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Dimensional Analysis


Some of the lectures will be accompanied byspecial lessons on Unit and Dimensional Analysis. While these lessons are notessential for following the video lectures, they will increase yourunderstanding of the physics and they will give you a different perspective onequations and formulas.


What makes an equation?


In school one often thinks about equationsas formulas. The aim of a formula is to shorten or condense information into anexpression so that you don't need to worry about the derivation. You want toknow the volume of a sphere, you look for the right formula (V=4πr3/3),you plug in the value for the radius, you plug in the value for π, and you aredone.

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,你找到正确的公式(V = 4πR3/ 3),你插入值为半径,你带入π的值,你就得出了结果。

The purpose of a formula is that you don'tneed 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.


Equations (as we will look at them in thiscourse) are different. Equations are something you want to contemplate and tounderstand how they were derived. Their fundamental role is not to shorten yourcomputations, but to express a relation, to equate two physical quantities. Theeasiest and most accessible way of contemplating an equation is to think aboutits dimensions.

Units and Dimensions


You are all familiar with physical units.You know that a length can be measured in meters, miles, inches, kilometersetc; a time can be measured in seconds, hours, minutes etc; or a velocity canbe measured in miles per hour, m/s, km/h etc.


Whenever a physical quantity is measured inunits of length we will say that its dimension is a length, and we will denotethat by L; whenever a quantity is measured in units of time, we will say itsdimension is time, and we will denote that by T.


The dimension of velocity can be expressedin terms of the fundamental dimensions of Length and Time. We know that, bydefinition, a velocity is a length divided by time. We will express that bysaying that the dimensions of velocity are L/T.


The purpose of dimensions is to do awaywith the clutter of numbers and units. Often, it is easy to forget that mph(miles per hour) and m are two fundamentally different units because they referto quantities which cannot be equated. A velocity will never be equal to alength; a volume cannot be equal to a length. The purpose of dimensions is tomake sure we are never comparing apples with oranges.


The advantage of dimensions over units isthat it really forces us to think about fundamental quantities. For example, isa light-year comparable to a meter, to a second or to a meter per second? Toanswer that question you must know that a light-year is a measure of distance.In this course we will say that a light-year has dimensions of distance.


To keep things, clear we will use squarebrackets to denote the dimensions of a physical quantity. For example, if thevelocity of a car is v=10 miles/hour, the dimensions of this quantity will be:

为了保持事物清晰,我们将使用方括号来表示物理量的维度。例如,如果汽车的速度是v = 10英里/小时,这个数量将是:

Notice that the numerical values do notaffect the dimensions of a quantity.


The Fundamental Rule of DimensionalAnalysis


Dimensions on the left = Dimensions on theright


As we said, the purpose of dimensionalanalysis is to make sure we are always comparing apples with apples and orangeswith oranges. Let’s say you want to write down the distance d covered by somecar that moves with constant velocity v in the time Δt. If you haven’t taken aphysics course in some time you might not remember the actual formula, and youcould scramble different answers:


Dimensional Analysis solves the problem foryou! Let’s compare the dimensions for each of these equations:


Plugging in the dimensions for thesequantities we get:


Dimensional Analysis tells us which equationis a possible candidate, and which equations can never be true.


The cool part about Dimensional Analysis isthat besides checking equations, you can also find equations (up to adimensionless constant, like 1/2 etc.). For example, let’s say we want todetermine all the possible equations we can find that relate a distance, anacceleration and time.


The most general formula we can write down(again, ignoring any extra numerical factor) is this:



Where m,p,q are some rational numbers. Ifwe re-write this relation in terms of dimension we get:


Matching the L and T terms imposes twoconditions m=p and q−2p=0. So the equation is now xp=apt2p.By taking a square root of p (the same way we can take a on both sides of anequation), we are left with x=at2. There is no other relation thatyou can write down that connects a distance, an acceleration and time.

匹配的LT条件规定了两个条件M = PQ−2P = 0。所以方程现在xp=apt2p。以P的平方根(我们可以取在一个等式两边),留给我们的是x = at2。没有任何其他关系,你可以写下来,连接距离,加速度和时间。