度量分析

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Conversion Factors
::换算系数

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Many quantities can be expressed in several different ways. The English system measurement of 4 cups is also equal to 2 pints, 1 quart, and ¼ of a gallon.
::许多量可以用几种不同的方式表示。英语系统测量的4个杯也等于2品脱、1夸特和1⁄4加仑。

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4 cups = 2 pints = 1 quart = 0.25 gallon
::4杯=2品脱=1夸特=0.25加仑

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Notice that the numerical component of each quantity is different, while the actual amount of material that it represents is the same. That is because the units are different. We can establish the same set of equalities for the metric system:
::注意每个数量的数字成分是不同的, 而它所代表的物质的实际数量是相同的。 这是因为各个单位是不同的。 我们可以为测量系统建立相同的等值 :

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1 meter = 10 decimeters = 100 centimeters = 1000 millimeters
::1米=10厘米=100厘米=1000毫米

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The metric system’s use of powers of 10 for all conversions makes this quite simple.
::衡量系统对所有转换使用10个功率使这一点非常简单。

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Whenever two quantities are equal, a ratio can be written that is numerically equal to 1. Using the metric examples above:
::当两个数量相等时,可以写出数字等于1的比率。

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$\begin{align*}\frac{1 \ \text{m}}{100 \ \text{cm}}=\frac{100 \ \text{cm}}{100 \ \text{cm}}=\frac{1 \ \text{m}}{1 \ \text{m}}=1\end{align*}$
::1 m100 厘米= 100 厘米100 厘米= 1 m1 m= 1

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The $\begin{align*}\frac{1 \ \text{m}}{100 \ \text{cm}}\end{align*}$ is called a conversion factor . A conversion factor is a ratio of equivalent measurements. Because both 1 m and 100 cm represent the exact same length, the value of the conversion factor is 1. The conversion factor is read as “1 meter per 100 centimeters”. Other conversion factors from the cup measurement example can be:
::1m100厘米称为换算系数。换算系数是等效测量之比。由于1m和100cm代表完全相同的长度,换算系数值为1。换算系数为“100厘米1米”。

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$\begin{align*}\frac{4 \ \text{cups}}{2 \ \text{pints}}=\frac{2 \ \text{pints}}{1 \ \text{quart}}=\frac{1 \ \text{quart}}{0.25 \ \text{gallon}}=1\end{align*}$
::4 杯2 品脱=2 品脱1 夸特=1 夸特=1 夸特0.25加仑=1

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Since the numerator and denominator represent equal quantities in each case, all are valid conversion factors.
::由于分子和分母在每种情况下代表相同数量,因此所有数字和分母均为有效的换算系数。

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Scientific Dimensional Analysis
::科学层面分析

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Conversion factors are used in solving problems in which a certain measurement must be expressed with different units. When a given measurement is multiplied by an appropriate conversion factor, the numerical value changes, but the actual size of the quantity measured remains the same. Dimensional analysis is a technique that uses the units (dimensions) of the measurement in order to correctly solve problems. Dimensional analysis is best illustrated with an example.
::转换系数用于解决必须用不同单位表示某种计量的问题。当某一计量乘以适当的换算系数时,数值变化,但所测量数量的实际大小保持不变。度量分析是一种技术,它使用测量单位(二元)来正确解决问题。度量分析最好用一个示例来说明。

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Sample Problem: Dimensional Analysis
::问题:多面分析

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How many seconds are in a day?
::一天有多少秒?

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

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Known
::已知已知

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• 1 day = 24 hours
  ::1天=24小时
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• 1 hour = 60 minutes
  ::1小时=60分钟
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• 1 minute = 60 seconds
  ::1分钟=60秒

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Unknown
::未知

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• 1 day = ? seconds
  ::1天=? 秒

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The known quantities above represent the conversion factors that we will use. The first conversion factor will have day in the denominator so that the “day” unit will cancel. The second conversion factor will then have hours in the denominator, while the third conversion factor will have minutes in the denominator. As a result, the unit of the last numerator will be seconds and that will be the units for the answer.
::上述已知数量代表我们将要使用的转换系数。 第一个转换系数在分母中有一天, 以便“ 日” 单位取消。 第二个转换系数在分母中有小时, 而第三个转换系数在分母中有分钟。 因此, 最后一个分子的单位将是秒, 也就是回答的单位 。

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Step 2: Calculate
::第2步:计算

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$$\begin{align*}1 \ \text{d} \times \frac{24 \ \text{h}}{1 \ \text{d}} \times \frac{60 \ \text{min}}{1 \ \text{h}} \times \frac{60 \ \text{s}}{1 \ \text{min}}=86, 400 \ \text{s}\end{align*}$$
::1 dx24 h1 dx60 分钟1 hx60 s1 分钟= 86 400 秒

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Applying the first conversion factor, the “d” unit cancels and 1 × 24 = 24. Applying the second conversion factor, the “h” unit cancels and 24 × 60 = 1440. Applying the third conversion factor, the “min” unit cancels and 1440 × 60 = 86,400. The unit that remains is “s” for seconds.
::应用第一个换算系数,“d”单位取消,1×24=24。应用第二个换算系数,“h”单位取消,24×60=1440。应用第三个换算系数,“min”单位取消,1440×60=86 400。剩下的单位为“s”秒。

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

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Seconds is a much smaller unit of time than a day, so it makes sense that there are a very large number of seconds in one day.
::秒比一天要小得多, 所以一天的秒数非常多是有道理的。

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Summary
::摘要

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• A conversion factor is a ratio of equivalent measurements.
  ::换算系数是等值测量之比。
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• Dimensional analysis is a technique that uses the units (dimensions) of the measurement in order to correctly solve problems.
  ::尺寸分析是一种技术,它使用测量单位(尺寸)来正确解决问题。

  

 

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Review
::回顾

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1. What is a conversion factor?
   ::什么是转换系数?
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2. What is dimensional analysis?
   ::什么是维学分析?
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3. How many meters are in 3.7 km?
   ::3.7公里内有多少米?
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4. How many kg in 12980 g?
   ::12980克中有多少公斤?