Graphs, errors, significant figures, dimensions and units
Errors, error bars and significant figures (误差，误差棒和有效数字)
We might be tempted to write t = 1.32 s, because the watch is capable of measurements with this precision. You will probably find, however, that you are not. Repeated measurements of the same period (eg the time it takes a second hand to pass 2 seconds on a clock) will not usually give the same answer, and the variation is probably about 0.1 s. So the measurement error here is (very roughly) about 0.1 s.
Now, if you write t = 1.32 s, you are implying that the '2' means something, that it is significant. You are implying a precision that you don't actually have. So, to avoid misleading the reader, you should retain only the two significant figures. Consequently, this should be written (as we say) to two significant figures, t = 1.3 s. To make the error explicit, we might write t = 1.3 ± 0.1 s. (If we had done a detailed study of our timing reproducibility and found its standard error to be 0.15 s, we could write t = 1.32 ± 0.15 s.)
How can we show the point (t,x) = (1.3 ± 0.1 s, 4.1 ± 0.2 m) on a graph? We do this by drawing a cross, whose vertical axis goes from 3.9 to 4.3 m and whose horizontal axis goes from 1.2 to 1.4 s. This is shown in red on the graph at right.
The above is only a quick sketch. For more details, see the notes Errors and handling them.
A good plotting program (好的绘图程序)
Michael Johnston, one of our past undergraduate physics students, was frustrated with the inability of such software to draw scientific graphs and to fit simple functions appropriately. So he wrote an application that does this. You'll note that it draws the error bars that you enter. If you ask to fit a simple function to the plot, it does a least-squares error-weighted fit: in other words, it gives more weight to the points with small errors than to those with large errors. Here is Michael's curve-fitting program., which we use to analyse an experiment in Projectiles and which we also used to make the graph at right.
Units and dimensions (单位和量度)
What is wrong with saying
- "My height is 45°C"? Or: 我的高度是４５摄氏度 或者
- "The game lasted 15 kg"? Or: 这场游戏持续了１５千克 或者
- "The displacement is given by the weight divided by the volume squared"? 位移定义为重量每平方体积
In each case, we mention two physical quantities that cannot possibly be equal. Let's take the first: a height cannot equal a temperature, so I cannot measure my height in degrees Celsius. Let's call my height h. I can say "h = 1.8 m". I can also say that h = V/A", where V is my volume and A is my average horizontal cross sectional area. In both cases, the quantity on the right hand side is a distance, as is the quantity on the left hand side. Both can be measured in metres.
在每一个情况下, 我们都提到了两个不可能对等的物理定量. 我们先来说一下这个: 高度不可能对等于温度, 所以我不可能测量我的身高用摄氏度. 假设我的身高是h. 我可以说h=1.8米. 我也可以说h=V/A, 这里V是我的体积，A是我的平均横截面. 在这两个情况下, 等式右边的定量始终是距离, 等同与等式左边的定量. 他们都可以用米来测量.
We shall see that this very simple idea can be quite important -- and also very useful.
Now the units on either side of an equation need not be the same. For instance, I may write
现在，等式两边的单位不一定相同. 例如, 我可以写.
1 inch = 25.4 mm.
This equation is true. (In fact, it is the definition of the inch, a unit of length in the old British system of units.) However, this equation is different from the silly examples given above, because both the millimetre and the inch measure length. We say that both sides of the equation have the dimensions of length. This condition must be satisfied for an equation to be true, or even to make sense. Further, if equations have different units with the same dimension, appropriate conversion factors must be included, as is the case above.
这个公式是正确的.(事实上, 这个是英寸的定义, 这个长度单位是英国的旧系统单位.) 可是, 这个方程是不同于我们上文所给的错误方程的, 因为毫米和英寸都可以衡量长度. 我们说等式的两边都有长度的量度. 一个正确的有意义的方程必须满足这个条件. 此外, 如果等式中含有同一量度不同的单位, 适当的转换因子应当被提供, 就像上诉情况一样.
Let's look at more interesting examples. When we write
F = ma ,
we are specifying that the dimensions of force are those of mass times acceleration. The dimensions of acceleration are length, which we write as [L], divided by time [T] squared, so we write, just for the dimensions:
我们声明力的量度是重量乘以加速度. 加速度的量度是长度（我们写作L）,除以时间（T）,的平方,所以我们写作, 仅仅是为了量度:
[F] = [M][L][T]−2.
In the units of the Système International, universally used in science, there are no conversion factors for the base units, so we can relate the newton, the unit of force, to other base units:
在国际系统单位里, 这个标准通常会在科学中使用, 基础的单位没有转换因子, 所以我们可以将牛顿表, 力的单位, 表现成其他基础单位:
1 N = 1 kg.m.s−2.
And to use our equation once more, we note that, while mass is a scalar, force and acceleration are both vectors, so our previous equation tells us not only that F = ma, but that F is parallel to a.
再用一次我们的方程, 我们发现, 重量是一个标量, 力和加速度是矢量, 所以我们前面的方程不仅表明了F = ma, 而且指明了F平行于a。
Let's now see how the method of dimensions can be useful, via this
Example: how does the frequency, of a pendulum depend on the length? (例子：一个钟摆的频率怎样取决于长度)
We know that this depends on the length, L -- a long one swings more slowly than does a short one. It also depends on the strength g of the gravitational field -- it won't swing at all without one. Does it also depend on the mass, m? On the temperature, T? Let's write for the frequency, f:
我们知道这个取决于长度, L -- 一个长的钟摆摆动比一个短的钟摆要慢. 它也取决于重力场强度g-- 没有这个它根本不会摆动. 它也取决于重量吗, m? 温度, T? 让我们写下频率, f:
f = N.La.gb.mc.Td
where N, a, b, c and d are numbers, yet to be determined. Of course, we can analyse the dynamics of the pendulum and determine them, but let's see how far we get just by considering the dimensions. Frequency has units of "per second" so it has the dimensions of reciprocal time, T−1 . So, setting the dimensions equal on both sides, we have:
这里N,a,b,c 和d是数字, 数值还没被决定. 当然, 我们能够通过分析钟摆的运动来推到出这些数字, 但是，让我们看看从量度的角度上我们可以走多远. 频率有一个单位‘每秒’所以它与时间有相反的量度, T−1 . 所以，通过调整两边的量度，我们得到了:
T−1 = N.La.(L.T−2)b.Mc.Temperatured
For this equation to be true, each dimension must occur, to the same power, on each side. So, considering each dimension, the exponent gives us an equation to be satisfied. If we start with time T, we see that it appears to the −1 power on the left, and to the −2b power on the right, so we have
为了让这个等式成立, 每一个量度在等式两边的作用要相同. 所以考虑到每个维度, 指数必须忙足方程. 我们从时间T开始, 我们看到在左边他的指数是-1, 在右边指数是-2b, 所以我们有
[T] ⇒ −1 = −2b
在这里箭头代表着’指代’. 对于其他的量度, 我们有
[L] ⇒ 0 = a + b
[M] ⇒ 0 = c
[Temperature] ⇒ 0 = d.
Now the last two shouldn't surprise us. A more massive pendulum experiences a greater gravitational force but it also requires more force to accelerate, so we should not be surprised that the dimensions of the problem tell us that c = 0, ie that these effects cancel out: the frequency does not depend* on the mass of the pendulum. Similarly, we see that d = 0, but we should not be too surprised that a hot pendulum and a cold one swing at the same frequency -- unless of course the temperature changes the length perceptibly.
现在最后两个等式不应该让我们奇怪. 一个更重的钟摆具有更大的重力但是它也需要更大的力去加速, 所以我们不应该惊奇于量度告诉我们c=0, 例如这些效果会相互抵消：钟摆的频率不取决于钟摆的质量. 于此相似, 我们看到d=0, 但是我们不应该太惊奇与一个热的钟摆与一个冷的钟摆有相同的摆动频率 -- 当然出了温度改变钟摆长度的时候.
The other two equations tell us that b = 1/2, and that a + b = 0, so a = −a = −1/2. So, substituting in our original equation for the frequency,
另外两个方程告诉我们b = 1/2, 并且a + b = 0, 所以 a = −a = −1/2. 所以, 带入我们原来的频率方程,
f = N.L−1/2.g1/2 = N(g/L)1/2.
We still don't know the value of the number N, and cannot get it from the information we have been given here. (It is 1/(2*π), in case you were wondering.) However, we do know that, all else equal, the frequency is proportional to the reciprocal of the square root of the length. To halve the frequency of pendulum, make it four times as long.
我们还是不知道N的数值, 并且不能从它已经给的信息中求得. 如果你好奇的话，我可以告诉你它是1/(2* π). 可是, 我们确实知道, 其他都是相等的, 频率是正比于长度平方根的倒数的. 要把钟摆的频率分一半, 就把他曾长四倍.
* I raise a couple of tiny caveats, to preempt the pedants. For a pendulum whose mass is comparable with the that of the planet upon which it is mounted, the pendulum mass does appear -- or at least the ratio of these two masses appears. Further, we have cheated a little on the temperature, because we could write temperature in units of energy. Doing so, the conversion factor would be Boltzmann's constant, whose very small size would give us the clue that temperature is only relevant in mechanics for objects of molecular size. And on this molecular scale we should often need to use quantum mechanics rather than Newtonian mechanics.
* 这里有几条附加说明, 用来应答一些学究. 对于一个重量可比拟于它所在的行星的重量的钟摆来说, 钟摆的重量M会出现在方程里-- 或者至少两个质量的比例会出现. 此外, 我们在温度方面的表示不完全正确, 因为我们可以把温度表现成能量). Doing so(这样做的话, 转换因子就是玻尔滋蔓常数, 这个非常小的常数向我们暗示着，仅仅当物体接近分子尺寸的时候温度才在机理中起作用. 在分子量级下，我们常常需要用量子力学而不是牛顿力学.
Other units (其他单位)
With rare exceptions, scientists use the SI system of units. (SI stands for Système International d'Unités.) This system is based on the kilogram for mass, the metre for length, the second for time, the ampere for electric current, the kelvin for temperature, the mole for chemical quantities and the candela for luminous intensity. Other systems are the British imperial system and natural units.
出了在个别情况下, 科学家都会使用SI单位系统(SI代表着国际标准单位).) 这个系统是基于千克（质量）, 米（长度), 秒（时间）, 安培（电流）, 开尔文（温度）, 摩尔（化学数量） 坎德拉（照明强度）. 其他的系统有英国度量系统和自然单位.
Physclips is a scientific presentation, and we use only the SI. If you encounter problems stated in other units, the simplest procedure is often to translate the problem into SI, solve it, then translate the back. This sounds like extra work, but it is usually much less than the extra work required in using the imperial system of units, which has internal conversion factors.
Physclips是一个科学介绍, 我们仅仅使用SI国际单位. 如果你在陈述其他单位的时候遇到了问题, 那么最简单的方法常常是把这个问题变成国际单位, 解决它, 然后再把单位变回来. 这听起来像是额外的工作, 但是这通常是比继续使用英制单位需要更少的额外工作, 英制单位具有内部转换因子.
In the United States of America, Liberia and Myanmar, the British imperial system is the official system. This system used to be much more widespread, and vestiges of it remain in other countries that are in the process of 'going metric', ie converting to the SI.
在美国, 利比亚和缅甸, 英制系统是官方系统. 这个系统过去有着很广泛的应用, 它在那些依然经过公制化的国际尚有余温, 也就是说转向国际单位.
Dealing with or converting from the imperial system usually involves just a multiplicative factor. For instance, the inch, an imperial unit of length, is officially defined to be equal to 25.4 mm. These multiplications can become awkward in some cases: consider this imperial unit of thermal conductivity, one British Thermal Unit per second per square foot per degree Fahrenheit per inch. One can see why it exists, but it is ugly and inconvenient. (For comparison, the SI unit thermal conductivity is W.m-1.K-1.)
处理或者转换英制系他通常需要的仅仅是一个乘数因子. 例如, 英寸, 一个英制长度单位, 官方定义是25.4毫米. 这些乘数因子在某些时候是相当麻烦的: 设想一下热传导率的英制单位, 一英制热每秒每平方英尺每华氏度每英寸. 它有存在的理由, 但是它的确很丑也不方便. 相比而言，国际单位热传导率是W.m−1K−1.
Some confusion arises, however, because of the different colloquial use of units of mass and force in the SI and imperial system. In the imperial system, the unit of force is the pound-force, or sometimes, as in many American physics text books, as just the pound. The unit of mass in the imperial system is the slug, which is a mass that is accelerated by one pound at one foot per second per second. The slug is 14.5939 kg. These equations, which are definitions, allow us to compare the units of mass and force:
可是，一些混淆出现了，因为对于质量单位和力量单位不同的习惯定义. 在英制系统中, 力的单位是磅力, 或者有些时候, 就像有些美国物理书中的, 仅仅是磅. 质量的英制单位是斯（斯勒格）, 这个单位是单位质量即被一磅力加速一英尺每平方秒). 斯是14.5939千克. 这些方程,它们是定义, 准许我们去比较质量和力的单位:
Unit of force(力的单位) = 1 newton = 1 kg.m.s−2
Unit of force(力的单位) = 1 pound = 1 slug.foot.second−2
In imperial units, the gravitational acceleration is 32 feet.second−2. Consequently, a slug weighs 32 pounds.
在英制单位里，重力加速度是32英尺每平方秒. 因此, 一斯是32磅.
The slug is very rarely used. Pound is used colloquially as a unit of quantity -- a pound of apples colloquially means a quantity of apples that weighs a pound-force (at the earth's surface). There is another imperial unit of force, the poundal. This is defined as the force required to accelerate at one foot.second−2 a mass whose weight is one pound. So a pound is 32 poundals.
斯很少被使用. 磅一般被用作数量单位——一磅苹果通常代表着一定量的苹果重力是一磅力（在地球表面上). 这里还有另一个英制力学单位, 磅达. 它被定义成一磅重的质量被加速到一英尺每平方秒. 所以一磅是32磅达.
The units mentioned above are related to features of the earth (its circumference originally determined the metre, and the second is related to the day) or of artifacts on earth, such as the standard kilogram, or of particular substances, especially water. The laws of physics and combinations of them yield natural units, which are used by some theoretical physicists, especially cosmologists. The speed of light, for instance, is taken as the unit for speed. Although this makes equations look simple these units are, in general, inconvenient for measurement. For instance, the natural units of length and time are inconveniently small (The Planck length is 1.6 x 10-35 metres, the Planck time is 5.4 x 10-44 seconds). See The Planck scale for more detail.
以上提到的单位是和地球的特征相结合的，地球的圆周根本上决定了米，而秒是和天有关或者地球上的人为定义, 例如标准的千米, 或者特定的物质, 特别是水. 将物理定律和它们结合起来就产生了自然单位, 这个单位会被一些理论物理学家使用, 特别是宇宙学家. 例如光速会被用为速度单位. 虽然这样会让公式变得简单，但是也会让测量变得不方便. 例如, 自然单位的长度和时间非常小，不方便. 普朗克长度是1.6 x 10-35 米, 普朗克时间是5.4 x 10-44 秒. 详见普朗克计量.