节: 衍生单位 | 3.10 衍生单位

章节大纲

A green farmland under a clear sky, illustrating agricultural land.

How has farming evolved?
::农业是如何演变的?

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.
::随着农业越来越昂贵和利润越来越低(至少对于小农场而言),许多家庭将把土地出售给想建立商业或住宅地产的建筑商,为了出售,需要有一个准确的地块,必须确定农场的面积并从这些方面计算面积。

Dimensional Analysis and Derived Units
::尺寸分析和衍生单位

Some units are combinations of . 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 :
::一些单位是.的组合。一个衍生单位是由SI基础单位的数学组合产生的单位。我们已经将体积和能量作为取出单位的两个例子讨论过。

Derived SI Units
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	$\begin{align}\frac{\text{mass}}{\text{volume}}\end{align}$
	c	moles/ liter	mol/L	$\begin{align}\frac{\text{amount}}{\text{volume}}\end{align}$
Speed (velocity)	v	meters/second	m/s	$\begin{align}\frac{\text{length}}{\text{time}}\end{align}$
Acceleration	a	meters/second/second	m/s ²	$\begin{align}\frac{\text{speed}}{\text{time}}\end{align}$
Force	F	newton	N	mass × acceleration
Energy	E	joule	J	force × length

Using 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.
::使用派生单位需要特别小心。当单位按面积或体积平方或立方时, 转换系数本身也必须平方。下面显示的是立方厘米和立方米的换算系数。

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

Because a cube has 3 dimensions (length, width and height) , each dimension 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:
::由于立方体有三个维度(长度、宽度和高度),每个维度可转换为1米至100厘米。由于100立方等于100万立方厘米(106),1立方米中有106立方厘米3。两个方便的体积单位是升,等于立方厘米,和毫立方厘米,等于立方厘米。转换系数是:

$\begin{align*}\left(\frac{1 \ \text{dm}}{10 \ \text{cm}}\right)^3 = \frac{1 \ \text{dm}^3}{1000 \ \text{cm}^3}=1\end{align*}$
:1 dm10 cm)3=1 dm31000 cm3=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
::1 dm3中有1000立方厘米3,这与1 立方厘米中有1000立方厘米相同。

A cube illustrating volume conversion between cubic centimeters and cubic meters.

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.
::由于每立方厘米3等于一毫升,每立方厘米3等于一毫升,每立方厘米3等于一升,我们可以说每立方厘米1千毫升。

Sample Problem: Derived Unit Conversion
::问题:衍生单位转换

Convert 3.6 mm ³ to mL.
::3.6毫米3改为毫升。

Step 1: List the known quantities and plan the problem.
::第1步:列出已知数量并规划问题。

Known
::已知已知

1 m = 1000 mm
::1米=1000毫米

1 ml = 1 cm ³
::1毫升=1厘米3

1 m = 100 cm
::1米=100厘米

Unknown
::未知

3.6 mm ³ = ? mL
::3.6毫米3 = 毫升

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.
::这一问题需要多个步骤和与衍生单位转换的技术。只需一步一步地进行: mm3至m3至cm3 = mL 。

Step 2: Calculate
::第2步:计算

$\begin{aligned} 3.6 {mm}^{3} \times {(\frac{1 m}{1000 mm})}^{3} \times {(\frac{100 cm}{1 m})}^{3} \times \frac{1 mL}{1 {cm}^{3}} = 0.0036 mL \end{aligned}$

$\begin{align*}3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{m}}{1000 \ \text{mm}}\right)^3 \times \left(\frac{100 \ \text{cm}}{1 \ \text{m}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}\end{align*}$
::3.6毫米3×(1米1000毫米)3×(100厘米1米)3×1毫L1厘米3=0.0036毫升

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.
::从数字上看,这些步骤是将3.6除以109,然后乘以106。你可能会发现,你可以一步一步地缩短问题,先确定从毫米到厘米的转换系数,然后使用该系数而不是首先转换成m。 1厘米中为10毫米。

$\begin{align*}3.6 \ \text{mm}^3 \times \left(\frac{1 \ \text{cm}}{10 \ \text{mm}}\right)^3 \times \frac{1 \ \text{mL}}{1 \ \text{cm}^3}=0.0036 \ \text{mL}\end{align*}$
::3.6毫米3×(1厘米10毫米)3×1毫升1厘米3厘米3=0.0036毫升

In this case 3.6 / 1000 gives the same result of 0.0036.
::在这种情况下,3.6/1000得出同样的0.0036结果。

Step 3: Think about your result.
::步骤3:想想你的结果。

Cubic millimeters are much smaller than cubic centimeters, so the final answer is much less than the original number of mm ³ .
::立方毫米比立方厘米要小得多,所以最终答案远低于毫米3的原数。

Summary
::摘要

A derived unit is a unit that results from a mathematical combination of SI base units.
::衍生单位是指由SI基单元的数学组合产生的单位。

Calculations involving derived units follow the same principles as other unit conversion calculations.
::涉及衍生单位的计算遵循与其他单位换算计算相同的原则。

Review
::回顾

What is a derived unit?
::什么是衍生单位?

Convert 0.00722 km ³ to m ³
::将0.00722平方公里转换为立方米

Convert 129 cm ³ to L
::将129厘米3转换为L

Convert 4.9 × 10 ⁵ μm ³ to mm ³
::转换 4.9 × 105 微米3 至毫米3

章节大纲

How has farming evolved? ::农业是如何演变的?

Dimensional Analysis and Derived Units ::尺寸分析和衍生单位

Summary ::摘要

Review ::回顾

How has farming evolved?
::农业是如何演变的?

Dimensional Analysis and Derived Units
::尺寸分析和衍生单位

Summary
::摘要

Review
::回顾