Dimensional Analysis

量纲分析


The Gravitational Constant

万有引力常数            

Newton's law of gravity is usuallyexpressed in terms of a universal constant, G:

牛顿的万有引力定律通常用宇宙常数G来表示:


基普朗克单位             

The universal constants c,h,G can be usedto re-express the fundamental units for mass, time and distance. That meansthat any physical quantity in mechanics can be expressed in terms of thesethree constants. In the case of thermodynamics or electrodynamics, there are afew other universal constants that would need to be included to have a completedescription.

通用常数chg可以用来表示基本单位的质量、时间和距离。这意味着力学中的任何物理量都可以用这三个常数来表示。在热力学或电动力学的情况下,还有一些其他的通用常数,需要包含完整的描述。

Recall that:

回顾

If we replace this in the relation weobtained in the above exercise, we get:

如果我们把它替换成我们在上面练习中得到的关系,我们得到:


So we can get a relation for the time dimension:

这样我们就可以得到时间维度的关系:


or:


Replacing the value for T in terms of[h],[c],[G] in the above equations we get:

我们用上述公式替换了t的值,在上述方程中得到了:


Based on these dimensional relations we canintroduce three new units for distance, time and mass, defined exclusively interms of these three universal constants:

基于这些量纲关系,我们可以引入三个新的距离、时间和质量单位,仅用这三个普遍常数来定义:


Because these units are defined in terms ofuniversal physical constants, we expect them to have a deeper physical meaningcompared to our standard units - the second, the meter and the kilogram:

因为这些单位是用通用物理常数来定义的,所以我们期望它们比我们的标准单位有更深的物理意义,即第二个单位,即米和千克:

the second has been defined as the 1/(243600) part of a mean solar day;

第二被定义为1 /243600)一个平均太阳日;            

the meter has been defined as the 1/40106 part of the circumference of the Earth through thepoles;

米被定义为地球周边1/40106部分通过极点;

the kilogram has been defined as the 1/1000part of a volume of one cubic meter of pure water at the temperature of meltingice.

在融化的冰的温度下,千克被定义为一立方米纯净水的1/1000部分。

While the precise definitions of thesecond, meter and kilogram have slightly changed, their values areapproximately the same. While they were designed to unify and standardizeprevious units, their definition is far from universal. These units impose ascale fitted for the human perception of space, time and weight.

虽然秒、米和千克的精确定义略有改变,但它们的值大致相同。虽然他们的目的是统一和规范以前的单位,他们的定义是远远不是普遍的。这些单位强加了一个适合人类感知空间、时间和重量的尺度。

The base Planck units impose a scale with amore profound physical meaning and implications.

基本普朗克单位施加一个更深刻的物理意义和含义的规模。