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# Derived Units

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

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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 below :

 Quantity Symbol Unit Unit Abbreviation Derivation Area A square meter m 2 length × width Volume V cubic meter m 3 length × width × height Density D kilograms/cubic meter kg/m 3 $\frac{\text{mass}}{\text{volume}}$ Concentration c moles/liter mol/L $\frac{\text{amount}}{\text{volume}}$ Speed (velocity) v meters/second m/s $\frac{\text{length}}{\text{time}}$ Acceleration a meters/second/second m/s 2 $\frac{\text{speed}}{\text{time}}$ 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.

$\left(\frac{1 \ \text{m}}{100 \ \text{cm}}\right)^3=\frac{1 \ \text{m}^3}{10^6 \ \text{cm}^3}=1$

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 (10 6 ), there are 10 6 cm 3 in 1 m 3 . 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:

$\left(\frac{1 \ \text{dm}}{10 \ \text{cm}}\right)^3 = \frac{1 \ \text{dm}^3}{1000 \ \text{cm}^3}=1$

There are thus 1000 cm 3 in 1 dm 3 , which is the same thing as saying there are 1000 mL in 1 L

There are 1000 cm 3 in 1 dm 3 . Since a cm 3 is equal to a mL and a dm 3 is equal to a L, we can say that there are 1000 mL in 1 L.

Sample Problem:  Derived Unit Conversion

Convert 3.6 × 10 8 mm 3 to mL.

Step 1: List the known quantities and plan the problem.

Known

• 1 m = 1000 mm
• 1 ml = 1 cm 3
• 1 m = 100 cm

Unknown

• 3.6 mm 3 = ? mL

This problem requires multiple steps and the technique for converting with derived units.  Simply proceed one step at a time: mm 3 to m 3 to cm 3 = mL.

Step 2: Calculate

$3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{m}}{1000 \ \text{mm}}\right)^3 \times \left(\frac{100 \ \text{cm}}{1 \ \text{m}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}$

Numerically, the steps are to divide 3.6 by 10 9 , followed by multiplying by 10 6 .  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.

$3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{cm}}{10 \ \text{mm}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}$

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 mm 3 .

#### 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

1. How many derived units are there?
2. Who established these units?
3. What derived unit gives rise to the definition of the watt?
4. What derived units are defined by the newton?

#### Review

1. What is a derived unit?
2. Convert 0.00722 km 2 to m 2
3. Convert 129 cm 3 to L
4. Convert 4.9 × 10 5 μm 3 to mm 3 .