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MEASUREMENTS AND CALCULATIONS
MEASUREMENTS AND
CALCULATIONS
METRIC SYSTEM
METRIC SYSTEM
For thousands of years, measurements were made using inexact methods.
For example, time might be measured by using a sundial or watching sand in
an hourglass.
In order to be able to accurately communicate scientific information with
others, though, a standard system of measurement needed to be created.
The system used by scientists around the world is known as the metric system.
The metric system is built around base units that describe different types of
measurements- length is measured in meters, mass is measured in grams,
volume is measured in liters, etc.
The metric system is a decimal system, meaning that measurements are
based on multiples of tens of the base units...
METRIC PREFIXES
Prefixes are used to describe multiples of the base units that are commonly
encountered.
For example, measuring distances commonly travelled by car in meters would
give very large numbers (a meter is slightly longer than a yard).
The prefix kilo- is used to describe a quantity of 1000, so a single kilometer is
equal to 1000 meters:
1 km = 1000 m
Again, since the metric system is based on the decimal system, understanding
the relationship between prefixes is fairly straightforward:
METRIC PREFIXES
Kilo-
Hecta-
Deca-
BASE UNIT (m, g, l, etc.)
Deci-
Centi-
Milli-
x 1000
x 100
x 10
x 1
x 1/10
x 1/100
x 1/1000
King
Henry
Drank
Beverages
During
Castle
Meetings
METRIC SYSTEM
Prefixes work the same for every type of metric measurement.
A kilometer (km) is a meter x 1000 = 1000 meters. A kilogram (kg) is a gram x
1000 = 1000 grams.
A millimeter (mm) is a meter x 1/1000 = 1/1000 meter. A milligram (mg) is a
gram x 1/1000 = 1/1000 grams.
There are other prefixes that describe even larger (mega-, giga-) and smaller
(micro-, nano-) multiples of base units, depending on the measurement being
made.
TEMPERATURE
Temperature is the average
amount of motion of molecules
that make up a substance.
Molecules that are moving faster
are perceived as warmer, while
molecules that are moving slower
are perceived as colder.
There have been several different
attempts to measure this motion,
leading to different temperature
scales:
TEMPERATURE
Fahrenheit is the preferred scale in
the US. But it’s not the only scale:
Anders Celsius used the boiling point
and freezing point of water as
reference points for the metric
measurement of temperature.
He used the freezing temperature of
water to define 0 degrees, and the
boiling point of water as the definition
of 100 degrees.
He then divided the distance between
these two temperatures by 100, and
created the Celsius temperature
scale...
TEMPERATURE
Lord Kelvin used a Celsius
degree as at the size of a degree
in his scale, but shifted 0 degrees
to represent no molecular motion
at all.
Called absolute zero, it is the
coldest temperature possible.
The Kelvin temperature scale is
the base unit of measurement in
the metric system.
A Celsius measurement is always
273 degrees higher than a Kelvin
measurement of the same
temperature (oK + 273 = oC).
DERIVED UNITS
Some units of measurement are actually combinations of other metric units.
These measurements are called derived units of measure.
A derived measurement requires other measurements to be taken first.
Examples of these include speed (m/s), volume (cm3), and density (g/cm3):
Speed is how far an object goes in a certain amount of time (2
measurements).
Volume is the amount of space an object occupies (3 measurements-
length x width x height).
Density is the amount of mass found in a certain volume (4 measurements!)
SCIENTIFIC NOTATION
When working with extremely large or small numbers (especially with ones
containing large amounts of zeros on either end), it is convenient to express
the numbers in scientific notation.
Scientific notation rewrites a numerical value as:
a number between 1 and 10,
multiplied by 10 raised to an exponential value.
Consider the number 602,200,000,000,000,000,000,000. Any calculations
using this number would be challenging, even with a calculator. Rewriting it in
scientific notation would result in 6.022 x 1023 (a much less cumbersome
number with which to work).
Here’s the process to convert a number to scientific notation:
SCIENTIFIC NOTATION
Move the decimal to a location directly behind the first non-zero number (if
there is no decimal present, it is assumed to be after the last digit).
How many numbers did you pass on the way to the decimal’s new home? In
this case, the decimal was moved 23 spaces. This is the exponent that we will
use raise our multiplier of ten.
Any zeros before the first non-zero number or after the last non-zero number
can now be dropped.
Moving the decimal to the left will increase the exponent used to raise the
multiplier, while moving it to the right will decrease the exponent.
How would you write 0.0000000000004302 in scientific notation?
. 602,200,000,000,000,000,000,
000
6.02,200,000,000,000,000,000,
000
6.022 x
1023
SCIENTIFIC NOTATION
0.0000000000004302
Follow the same steps:
Move the decimal to a location directly behind the first non-zero number (if
there is no decimal present, it is assumed to be after the last digit) =
00000000000004.302
How many numbers did you pass on the way to the decimal’s new home? In
this case, the decimal was moved 13 spaces. This is the exponent that we will
use raise our multiplier of ten.
Any zeros before the first non-zero number or after the last non-zero number
can now be dropped = 4.302 x 10-13
Remember: moving the decimal to the left will increase the exponent used to
raise the multiplier, while moving it to the right will decrease the exponent.
DIMENSIONAL ANALYSIS
The length of a football field can be described as 100 yards long. It can also
be described as 300 feet long. It can also be described as being 3600 inches
long.
These three measurements describe the same length, but in different units.
You are probably familiar with these lengths because they have been
described to you many times, but suppose you needed to convert something
less familiar?
The process of converting a measurement from one unit to another is called
dimensional analysis.
The basic idea is to multiply the original measurement by equivalence
statements, which are fractions made of two different units with equal value,
(which makes the fraction equal to 1) which allow the original unit of
measurement to be cancelled out in favor of a new unit of measure.
EQUIVALENCE
STATEMENTS
1/1 = 1
2/2 = 1
3/3 = 1
12/12 = 1
138/138 = 1
(4-2)/2 = 1
2/(4-2) = 1
6/1 half dozen = 1
3/1 trio = 1
1 pair/2 = 1
12 inches/1 foot = 1
3 feet/1 yard = 1
1 yard/3 feet = 1
5280 feet/1 mile = 1
Any measurement
divided by another
measurement of the
same value will be
equal to 1, and is
considered an
equivalence
statement.
DIMENSIONAL ANALYSIS
A football field is 100 yards long. Express this distance in feet. Then, express it in
inches.
100 yds = ? ft = ? in
Place the 100 yards over 1 to make a fraction:
100 yds = ? ft
1
We want to multiply 100 yards by a fraction (equal to 1) that will allow us to cancel out
yards and replace it with another measurement- this will be our equivalence statement.
A fraction containing yards and feet, that equals 1, could be either:
1 yd or 3 ft
3 ft 1 yd
Which one should we use?
DIMENSIONAL ANALYSIS
The idea is to cancel out the original unit (in this case, “yards”) and replace it
with another unit. In order to do this, the unit in the numerator (top) of the first
fraction must be in the denominator (bottom) of the next fraction:
100 yds x 3 ft = ? ft
1 1 yd
The unit “yards” cancels out, leaving us with only one unit left- feet. If this unit is
the one needed for the answer, then you don’t need any more fractions.
Multiply numerators together to get a final numerator, then multiply
denominators together to get a final denominator:
100 x 3 ft = 300 ft
1 1
DIMENSIONAL ANALYSIS
To convert to inches, do the same thing:
100 yds x 3 ft = ? in
1 1 yd
The unit “yards” cancels out, leaving us with only one unit left- feet. But our
problem wants us to find the measurement in inches, so we multiply another
equivalence statement, this one converting feet to inches:
100 yds x 3 ft x 12 inches = ? in
1 1 yd 1 ft
Again, cancel out all units that appear in the numerator and the denominator until
you have just one unit left- make sure it’s the correct unit!- then multiply:
100 x 3 x 12 inches = 3600 in
1 1 1
DIMENSIONAL ANALYSIS
If there is a direct equivalence statement, the process can be shortened:
100 yds x 36 in = ? in
1 1 yd
The unit “yards” cancels out, leaving us with only one unit left- inches.
Multiply numerators together to get a final numerator, then multiply
denominators together to get a final denominator:
100 x 36 in = 3600 in
1 1
DIMENSIONAL ANALYSIS
Converting between metric units is exactly the same process:
274 cm = ? km
Use “King Henry” to create equivalence statements (remember- smaller units
get bigger numbers), and make fractions from them:
1 meter = 100 cm 1000 m = 1 km
274 cm x 1 m x 1 km = ? km
1 100 cm 1000 m
Cancel out units until only one unit is left, make sure that it is the correct unit,
then multiply:
274 x 1 x 1 km = 274 km = 0.00274 km
1 100 1000 100000
DIMENSIONAL ANALYSIS
Converting between derived units is handled exactly the same way:
100 km/hr = ? m/s
Use “King Henry” to create equivalence statements (remember- smaller units
get bigger numbers), and make fractions from them:
1000 meters = 1 km 60 min = 1 hr 60 s = 1 min
100 km x 1000 m x 1 hr x 1 min = ? m/s
1 hr 1 km 60 min 60 s
Cancel out units until only one unit is left, make sure that it is the correct unit,
then multiply:
100 x 1000 m x 1 x 1 = 100,000 m = 27.8 m/s
1 1 60 60 s 3600 s
DIMENSIONAL ANALYSIS
In summary:
Place the original measurement over “1” to make a fraction (with a complex
unit such a g/ml, the top unit goes in numerator and the bottom goes in
denominator after the “1”)
Multiply by an equivalence statement to cancel out the original unit and replace
it with a new unit,
Keep multiplying by equivalence fractions until the desired unit is reached.
ACCURACY VS. PRECISION
• Suppose you went to a doctor because you had a pain in your foot. The doctor
spends several minutes examining your hands, and finally says, “I don’t see
anything wrong.”
• Your doctor was wayyy off target with his exam. In science, hitting/missing the
target is called accuracy. In order to measure accuracy, a measurement taken
by a scientist must be compared to the actual measurement in question.
• For example, if you measure a football field (which is defined in the rulebook as
being 91.44 meters), and your measurement is 91.5 meters, then you were
pretty accurate!
• However, if your measurement was 81.5 meets, you missed the target…badly.
You were NOT very accurate.
ACCURACY VS. PRECISION
• So, you point to your foot and say, “The pain is down here!”. The doctor says,
“Oh!”, and begins examining your foot. He finds a splinter deep in your heel and
says, “I’m going to remove this for you; let me go get my chain saw…”
• While it is true that the doctor will be operating in the correct location, a chain saw is
going to cause more damage to the rest of your foot than it will fix. Chain saws are
NOT good for removing splinters, a scalpel and tweezers are more precise- they
focus the work on a very small area of your foot.
• In science, it is STRONGLY recommended that a measurement be taken several
times (to make sure that the device is read correctly). Multiple measurements that
are very close to each other are precise; measurements with a lot of variation are
NOT precise.
• If you measure the football field three times, and the measurements are 91.5
meters, 91.3 meters, and 91.4 meters, you have been very precise in your
measurements.
• If your measurements are 91.5 meters, 93.7 meters, and 90.1 meters, well… NOT
very precise measurements.
ACCURACY VS. PRECISION
• Can measurements be precise, but not accurate?
• If your measurements of a football field are 87.5 meters, 87.6 meters, and 87.3
meters, they ARE precise (they are very close to each other), but NOT very
accurate (none of them are close to 91.44 meters).
• Can measurements be accurate, but not precise?
• 90.7 meters, 91.5 meters, and 92.1 meters are all close to 91.44 (accurate), but
there is a lot of variation between the three measurements- NOT very precise.
