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:mprQvinq j'llath Skills
. HOW L-~ //' 00 Unit Conversions
- - _.. . .V
When you measure something, you always need to indicate what units
you are using. For example, suppose someone told you that her cat had a
weight of "20." That doesn't mean much without units. Does the cat weigh
20 pounds, 20 newtons, 20 ounces, or 20 tons? Distance measurements
(length) also need units such as feet, inches, meters, kilometers, and miles.
If you are measuring time, you use units such as seconds, minutes, hours,
and years.
Converting between units is also very important. For example, what if
a friend told you that he would phone you in 86,400 seconds (sec)? When
would that be? Aher this activity, you will be able to show that this is the
same as 1 day. Unit conversions are also important for comparing two
measurements made with different units. For example, suppose a person
who is 5 feet (ft) 8 inches (in) tall has a hat on that is 0.30 meters (m) tall.
What is the total height of the person, including the hat? Unfortunately,
you cannot simply add the lengths. You must convert all of them to the
same unit, and then you can add the lengths. You may have to convert
again to a more reasonable unit. The total height of the person would be
79.8 in, 6 ft 7.8 in, or 2.03 m tall.
How do you make these conversions? The method is called unit
analysis (or dimensional analysis). These terms may sound complicated,
but the method is pretty simple. The method uses conversion factors
to convert units step-by-step, canceling units at each step. Using these
guidelines, unit analysis is simple.
Unit Analysis Guidelines
) Conversion factors relate different units and are differ~nt ways of
expressing the number-I. For example~ there are 12 in inl ~ or
12 in = 1 ft, or
C;:;)
= 1.
. ) Conversion factors can be flipped (inverted) as long as the units stay
with the number. For example, you can write
r; ~n)
= 1, or ( 1~ ~~) = l.
This is the same thing.
~) Units behave as numbers do when you multiply fractions. The units
in the numerator of fractions will cancel the same units in the
12
denominator of fractions. For example, ( 3 jz( = 4.
jri)
';) In unit analysis, your goal is to cancel the same units in the numerators
and denominators until you end up with the units you want.
c.) When you convert between units, follow these steps.
a. Identify the units that you have.
b. See which units you want.
c. Note the conversion factors that get you from Steps Sa and Sb.
(.) Work through the following example to practice using the guidelines
for unit analysis from Steps 1-5.
How many inches are there in 1 mile (mi)?
Conversion steps (from Step 5)
Answers
What unit do you have now?
1 ml
What unit do you want?
"How many inches"
I
What are the conversion factors?
1 mi
1
=5,280 ft
or
(5,28~ft) =
= 12 in
or
(12In)
=1
1 ft
1 ft
1 ml
, figure H5B.1 Conversion table for mil. to Inch••
To convert miles to inches, start with what you know (1 mi). Then use
conversion factors (figure HSB.1) to cancel units as you go until you get
to the units that you want (inches). When the same units are on both the
bottom and the top, they cancel. Work with your teacher to see how to
cancel these units on the top and bottom.
The units cancel, so you are left with units of inches. You can then
multiply the numerator numbers together for the answer in inches.
~.
--""';111_,
_-. t'·_..... olIIti..
.,~lAIMl&·
_
••
......__M"-"'__;iW
_
·~"",,"-
.) Work through a more complicated example using dimensional
analysis. Suppose that you want to convert 75 miles per hour (mph)
into feet per second (ftJsec).
Conversion steps (from Step 5)
Answers
What unit do you. have now?
mi
75 hr (mph)
What unit do you want?
What are the conversion factors?
ft
sec (ft Isec)
1 hour (hr)
1 minute (min)
1 mi
=
=
=
60 minutes (min)
60 seconds (sec)
Now what? Take your conversions (figure H5B.2) one step at a
time, canceling units as you go until you arrive at the units you want.
X
e7~t) X (6~~) X
Ut:)
=
ft) -_(110
ft) _110 Asec
1 sec -
75 x 5,280 it X 1 X 1) _ (396,000
( 1 X 1 X 60 x 60 sec 3,600 sec
o
-4
Try the conversions in Steps 8a-e on your own. Use the following
conversion factors, and show your calculations for each conversion.
1 slink = 7 zips
1 sHff = 5 zips
4 voles = 3 sliffs
8 lampos = 7 flies
12 voles = 1lampo
a. How many sliffs are in 1 tam po?
b. One vole is how many zips?
c. How many flies are in 1 slink?
JViII"- 5 " .... t. De UIII c.wrsteII
I
~
5,280 ft
Figure H58.2 CORVlnion talill for mila per hour to lee. per sicond.
(7;;)
-,
~
Toolbox Activities
:Jnversions
C"e.4.f~ o,.,e.w jo"t"~' ~r\~ q..OH$w~c- CAli G~ be/OkJ'
. You must show
all vour work and cross out units in the numerator and the denominator that cancel.
Decide when you wish to convert an answer to scientific notation.
1. Use the following information in the conversions. There are
60 seconds in 1 minute (60 sec = 1 min)
60 minutes in 1 hour (60 min
= 1 hr)
24 hours in 1 day (24 hr = 1 day)
365.25 days in 1 year (365.25 days = 1 yr)
a. How many seconds are in 1 hr?
b. How many seconds are
c. How many seconds are
d. How many seconds are
e. How many seconds are
f. How many seconds are
2. Use the conversion factors
a. How many "minutes are
(10~)
x
(1
b.
c.
d.
e.
How
How
How
How
in 4 hr?
in 1 day?
in 23 days?
in 1 yr?
in 100 yr?
from question 1.
in 10 sec?
min') = 0.16 min
60 >d)
many
many
many
many
hours are in 60 min?
hours are in 30 min?
years are in 100 days?
days are in 12 hr?
3. Use 1 in = 2.54 cm for the following distances. Show all units canceling.
a. How many centimeters (cm) are in 1 inch (in)?
b. How many centimeters are in 6 in?
c. Twenty-three inches are how many feet (ft)?
d. How many inches are in 1 meter (m)?
(1
~)
x (100
yr6) x ( 2.549ri
1 in )
1m
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