Derived Units
Ck12 Science
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Printed: November 17, 2015
AUTHOR
Ck12 Science
www.ck12.org
C HAPTER
Chapter 1. Derived Units
1
Derived Units
• Define derived unit.
• Carry out unit conversions using derived units.
How has farming evolved?
As farming becomes more expensive and less profitable (at least for small farms), many families will sell the land
to builders who want to erect either commercial or residential properties. In order to sell, an accurate property tile is
needed. The dimensions of the farm must be determined and the acreage calculated from those dimensions.
Dimensional Analysis and Derived Units
Some units are combinations of SI base units. A derived unit is a unit that results from a mathematical combination
of SI base units. We have already discussed volume and energy as two examples of derived units. Some others are
listed in the Table 1.1:
TABLE 1.1: Derived SI Units
Quantity
Area
Volume
Symbol
A
V
Unit
square meter
cubic meter
Unit Abbreviation
m2
m3
Density
D
kg/m3
Concentration
c
kilograms/cubic
meter
moles/liter
Speed (velocity)
v
meters/second
m/s
mol/L
Derivation
length × width
length × width ×
height
mass
volume
amount
volume
length
time
1
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TABLE 1.1: (continued)
Acceleration
Force
Energy
a
F
E
speed
time
mass × acceleration
force × length
meters/second/second m/s2
newton
N
joule
J
Using dimensional analysis with derived units requires special care. When units are squared or cubed as with area
or volume, the conversion factors themselves must also be squared. Shown below is the conversion factor for cubic
centimeters and cubic meters.
3
3
1m
= 1016 m
100 cm
cm3 = 1
Because a cube has 3 sides, each side is subject to the conversion of 1 m to 100 cm. Since 100 cubed is equal to
1 million (106 ), there are 106 cm3 in 1 m3 . Two convenient volume units are the liter, which is equal to a cubic
decimeter, and the milliliter, which is equal to a cubic centimeter. The conversion factor would be:
3
3
1 dm
1 dm
=
=1
10 cm
1000 cm3
There are thus 1000 cm3 in 1 dm3 , which is the same thing as saying there are 1000 mL in 1 L
FIGURE 1.1
There are 1000 cm3 in 1 dm3 . Since a
cm3 is equal to a mL and a dm3 is equal
to a L, we can say that there are 1000 mL
in 1 L.
Sample Problem: Derived Unit Conversion
Convert 3.6 × 108 mm3 to mL.
Step 1: List the known quantities and plan the problem.
Known
• 1 m = 1000 mm
• 1 ml = 1 cm3
• 1 m = 100 cm
Unknown
• 3.6 mm3 = ? mL
This problem requires multiple steps and the technique for converting with derived units. Simply proceed one step
at a time: mm3 to m3 to cm3 = mL.
2
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Chapter 1. Derived Units
Step 2: Calculate
1m
3.6 mm ×
1000 mm
3
3
100 cm
×
1m
3
×
1 mL
= 0.0036 mL
1 cm3
Numerically, the steps are to divide 3.6 by 109 , followed by multiplying by 106 . You may find that you can shorten
the problem by a step by first determining the conversion factor from mm to cm and using that instead of first
converting to m. There are 10 mm in 1 cm.
3 1 mL
3.6 mm3 × 101 cm
mm × 1 cm3 = 0.0036 mL
In this case 3.6 / 1000 gives the same result of 0.0036.
Step 3: Think about your result.
Cubic millimeters are much smaller than cubic centimeters, so the final answer is much less than the original number
of mm3 .
Summary
• A derived unit is a unit that results from a mathematical combination of SI base units.
• Calculations involving derived units follow the same principles as other unit conversion calculations.
Practice
Questions
Use the link below to answer the following questions:
http://www.unc.edu/~rowlett/units/siderive.html
1.
2.
3.
4.
How many derived units are there?
Who established these units?
What derived unit gives rise to the definition of the watt?
What derived units are defined by the newton?
Review
Questions
1.
2.
3.
4.
What is a derived unit?
Convert 0.00722 km2 to m2
Convert 129 cm3 to L
Convert 4.9 × 105 µm3 to mm3 .
• derived unit: A unit that results from a mathematical combination of SI base units.
References
1. User:Joegrimes/Wikipedia. http://commons.wikimedia.org/wiki/File:Farmlandlysander.JPG .
2. CK-12 Foundation - Christopher Auyeung. .
3