 3 significant figures in the length 28.2 cm. The value 28.2 cm implies that the measurement was made to an accuracy of tenths of a centimeter. It would be incorrect to quote the result as 28.20 cm because that would imply the measurement was accurate to 4 significant figures. If the ruler used is not accurate, the value should not include as many significant figures. If the ruler measures more accurately than this, the value should be reported with the appropriate precision.

NOTE: The number of significant figures is the number of reliably known digits in a measurement.

The significant figures are also linked to the precision of the instrument used for measurements. If you measure temperature with a thermometer that is precise to the nearest degree, your measurement will not comprise any decimals. If it is precise to the nearest 10^th of a degree, you will have one decimal. In Lab 1, since the data you were dealing with involved only one decimal place, you should have given your answers for the Normals and the means with only one decimal place.

You have to be careful with measurements that involve high precision when the accuracy could be questioned. This is very important to take into account when you perform conversions.

EXAMPLE:

A measured distance of 11 inches on a map is converted to centimeters: 11 in = 27.94 cm

27.94 cm implies high accuracy, which you did not have with the original measurement. The solution is to round the answer to about the same number of significant figures as the original measurement: 28 cm in this case.

If we have the value 200 m, we do not know its absolute precision. Is it precise to the nearest m, 10 m, or 100 m? If we use scientific notation, we can solve this problem:

NOTE: When you have very large or very small numbers the zeros will act to hold position if the number is not written in scientific form but in full. The location of the decimal point does not affect the determination of significant figures. The numbers 28.2 and 0.000282 both have three significant figures. In scientific notation these would be written 2.82 x 10¹ and 2.82 x 10^-4 respectively.

The following are guidelines for significant figures:

Keep in mind the two following rules in your calculations:

Data may be read off from graphs either by:

1. Interpolation: the new information that is to be gathered lies between known data points, i.e., within the range of observations.

2. Extrapolation: the new information is to be gathered from beyond the range of known data points, i.e., it can be used to predict data. Thus the relation must be extended either graphically (i.e., by extrapolating the best-fit line) or mathematically using the equations 1.2 and 1.3.

Obviously because extrapolation demands that the relation be extended beyond the range of known data there may be problems and caution should be exercised when doing so. We can also use interpolation and extrapolation in a different context, isolines.

Lab 2 Exercise

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