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# Derived Units

## A combination of SI base units. Example: Joules.

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Derived Units

Credit: User:Joegrimes/Wikipedia
Source: http://commons.wikimedia.org/wiki/File:Farmlandlysander.JPG

#### 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 Table below:

 Quantity Symbol Unit Unit Abbreviation Derivation Area A square meter m2 length × width Volume V cubic meter m3 length × width × height Density D kilograms/cubic meter kg/m3 massvolume\begin{align*}\frac{\text{mass}}{\text{volume}}\end{align*} Concentration c moles/liter mol/L amountvolume\begin{align*}\frac{\text{amount}}{\text{volume}}\end{align*} Speed (velocity) v meters/second m/s lengthtime\begin{align*}\frac{\text{length}}{\text{time}}\end{align*} Acceleration a meters/second/second m/s2 speedtime\begin{align*}\frac{\text{speed}}{\text{time}}\end{align*} Force F newton N mass × acceleration Energy E joule J force × length

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.

(1 m100 cm)3=1 m3106 cm3=1\begin{align*}\left(\frac{1 \ \text{m}}{100 \ \text{cm}}\right)^3=\frac{1 \ \text{m}^3}{10^6 \ \text{cm}^3}=1\end{align*}

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:

(1 dm10 cm)3=1 dm31000 cm3=1\begin{align*}\left(\frac{1 \ \text{dm}}{10 \ \text{cm}}\right)^3 = \frac{1 \ \text{dm}^3}{1000 \ \text{cm}^3}=1\end{align*}

There are thus 1000 cm3 in 1 dm3, which is the same thing as saying there are 1000 mL in 1 L

Credit: CK-12 Foundation - Christopher Auyeung

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.[Figure2]

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.

Step 2: Calculate

3.6 mm3×(1 m1000 mm)3×(100 cm1 m)3×1 mL1 cm3=0.0036 mL\begin{align*}3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{m}}{1000 \ \text{mm}}\right)^3 \times \left(\frac{100 \ \text{cm}}{1 \ \text{m}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}\end{align*}

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.

3.6 mm3×(1 cm10 mm)3×1 mL1 cm3=0.0036 mL\begin{align*}3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{cm}}{10 \ \text{mm}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}\end{align*}

In this case 3.6 / 1000 gives the same result of 0.0036.

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.

### Review

1. What is a derived unit?
2. Convert 0.00722 km2 to m2.
3. Convert 129 cm3 to L.
4. Convert 4.9 × 105 μm3 to mm3.

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