Physical Quantities & Units
A-Level Physics Topic 1 22:04 English narration · English + 中文 subtitles burned in
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In 1999, a spacecraft reached Mars after a journey of ten months.
1999年，一艘飞船经过十个月的旅程抵达火星。
It had cost over three hundred million dollars.
它耗资超过三亿美元。
But as it arrived, it was lost forever.
但就在它到达时，却永远失去了联系。
The cause was not a broken engine.
原因不是引擎故障。
One team had used metric units. Another team read them as imperial units. Nobody converted between the two.
一个团队使用了公制单位， 另一个团队却把数据当成英制单位来读，没有人做换算。
The numbers were wrong by a factor of about four, and the spacecraft was destroyed.
数值相差约四倍，飞船因此被摧毁。
This is why physics begins with quantities and units.
这正是物理学要从物理量和单位开始的原因。
Get them right, and everything else follows.
把它们弄对，其余的一切就顺理成章。
Today, we learn how to measure the world, how to handle the errors in every measurement, and how to work with directions.
今天，我们学习如何测量世界、如何处理每一次测量中的误差，以及如何处理方向。
Let's begin.
让我们开始吧。
Every physical quantity has two parts: a number, and a unit.
每一个物理量都有两个部分：一个数值和一个单位。
The number alone means nothing.
单看数值毫无意义。
If I say the length is one point five, you must ask: one point five what?
如果我说长度是一点五，你一定会问：一点五什么？
One point five metres is a complete physical quantity.
一点五米才是一个完整的物理量。
The number is its magnitude.
这个数值是它的大小。
The unit tells you what the number is counting.
单位告诉你这个数值在数什么。
You should also be able to estimate everyday quantities.
你还应该能够估算日常物理量。
An adult has a mass of about seventy kilograms.
一个成年人的质量约为七十千克。
Sound travels at about three hundred and forty metres per second.
声音在空气中的速度约为每秒三百四十米。
And near the ground, things fall with an acceleration of about ten metres per second squared.
在地面附近，物体下落的加速度约为每秒每秒十米。
A good estimate has the right order of magnitude — the right power of ten.
一个好的估算要有正确的数量级——正确的十的幂次。
Seventy kilograms is close. Seven kilograms is not.
七十千克很接近，七千克则不对。
One to carry with the others: atmospheric pressure is about one times ten to the fifth pascals.
还有一个要和其他估算值一起记住的：大气压强大约是一乘以十的五次方帕斯卡。
It turns up in pressure questions as the baseline everything else is measured against.
它在压强题里经常出现，是其他一切压强的基准。
Learn a few more rough values.
再记几个粗略值。
An adult's weight is about seven hundred newtons — that is mass times ten, near the ground.
成年人的重力约为七百牛顿——在地面附近就是质量乘以十。
An adult has a height of about one point seven metres.
成年人的身高约为一点七米。
An apple has a mass of about zero point one kilograms, so its weight is about one newton.
一个苹果的质量约为零点一千克，所以它的重力约为一牛顿。
And the speed of light in a vacuum is three times ten to the eight metres per second.
真空中的光速是三乘以十的八次方米每秒。
Keep these ready for estimation questions.
估算题时这些要随时能用。
All units are built from just a few. The system we use is the S-I system.
所有单位都由少数几个基本单位构成。
In this topic, five base quantities matter.
我们使用的是国际单位制。
Mass, in kilograms. Length, in metres. Time, in seconds.
在本专题中，有五个基本量很重要。
Current, in amperes. And temperature, in kelvin. Learn these five.
质量，用千克；长度，用米；时间，用秒； 电流，用安培；温度，用开尔文。
Every other unit in the course is made from them.
记住这五个，课程里其余每个单位都由它们构成。
A derived unit is built by multiplying or dividing base units.
导出单位是由基本单位相乘或相除得到的。
Take force.
以力为例。
Force equals mass times acceleration, so its base units are kilogram metre per second squared.
力等于质量乘以加速度， 所以它的基本单位是千克米每二次方秒。
That whole group we call the newton.
这一整组我们称为牛顿。
In the same way, the joule is the unit of energy, the watt is the unit of power, and the pascal is the unit of pressure.
同样地，焦耳是能量的单位，瓦特是功率的单位，帕斯卡是压强的单位。
Each name hides a small group of base units inside it.
每一个名称里都藏着一小组基本单位。
Build every derived unit from its defining equation.
从定义方程出发建立每一个导出单位。
Speed is distance over time, so the unit is metres per second.
速率是距离除以时间，所以单位是米每秒。
Acceleration is the change in velocity over time, so metres per second squared.
加速度是速度的变化除以时间，所以是米每二次方秒。
Work and energy equal force times distance, so kilogram metre squared per second squared — that is one joule.
功和能量等于力乘以距离， 所以是千克米二次方每二次方秒——也就是一焦耳。
Power is energy over time, so kilogram metre squared per second cubed — one watt.
功率是能量除以时间， 所以是千克米二次方每三次方秒——也就是一瓦特。
Pressure — and stress — are force over area, so kilogram per metre per second squared — one pascal.
压强是力除以面积， 所以是千克每米每二次方秒——也就是一帕斯卡。
When a question asks for base units, replace each named unit and simplify.
题目要基本单位时，把每个命名单位换掉再化简。
Units give us a powerful check.
单位给了我们一个强大的检验方法。
An equation must be homogeneous: both sides must have the same base units.
一个方程必须量纲一致：两边必须有相同的基本单位。
Take an equation of motion, v squared equals u squared plus two a s.
以一个运动学方程为例，速度的平方等于初速度的平方加上二倍加速度乘位移。
Write each term in base units.
把每一项写成基本单位。
The left side gives metres squared per second squared.
左边是二次方米每二次方秒。
Every term on the right gives the same.
右边每一项都相同。
The units match, so the equation could be right.
单位一致，所以方程可能是对的。
But be careful — matching units do not prove it.
但要小心——单位一致并不能证明它正确。
Yet if the units do not match, the equation is wrong for certain.
然而，如果单位不一致，方程就一定是错的。
Matching units never prove the whole equation is correct.
单位一致永远不能证明整个方程正确。
It could still have a wrong number, or a missing factor of two.
它仍可能有错误的数字，或少了一个二的因数。
The check only rules out impossible combinations of units.
这项检查只能排除不可能的单位组合。
If the units do not match, stop and fix the equation before you do any arithmetic.
如果单位不一致，先停下来改方程，再做任何计算。
Numbers in physics can be huge or tiny, so we use prefixes.
物理中的数值可以极大或极小，所以我们使用词头。
Each prefix is a jump by a power of ten.
每个词头都是十的幂次的跳跃。
Kilo means a thousand.
千代表一千。
Mega, a million.
兆，一百万。
Milli means one thousandth.
毫代表千分之一。
Micro, one millionth.
微，百万分之一。
They climb in steps of a thousand, from pico, far below one, up to tera, far above.
它们以一千为一档往上升，从远小于一的皮，一直到远大于一的太。
When you convert, replace each prefix with its power of ten.
换算时，把每个词头换成它对应的十的幂次。
And take care with squared units — if the unit is squared, you must square the factor too.
要小心平方单位——如果单位是平方，因数也要平方。
Here is the full ladder you must know.
这是你必须掌握的完整阶梯。
Tera is ten to the twelve.
太是十的十二次方。
Giga, ten to the nine.
吉，十的九次方。
Mega, ten to the six.
兆，十的六次方。
Kilo, ten to the three.
千，十的三次方。
Then below one: deci is one tenth, centi is one hundredth, milli is one thousandth.
小于一的：分是十分之一，厘是百分之一，毫是千分之一。
Micro is ten to the minus six.
微是十的负六次方。
Nano, ten to the minus nine.
纳，十的负九次方。
Pico, ten to the minus twelve.
皮，十的负十二次方。
Learn the symbols too: capital T, G, M for the large ones; k, d, c, m, and the Greek mu for micro, then n and p for the smallest.
符号也要记： 大写的太、吉、兆；小的是千、分、厘、毫，以及微的希腊字母缪，然后是纳和皮。
Worked example.
例题。
Change zero point two five kilo-newtons per square millimetre into newtons per square metre.
把零点二五千牛每平方毫米换成牛顿每平方米。
First, kilo is ten to the three, so one kilo-newton is one thousand newtons.
首先，千是十的三次方， 所以一千牛等于一千牛顿。
Next, milli is ten to the minus three, and the unit is squared, so square that factor: ten to the minus six square metres.
其次，毫是十的负三次方，单位是平方，所以要把因数平方： 得到十的负六次方平方米。
Divide one thousand by ten to the minus six, then multiply by zero point two five.
用一千除以十的负六次方，再乘以零点二五。
The result is two point five times ten to the eight newtons per square metre.
结果是二点五乘以十的八次方牛顿每平方米。
Every measurement has some uncertainty — we are never fully sure of the value.
每一次测量都有某种不确定度——我们永远无法完全确定数值。
A good experimenter knows where the uncertainty comes from.
好的实验者知道不确定度从哪里来。
Two common length tools are the vernier caliper and the micrometer screw gauge.
两种常用的长度工具是游标卡尺和螺旋测微器。
The caliper measures to about one tenth of a millimetre.
卡尺大约能测到零点一毫米。
The micrometer is finer: to about one hundredth of a millimetre.
螺旋测微器更精细：大约零点零一毫米。
On the micrometer, the anvil and spindle close onto the object, the sleeve carries the main scale, and the thimble carries the fine scale.
在螺旋测微器上，测砧和测微螺杆夹住物体， 固定套筒带主尺，微分筒带细标尺。
Reading a micrometer is two steps.
读螺旋测微器分两步。
First, read the main scale on the sleeve — whole millimetres and half-millimetres past the thimble edge.
先读固定套筒上的主尺——微分筒边缘经过的整毫米和半毫米。
Second, add the thimble scale reading where it lines up with the datum line.
再把与基准线对齐的微分筒读数加上。
In panel a, both scales read zero, so the zero reading is zero point zero zero millimetres.
图甲中两尺都是零，所以零点读数是零点零零毫米。
In panel b, the main scale shows five point five millimetres, and the thimble shows twenty-eight hundredths, giving five point seven eight millimetres.
图乙中主尺是五点五毫米，微分筒是二十八个百分之一毫米，合起来是五点七八毫米。
A vernier caliper works the same idea.
游标卡尺是同样的思路。
Read whole millimetres from the main scale at the vernier zero.
在游标零刻度处从主尺读整毫米。
Then find the one vernier line that lines up with a main-scale line — that gives the tenths of a millimetre.
然后找与主尺某一刻度对齐的那条游标线—— 那就是十分之几毫米。
Here the main scale shows fifteen millimetres, and vernier line four lines up, so the reading is fifteen point four millimetres.
这里主尺是十五毫米，游标第四条线对齐，所以读数是十五点四毫米。
Always check for a zero error before you trust either instrument.
信任任何仪器之前，都要先检查零点误差。
No measurement is perfect.
没有任何测量是完美的。
Errors come in two kinds.
误差有两类。
A systematic error shifts every reading the same way — like a scale that does not read zero when it should.
系统误差让每一次读数都朝同一方向偏移—— 就像一个在应当为零时却不指零的秤。
This is a zero error, and repeating the measurement will not remove it.
这叫零点误差，重复测量并不能消除它。
A random error scatters readings above and below the true value, with no pattern.
随机误差让读数在真实值上下无规律地散布。
Here, repeating helps: take the mean of many readings, and random errors partly cancel.
这时重复会有帮助： 取许多次读数的平均值，随机误差就会部分抵消。
One source you cannot read off a display: on a hand-timed measurement your own reaction time matters more than the instrument does.
有一种误差是你从仪表上读不出来的：手动计时的时候， 你自己的反应时间比仪器的影响更大。
Allow about zero point one to zero point two seconds for it, and time many oscillations instead of one, so that the same uncertainty is shared between all of them.
给它留出大约零点一到零点二秒， 而且要计很多次振动而不是一次，这样同一个不确定度就被分摊到所有振动上。
Look at this ammeter.
看这只电流表。
No current is flowing, so the needle should rest on zero.
没有电流通过，指针应当停在零。
Instead it sits just below zero.
但它却停在零的下方一点点。
That gap is a zero error.
这个间隙就是零点误差。
Every later reading will be shifted by the same amount.
之后每一次读数都会偏同样的量。
You cannot fix it by taking more readings.
多测几次不能消除它。
You either adjust the zero, or you subtract the error from each reading.
你要么调零，要么从每次读数中减去这个误差。
A calibration error is similar: the scale itself is wrong, so every value is systematically off.
校准误差类似：刻度本身就不对， 所以每个值都系统性地偏离。
Another systematic cause is parallax.
另一个系统原因是视差。
A pointer sits above a scale.
指针在标尺上方。
If your eye is straight above, you read the true mark.
如果你的眼睛正上方对准，你读到真实刻度。
If your eye is always too far left, you always read too low.
如果眼睛总是偏左，你总是读得偏低。
If it is always too far right, you always read too high.
如果总是偏右，你总是读得偏高。
Because the bias has a direction, it is systematic — it hurts accuracy, not precision.
因为偏差有固定方向，它是系统性的——损害准确度，而不损害精密度。
Random error, by contrast, comes from careful-or-not scale reading, changing conditions, and the instrument's smallest step.
相比之下，随机误差来自读尺是否仔细、条件是否变化，以及仪器的最小分度。
Two words are often confused: precision and accuracy.
有两个词常被混淆：精密度和准确度。
Think of throwing darts.
想象扔飞镖。
Accuracy means your darts land near the target — the true value.
准确度是指你的飞镖落在靶心附近——也就是真实值。
Precision means your darts land close together, wherever they land.
精密度是指你的飞镖彼此靠得很近， 不管落在哪里。
You can be precise but not accurate: tightly grouped, but off to one side — a systematic error.
你可以精密但不准确：扎得很紧，却偏向一边——这是系统误差。
And you can be accurate but not precise: spread out, yet centred on the truth.
你也可以准确但不精密：散得很开，但中心落在真实值上。
The best measurements are both.
最好的测量两者兼备。
On a graph the two look different, and that difference is examinable.
在图上这两种误差看起来不一样，而这个区别是会考的。
A systematic error moves every point by the same amount, so the line of best fit keeps its gradient but no longer passes through the origin — the intercept is shifted.
系统误差把每个点都平移同样的量， 所以最佳拟合直线的斜率不变，但不再过原点——截距被移动了。
A random error scatters the points either side of the line without moving the line itself.
随机误差让点散布在直线两侧，而直线本身不动。
Here is the same idea as distribution curves.
这里用分布曲线表示同样的意思。
Both curves peak on the true value T, so both sets can be accurate.
两条曲线的峰值都在真实值T上，所以两组都可以准确。
The top curve is narrow and tall — the readings cluster tightly, so they are precise.
上面的曲线又窄又高——读数聚得很紧，所以精密。
The bottom curve is wide and low — the readings spread out, so they are imprecise.
下面的曲线又宽又低——读数散开，所以不精密。
Precision is about how narrow the distribution is.
精密度看分布有多窄。
When a table of repeats is given, look at the spread for precision, and the mean for accuracy, separately.
题目给了重复读数表时，分别看离散程度（精密度）和平均值（准确度）。
Now the peaks sit to the right of T.
现在峰值都在T的右侧。
The top curve is still narrow — precise but not accurate, the classic systematic offset.
上面的曲线仍然很窄——精密但不准确，这是典型的系统偏移。
The bottom curve is wide and also offset — neither precise nor accurate.
下面的曲线又宽又偏移——既不精密也不准确。
Systematic error pulls the whole peak away from the true value.
系统误差把整个峰值从真实值拉开。
Random error only widens the spread.
随机误差只是加宽散布。
Enough repeats can still leave the mean accurate if there is no systematic shift.
如果没有系统偏移，足够多次重复仍可能让平均值准确。
A measurement is often written as x plus or minus delta x.
一次测量常写成x加减德尔塔x。
Delta x is the absolute uncertainty — the raw plus-or-minus amount, in the same unit as x.
德尔塔x是绝对不确定度——原始的加减量，单位与x相同。
The percentage uncertainty is delta x over the size of x, times one hundred percent.
百分比不确定度是德尔塔x除以x的大小，再乘以百分之一百。
Absolute uncertainty tells you the range.
绝对不确定度告诉你范围。
Percentage uncertainty is better when you multiply or divide, because those rules work with percentages.
相乘或相除时百分比不确定度更方便，因为那些规则用的是百分比。
Where the numbers come from: a single reading on an analogue scale is half the smallest division — a metre rule reads to one millimetre, so one reading is plus or minus zero point five millimetres.
这些数字是怎么来的：模拟刻度上的一次读数，不确定度是最小分度的一半—— 米尺的分度是一毫米，所以一次读数是正负零点五毫米。
A length needs two readings, one at each end, so its uncertainty is one millimetre, not half.
量一个长度需要读两次，两端各一次， 所以它的不确定度是一毫米，不是零点五。
We write a measurement as a value, plus or minus an uncertainty.
我们把一次测量写成一个数值，加或减一个不确定度。
To combine them, follow simple rules.
要把它们合并，遵循简单的规则。
When you add or subtract quantities, add their absolute uncertainties.
当你把物理量相加或相减时，把它们的绝对不确定度相加。
When you multiply or divide, add their percentage uncertainties.
当你相乘或相除时， 把它们的百分比不确定度相加。
And for a power, multiply the percentage by that power.
而对于幂，则把百分比乘以那个幂次。
Here is an example.
举个例子。
A ball's diameter is measured to about zero point four percent.
一个球的直径测得的不确定度约为百分之零点四。
Its volume depends on the diameter cubed.
它的体积取决于直径的立方。
So the uncertainty in the volume is three times that — about one percent.
所以体积的不确定度是它的三倍——约百分之一。
Practical questions often ask whether two calculated values of a constant agree well enough to support a suggested relationship, and "they are close" earns nothing.
实验题经常问：算出来的两个常数值是否足够接近，能支持题目提出的关系， 而「它们很接近」一分都拿不到。
Work out the percentage difference between them and compare that with the percentage uncertainty in the experiment: if the difference is smaller than the uncertainty, the values agree, and you say so with the numbers.
要算出两者之间的百分比差异，再和实验的百分比不确定度作比较： 如果差异小于不确定度，两个值就是一致的，而且你要用数字把这话说出来。
That comparison is how you justify the conclusion rather than assert it.
这个比较，就是把结论论证出来而不是断言出来的方法。
Work the full example.
把完整例题做完。
A ball's diameter is five point two six plus or minus zero point zero two centimetres.
球的直径是五点二六加减零点零二厘米。
The percentage uncertainty in d is zero point zero two over five point two six, times one hundred — about zero point three eight percent.
d的百分比不确定度是 零点零二除以五点二六再乘一百——约百分之零点三八。
Volume goes as d cubed, so the percentage in V is three times this — about one point one four percent.
体积与d的立方成正比， 所以V的百分比是它的三倍——约百分之一点一四。
The volume is about seventy six point two cubic centimetres.
体积约七十六点二立方厘米。
Absolute uncertainty is zero point zero one one four times seventy six point two — about zero point nine.
绝对不确定度是零点零一一四乘以七十六点二——约零点九。
So V is seventy six point two plus or minus zero point nine cubic centimetres.
所以V是七十六点二加减零点九立方厘米。
When you write a calculated quantity, give it the same number of significant figures as the least precise measurement you used — usually two or three in this syllabus.
写出计算得到的量时，有效数字位数要和你用过的最不精密的测量相同——本大纲通常是两位或三位。
Too many significant figures make the answer look more exact than it really is.
有效数字太多，会让答案看起来比实际更精确。
Too few lose useful information.
太少则会丢掉有用信息。
Match the precision of your data, not the digits your calculator shows.
要匹配数据的精密度，而不是计算器显示的位数。
Some quantities need a direction; some do not.
有些物理量需要方向，有些不需要。
A scalar has size only.
标量只有大小。
Mass, time, temperature, energy, and speed are all scalars.
质量、时间、温度、能量和速率都是标量。
A vector has both size and direction.
矢量既有大小又有方向。
Displacement, velocity, acceleration, force, and momentum are all vectors.
位移、速度、加速度、力和动量都是矢量。 这里有一个快速判断法。
Here is a quick test. If it makes sense to ask, in which direction, then the quantity is a vector.
如果问"朝哪个方向"是有意义的，那么这个量就是矢量。
You cannot ask the direction of a temperature. But you can ask the direction of a velocity.
你不能问温度的方向，但你可以问速度的方向。
The syllabus list is longer.
大纲上的名单更长。
Scalars also include work, power, distance, pressure, density and electric charge.
标量还包括功、功率、距离、压强、密度和电荷。
Vectors also include weight — weight is a force, so it has a direction, straight down.
矢量还包括重力——重力是力，所以有方向，竖直向下。
Density and electric charge have size only, so they stay scalars.
密度和电荷只有大小，所以仍是标量。
Keep the lists separate: speed is a scalar; velocity is the vector version with direction.
把名单分开记：速率是标量；速度是带方向的矢量版本。
Draw a vector as an arrow.
把矢量画成箭头。
The direction of the arrow is the direction of the quantity.
箭头的方向就是量的方向。
The length of the arrow, drawn to scale, is the magnitude.
按比例画出的箭头长度就是大小。
Here arrow a points east and is three units long for fifteen metres per second.
这里箭头甲指向东，三格代表每秒十五米。
Arrow b points south and is two units long for ten metres per second.
箭头乙指向南，两格代表每秒十米。
Always state the scale — for example, one unit equals five metres per second — so the diagram can be measured.
一定要标明比例——例如一格等于每秒五米——这样图才能被量出来。
To add two vectors, draw them tip to tail.
要把两个矢量相加，就把它们首尾相接地画出来。
The resultant runs from the very start to the very end.
合矢量从最开始一直连到最末端。
When the two vectors are at right angles, the resultant is easy.
当两个矢量互相垂直时，合矢量很容易求。
Its size is the square root of the sum of the squares — just like the sides of a right triangle.
它的大小是平方和的平方根—— 就像直角三角形的两条边一样。
For example, a swimmer heads north at one point two metres per second, across a river flowing east at zero point five.
举个例子：一个游泳者以每秒一点二米向北游， 横渡一条以每秒零点五米向东流的河。
The resultant is about one point three metres per second, aimed a little east of north.
合速度约为每秒一点三米，方向偏向北略微朝东。
For parallel vectors the same tip-to-tail rule works.
对平行矢量，同样的首尾相接规则也成立。
Row a: twenty newtons up plus thirty newtons up stacks to a fifty-newton resultant up.
甲行：向上二十牛顿加向上三十牛顿，叠成向上五十牛顿的合矢量。
Row b shows subtraction.
乙行是减法。
To find X minus Y, add the reverse of Y — same size, opposite direction.
求X减Y，就加上Y的反向——大小相同，方向相反。
Twenty newtons up minus thirty newtons up is drawn as twenty up tip to tail with thirty down, giving ten newtons down.
向上二十牛顿减去向上三十牛顿，画成向上二十与向下三十首尾相接，得到向下十牛顿。
Coplanar vectors live in the same flat plane; tip to tail still finds the resultant.
共面矢量在同一平面内；首尾相接仍然求出合矢量。
Back to the swimmer as a vector triangle.
再看游泳者的矢量三角形。
One point two north and zero point five east, tip to tail.
向北一点二与向东零点五首尾相接。
The resultant size is the square root of one point two squared plus zero point five squared, which is the square root of one point six nine — exactly one point three metres per second.
合矢量大小是一点二的平方 加零点五的平方再开方，也就是一点六九的平方根——正好是每秒一点三米。
The angle east of north comes from tan theta equals zero point five over one point two, about twenty-three degrees.
偏东的角度来自tanθ等于零点五除以一点二，约二十三度。
Size and direction together fully describe the vector.
大小和方向一起完整描述这个矢量。
One more pattern.
还有一种模式。
If two vectors have the same size F, with an angle of two alpha between them, the resultant has size two F cos alpha.
如果两个矢量大小同为F，夹角为二倍阿尔法，合矢量大小就是二F乘以cos阿尔法。
It lies along the line that cuts the angle in half — the angle bisector.
它沿平分夹角的那条线——角平分线。
This is handy when two equal forces or two equal velocities meet at a known angle.
当两个等大的力或速度以已知夹角相遇时，这个公式很方便。
The reverse is just as useful: any vector can be split into two perpendicular parts.
反过来同样有用：任何矢量都能分解成两个互相垂直的部分。
Watch this force break into a horizontal part and a vertical part.
看这个力分解成一个水平部分和一个竖直部分。
The horizontal part is the size times the cosine of the angle.
水平部分等于大小乘以角度的余弦。
The vertical part is the size times the sine of the angle.
竖直部分等于大小乘以角度的正弦。
Splitting a vector this way lets you handle each direction on its own.
这样分解一个矢量，你就能分别处理每一个方向。
On the diagram, force F sits at angle theta to the horizontal.
图中力F与水平方向成角θ。
The horizontal component is F cos theta.
水平分量是F cosθ。
The vertical component is F sin theta.
竖直分量是F sinθ。
Together with F they form a right-angled triangle.
它们与F一起构成直角三角形。
Choose the directions that make the problem easiest — horizontal and vertical, or along and across a surface.
选择让问题最容易的方向——水平与竖直，或沿表面与垂直于表面。
You resolve whenever you need how much of a vector acts in one direction, for example the parts of a thrown ball's velocity.
每当你需要知道矢量在某一方向上有多少时就分解，例如抛出后球速度的各个分量。
On a slope — an inclined plane — split the weight into one part along the slope and one part at right angles to it.
在斜面上——倾斜平面——把重力分解成沿斜面的一部分和垂直于斜面的一部分。
If theta is the angle of the slope to the horizontal, the parallel part is W sin theta, and the perpendicular part is W cos theta.
如果θ是斜面与水平面的夹角，平行分量是W sinθ，垂直分量是W cosθ。
The parallel part tries to slide the object down.
平行分量试图让物体滑下。
The perpendicular part presses into the surface.
垂直分量压向表面。
Choosing these axes makes the free-body diagram clean.
选这两个轴会让受力图更清晰。
Before you go, the marks students lose most.
结束之前，说说学生最常失分的地方。
First, give every answer a unit, and check that both sides of your equation have the same base units.
第一，给每个答案都写上单位， 并检查方程两边是否有相同的基本单位。
Second, keep random and systematic errors clear in your mind — only random ones shrink when you repeat.
第二，把随机误差和系统误差分清楚—— 只有随机误差会在重复测量时减小。
Third, when combining uncertainties, add the absolute ones for adding, and the percentage ones for multiplying.
第三，合并不确定度时，相加就把绝对不确定度相加， 相乘就把百分比不确定度相加。
Master these, and this topic is yours.
掌握这些，这个专题就是你的了。
The fixed-wording definitions, one answer only.
固定措辞的定义，只给一个答案。
A physical quantity: a numerical magnitude together with a unit.
物理量：一个数值大小连同一个单位。
A homogeneous equation: one in which every term has the same base units.
量纲齐次的方程：每一项的基本单位都相同的方程。
A systematic error shifts every reading in the same direction by the same amount and is NOT reduced by repeating; a random error scatters readings above and below the true value and IS reduced by repeating and averaging.
系统误差让每次读数朝同一方向偏移同样的量，重复测量不能减小它； 随机误差让读数散布在真值上下，重复并取平均可以减小它。
Zero error: the reading an instrument shows when the true value is zero.
零误差：真值为零时仪器显示的读数。
Precision: how close repeated readings are to each other.
精密度：重复读数彼此之间有多接近。
Accuracy: how close a reading, or the mean of several, is to the true value.
准确度：一次读数，或者若干次读数的平均，与真值有多接近。
Absolute uncertainty: the range in which the true value is expected to lie, in the units of the quantity.
绝对不确定度：真值预期所在的范围，用该量的单位表示。
Percentage uncertainty: the absolute uncertainty divided by the value, times one hundred.
百分比不确定度：绝对不确定度除以数值，再乘以一百。
A scalar has magnitude only; a vector has magnitude and direction.
标量只有大小；矢量既有大小又有方向。
The traps.
陷阱。
Convert the prefix before substituting — five centimetres is nought point nought five metres.
代入之前先换算前缀——五厘米是零点零五米。
Squaring a quantity DOUBLES its percentage uncertainty and a square root halves it.
一个量取平方，它的百分比不确定度加倍；取平方根则减半。
Percentage and absolute uncertainty are not the same number.
百分比不确定度和绝对不确定度不是同一个数。
Do not quote the smallest division as the uncertainty of a reading that was hard to take.
一次很难读准的读数，不要拿最小分度当作它的不确定度。
And never offer two answers to a definition question — commit to one.
另外定义题绝不要给两个答案——只给一个。
Two more exam habits.
还有两个考试习惯。
Distinguish precision from accuracy: small spread versus close to the true value.
把精密度和准确度分开：小的离散与靠近真实值。
Resolve a vector into perpendicular components — F cos theta and F sin theta — or add vectors tip to tail or by components.
把矢量分解成互相垂直的分量——F cosθ和F sinθ——或者首尾相接或用分量相加。
Write the unit on every final line.
每一行最终答案都写单位。
Check homogeneity before you trust the algebra.
信任代数之前先检查量纲一致。
These habits turn this topic into reliable marks.
这些习惯把本专题变成稳拿的分。
On a graph with uncertainties, plot each point with an error bar whose length shows the absolute uncertainty in that value.
在带不确定度的图上，每个点都要画误差棒，长度表示那个值的绝对不确定度。
Draw the line of best fit through the points, then the worst acceptable line: the steepest, or shallowest, straight line that still passes through every error bar.
先画穿过这些点的最佳拟合直线，再画最差可接受直线： 就是仍然能穿过每一根误差棒的、最陡或者最平的那条直线。
The difference between the two gradients is the uncertainty in your gradient, and that is where the marks are.
两条线斜率之差就是你所求斜率的不确定度，分数就在这里。