What makes an Equation? 

方程式是什么?


What makes an equation? 

方程式是什么?

In school one often thinks about equations as formulas. The aim of a formula is to shorten or condense information into an expression so that you don't need to worry about the derivation. You want to know the volume of a sphere, you look for the right formula (), you plug in the value for the radius, you plug in the value for π, and you are done. 

在学校里,人们常常把方程看成公式。公式的目的是将信息缩短或浓缩到表达式中,这样就不必担心派生问题。你想知道一个球体的体积,你寻找正确的公式(),你插入半径的值,你插入π的值,你就完成了。

The purpose of a formula is that you don't need to worry about it. Once you know what formula to use things are easy, and you can worry about a different part of the problem. 

公式的目的是让你不必担心它。一旦你知道使用什么样的公式是容易的,你可以考虑问题的不同部分。

Equations (as we will look at them in this course) are different. Equations are something you want to contemplate and to understand how they were derived. Their fundamental role is not to shorten your computations, but to express a relation, to equate two physical quantities.

 方程式(我们在这门课中看他们)是不同的。方程是你想思考并理解它们是如何派生出来的东西。它们的基本作用不是缩短计算,而是表达一个关系,使两个物理量相等。

The easiest and most accessible way of contemplating an equation is to think about its dimensions. On both sides of the equal sign you need to have the same dimensions. Let's take one of the most famous equations in physics (we will talk about a way of deriving it in Lesson 3). 

考虑方程的最简单、最容易获得的方法是考虑它的维度。在等号的两边,你需要有相同的维度。让我们用物理学中最著名的方程之一(我们将讨论在第3课中推导它的方法)。


We know that E is an energy, so  must have dimensions of energy. If you recall from high school that the kinetic energy of a particle is  then this is not surprising. Because if: 

我们知道,E是能量,所以有一定维度的能量。如果你还记得高中,一个粒子的动能是那么这并不令人意外。因为如果:


Then:

那么:


But what if you can't remember that result? Then you would need to start from a quantity that you consider to be the definition of energy. Here, you can play Newton and construct your own mechanics. You can intuit that energy has to do with force, but it is not equal to force. You need to use energy to lift a heavy body. The heavier the body or the higher the lift, the more energy you need to use. So you could expect that the energy would be proportional to the weight of the body (i.e. the force) and the height over which you are lifting (i.e. the distance). So you could write something like: 

但是如果你不记得那个结果怎么办?然后你需要从一个你认为是能量定义的量开始。在这里,你可以扮演牛顿和建造你自己的机械。你可以想象能有力量,但它不等于力。你需要用能量来举起一个沉重的身体。车身越重,升力越高,你需要用到的能量就越多。所以你可以知道能量与身体的重量(即力)和你举起的高度成比例(即距离)。所以你可以写类似的东西:

energy=force×distance

Now, you might worry that not all the energy you put in is used for lifting the body. Maybe something is lost through friction, and you should add something. Or maybe only a fraction of the energy you put in is being used, so you need multiply by some number. So maybe you will write:

energy you put in=1.1×force×distance

现在,你可能会担心,不是所有你投入的能量都是用来提升身体的。也许有些东西是通过摩擦而失去的,你应该添加一些东西。或者,只有一小部分的能量,你投入使用,所以你需要乘以一些数字。也许你会写:

Or

或者

energy you put in=force×distance+frictionforce×some distance

But either way, the dimensions would be the same. Notice that if you think that some energy is lost through friction, you cannot simply add a factor corresponding to the friction forces. You need to multiply the force by some factor that has dimensions of distance, otherwise you would be adding apples and oranges. You will never be able to write: energy+force! You can only write: energy+force×something that has dimension of distance. 

但无论哪种方式,维度都是相同的。注意,如果你认为某些能量通过摩擦而失去,你不能简单地添加一个与摩擦力相对应的因素。你需要把力乘以某个距离大小的因子,否则你会增加苹果和橙子。你永远不会写:能量+力量!你可以只写:能+力×有时空的距离。

Because we only care about dimensions, let's simplify our notation:

因为我们只关心维度,让我们简化我们的符号:

dimension of energy=dimension of force×dimension of distance

Or, using the notation from the lessons:

或者,从课程中使用符号:

 [E]=[F]×[d]

But what are the dimensions of force? Newton's second law tells you that force is mass times acceleration. No force, no acceleration. No acceleration, no force. And of course, you need to include the mass term because you are always accelerating some matter. If you were to talk only about acceleration you would be describing only the motion of a particle (its kinematics), but we also want to talk about how that motion is being produced, so the cause of the acceleration (its dynamics). So we can write: 

但是力的维度是多少?牛顿的第二定律告诉你力是质量乘以加速度。没有力量,没有加速度。没有加速度,没有力。当然,你还需要包含大众术语,因为你总是在加速一些事情。如果你只谈论加速度,你只会描述一个粒子的运动(它的运动学),但是我们也要讨论这个运动是如何产生的,所以加速度的原因(它的动力学)。所以我们可以写:

F=m⋅a

But because we only care about the dimensions we have: 

但因为我们只关心我们拥有的维度:

[F]=[m]⋅[a]

Now what are the dimensions of acceleration? Acceleration is the change of a velocity with time, so it must be velocity over time: 

加速度的维度是多少?加速度是速度随时间的变化,所以它必须是随时间变化的速度:


Let's put the things together:

让我们把这些东西放在一起


So what we have showed is that if you defined energy to be a force times a distance, and the force to be acceleration times mass, then energy will have dimensions of mass times velocity squared. This doesn't mean that we have proved the formula for the kinetic energy or for Einstein's equation E=mc2. But it does mean that whenever we encounter a quantity formed from a mass times a velocity squared, that has dimensions of energy. And we shouldn't be too surprised if we can actually express that quantity as some form of energy.

我们所展示的是,如果把能量定义为一个力乘以一个距离,而力为加速度乘以质量,那么能量将具有质量乘以速度平方的尺寸。这并不意味着我们已经证明为动能或爱因斯坦的公式E = mc2公式。但它确实意味着每当我们遇到一个质量乘以质量平方的速度时,它就有能量的大小。如果我们能以某种能量来表达这个量,就不应该太惊讶了。

Dimensional analysis can't prove an equation, but it can help you disprove certain types of equations. By now you should be able to see why

量纲分析不能证明一个方程,但它可以帮助你反驳某些类型的方程。现在你应该知道为什么了。


or:

或者:


can never be acceptable equations. 

是永远不能被接受的方程。