Errors and uncertainties · 误差与不确定度
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| uncertainty/ʌnˈsɜːtənti/ | 不确定度 | bùquèdìngdù |
| micrometer/maɪˈkrɒmɪtə/ | 螺旋测微器 | luó xuán cè wēi qì |
| vernier calipers/ˈvɜːnɪə ˈkælɪpəz/ | 游标卡尺 | yóu biāo kǎ chǐ |
| systematic error/ˌsɪstəˈmætɪk ˈerə/ | 系统误差 | xìtǒng wùchā |
| zero error/ˈzɪərəʊ ˈerə/ | 零点误差 | língdiǎn wùchā |
| calibration/ˌkælɪˈbreɪʃn/ | 校准 | jiàozhǔn |
| parallax/ˈpærəlæks/ | 视差 | shìchā |
| random error/ˈrændəm ˈerə/ | 随机误差 | suíjī wùchā |
| precision/prɪˈsɪʒn/ | 精密度 | jīngmìdù |
| accuracy/ˈækjʊrəsi/ | 准确度 | zhǔnquèdù |
| absolute uncertainty/ˈæbsəluːt ʌnˈsɜːtənti/ | 绝对不确定度 | juéduì bùquèdìngdù |
| significant figures/sɪɡˈnɪfɪkənt ˈfɪɡəz/ | 有效数字 | yǒuxiào shùzì |
How sure are you?
- Measure a wire twice and you might get $5.77\ \text{mm}$, then $5.79\ \text{mm}$.
- Neither is "wrong" — every measurement has some uncertainty 不确定度.
- Good physics means knowing how much you are unsure.
你有多确定?
- 量一根金属丝两次,你可能得到 $5.77\ \text{mm}$,然后是 $5.79\ \text{mm}$。
- 两个都不“错”——每次测量都有一些 不确定度(uncertainty)。
- 好的物理意味着知道你有多不确定。
Uncertainty lab · 不确定度实验
Classify measurement problems by how they affect data. · 根据测量问题对数据的影响对其进行分类。
Reading an instrument
- Every scale can only be read so finely.
- A micrometer 螺旋测微器 reads to $0.01\ \text{mm}$; vernier calipers 游标卡尺 to $0.1\ \text{mm}$.
- The smallest step the instrument shows sets a limit on your reading.
A vernier caliper measures length precisely, with a small uncertainty
Read the whole millimetres from the main scale, then add the tenths where the sliding scale lines up
读取仪器
- 每个刻度只能读到一定的精细程度。
- 千分尺(micrometer) 可读到 $0.01\ \text{mm}$;游标卡尺(vernier calipers) 读到 $0.1\ \text{mm}$。
- 仪器能显示的最小一格,就限制了你读数的精度。
一把游标卡尺精确地测量长度,带有很小的不确定度
先从主尺读整毫米,再在游标尺对齐处加上十分位
A careful enough measurement can be exactly correct, with zero uncertainty. · 足够精确的测量可以是完全正确的，没有不确定度。
No real measurement is exact — every reading has some uncertainty. The skill is estimating how big it is. · 没有真实的测量是精确的——每个读数都有某种不确定度。关键技能在于估计其大小。
Systematic errors 系统误差
- A systematic error shifts every reading by the same amount, the same way.
- Causes: a zero error 零点误差, a wrong calibration 校准, or parallax 视差 (reading from the side).
- Repeating the measurement does not remove it.
A micrometer screw gauge measures to the nearest 0.01 mm
Parallax: viewing a scale from an angle shifts the reading — a systematic error
系统误差
- 系统误差(systematic error) 让每一个读数都朝同一方向偏移同样的量。
- 原因:零点误差(zero error)、错误的 校准(calibration),或 视差(parallax)(从侧面读数)。
- 重复测量不能消除它。
螺旋测微器测量到最近的 0.01 mm
视差:从一个角度看刻度会使读数偏移——一种系统误差
Which of these is a systematic error? · 以下哪项是系统误差？
A zero error shifts every reading by the same amount in the same direction — that is systematic. The others vary randomly from reading to reading. · 零点误差使每个读数向同一方向偏移相同的量——这是系统误差。其他读数则从一次到另一次随机变化。
Random errors 随机误差
- A random error makes readings jump above and below, with no pattern.
- Repeat the measurement many times and take the mean — the scatter partly cancels.
Precision 精密度: how narrow the distribution is around the true value T
随机误差
- 随机误差(random error) 让读数无规律地忽高忽低。
- 多次重复测量再取 平均值(mean)——散布会部分抵消。
精密度:分布在真值 T 周围有多窄
To reduce a random error, repeat the measurement and take the . · 要减小随机误差，请重复测量并取。
Random errors scatter both ways, so the mean · 平均值 of many repeats partly cancels them. (A systematic error survives averaging.) · 随机误差向两个方向散射，因此多次重复的均值可以部分抵消它们。（系统误差无法通过平均消除。）
Precision vs accuracy 准确度
- Precision = how tightly repeated readings cluster.
- Accuracy = how close they are to the true value.
精密度与准确度
- 精密度(precision) = 重复读数聚集得有多紧。
- 准确度(accuracy) = 它们离真值有多近。
Match each term to the definition the examiner marks. · 将每个术语与考官标记的定义匹配。
Repeating fixes only the random half. A zero error makes readings precise and wrong, which is why the two words are not interchangeable. · 重复只能修正随机部分。零点误差使读数精密但不准确，这也是这两个词不可互换的原因。
Which error hurts which?
- A systematic error moves the whole cluster off-target → worse accuracy.
- A random error spreads the cluster out → worse precision.
哪种误差影响哪个?
- 系统误差把整个聚集移离目标 → 准确度变差。
- 随机误差把聚集摊开 → 精密度变差。
Match each pattern of repeated readings to its description. · 将每种重复读数的模式与其描述相匹配。
Precision is how tight the cluster is; accuracy is how near the true value (the bullseye) it sits. · 精密度指簇的紧密程度；准确度指其距离真值（靶心）的远近。
Writing an uncertainty
- We write a result as $x \pm \Delta x$, where $\Delta x$ is the absolute uncertainty 绝对不确定度.
- The percentage uncertainty is $\dfrac{\Delta x}{|x|} \times 100\%$.
写出不确定度
- 我们把结果写成 $x \pm \Delta x$,其中 $\Delta x$ 是 绝对不确定度(absolute uncertainty)。
- 百分不确定度(percentage uncertainty) 是 $\dfrac{\Delta x}{|x|} \times 100\%$。
A length is $(20.0 \pm 0.5)\ \text{cm}$. What is its percentage uncertainty? · 某长度为 $(20.0 \pm 0.5)\ \text{cm}$。其百分比不确定度是多少？
$\dfrac{\Delta x}{|x|} \times 100\% = \dfrac{0.5}{20.0} \times 100\% = 2.5\%$.
Combining: add or subtract
- If $y = a + b$ or $y = a - b$, add the absolute uncertainties.
- $\Delta y = \Delta a + \Delta b$.
合成:加法或减法
- 如果 $y = a + b$ 或 $y = a - b$,把绝对不确定度相加。
- $\Delta y = \Delta a + \Delta b$。
Combining: multiply or divide
- If $y = \dfrac{a \cdot b}{c}$, add the percentage uncertainties.
- $\dfrac{\Delta y}{|y|} = \dfrac{\Delta a}{|a|} + \dfrac{\Delta b}{|b|} + \dfrac{\Delta c}{|c|}$.
合成:乘法或除法
- 如果 $y = \dfrac{a \cdot b}{c}$,把百分不确定度相加。
- $\dfrac{\Delta y}{|y|} = \dfrac{\Delta a}{|a|} + \dfrac{\Delta b}{|b|} + \dfrac{\Delta c}{|c|}$。
For · 支持 $y = \dfrac{a \cdot b}{c}$, how do you combine the uncertainties? · 对于 $y = \dfrac{a \cdot b}{c}$，如何合并不确定度？
For multiplying and dividing, add the percentage uncertainties: $\dfrac{\Delta y}{|y|} = \dfrac{\Delta a}{|a|} + \dfrac{\Delta b}{|b|} + \dfrac{\Delta c}{|c|}$. · 对于乘法和除法，相加百分比不确定度：$\dfrac{\Delta y}{|y|} = \dfrac{\Delta a}{|a|} + \dfrac{\Delta b}{|b|} + \dfrac{\Delta c}{|c|}$。
Which rules for combining uncertainties are correct? Select all · 所有 that apply. · 关于合并不确定度的规则哪些是正确的？选择所有适用项。
Uncertainties never cancel. Subtracting two similar values keeps the sum of their uncertainties over a much smaller result, which is why such a difference is so imprecise. · 不确定度永远不会相互抵消。减去两个相似值会使它们的总不确定度落在一个更小的结果上，这就是为什么这种差值非常不精确。
Combining: powers
- If $y = a^{n}$, multiply the percentage uncertainty by the power.
- $\dfrac{\Delta y}{|y|} = |n| \cdot \dfrac{\Delta a}{|a|}$.
合成:幂
- 如果 $y = a^{n}$,把百分不确定度乘以这个幂。
- $\dfrac{\Delta y}{|y|} = |n| \cdot \dfrac{\Delta a}{|a|}$。
A radius is measured with a $0.4\%$ uncertainty. A circle's area is $A = \pi r^{2}$. What is the percentage uncertainty in $A$? · 半径测量有 $0.4\%$ 的不确定度。圆的面积为 $A = \pi r^{2}$。$A$ 的百分比不确定度是多少？
For a power, multiply by the power: $A \propto r^{2}$, so the % uncertainty is $2 \times 0.4\% = 0.8\%$. · 对于幂运算，乘以指数：$A \propto r^{2}$，因此百分比不确定度为 $2 \times 0.4\% = 0.8\%$。
A period T is measured with a 2.0% uncertainty. What is the percentage uncertainty in T squared? · 周期 T 的测量不确定度为 2.0%。T 平方的百分比不确定度是多少？
Squaring doubles the percentage uncertainty; a square root halves it. Quoting 2.0% for T^2 is one of the commonest errors in this topic. · 平方会使百分比不确定度加倍；开方则使其减半。对 T^ 2.0% 的表述是此主题中最常见的错误之一。2 是此主题中最常见的错误之一。
Worked example: volume of a ball
- Diameter $d = (5.26 \pm 0.02)\ \text{cm}$, and volume $V \propto d^{3}$.
- % uncertainty in $d$: $\dfrac{0.02}{5.26} \times 100\% \approx 0.38\%$.
- Power rule: % uncertainty in $V = 3 \times 0.38\% \approx 1.1\%$.
- So $V = (76.2 \pm 0.9)\ \text{cm}^{3}$.
例题:球的体积
- 直径 $d = (5.26 \pm 0.02)\ \text{cm}$,体积 $V \propto d^{3}$。
- $d$ 的百分不确定度:$\dfrac{0.02}{5.26} \times 100\% \approx 0.38\%$。
- 幂规则:$V$ 的百分不确定度 $= 3 \times 0.38\% \approx 1.1\%$。
- 所以 $V = (76.2 \pm 0.9)\ \text{cm}^{3}$。
Significant figures 有效数字
- Give an answer the same significant figures as the least precise measurement.
- Usually 2 or 3 in this course — too many makes it look more exact than it is.
有效数字
- 给答案 和最不精确的测量相同的有效数字。
- 本课程里通常是 2 或 3 位——太多会让结果显得比实际更精确。
You've got it
- systematic error → shifts every reading (worse accuracy); random → scatter (worse precision)
- repeat and take the mean to beat random error
- combine uncertainties: add absolute for $+\,-$, add % for $\times\,\div$, × the power for $a^{n}$
你掌握了
- 系统误差 → 移动每个读数(准确度变差);随机 → 散布(精密度变差)
- 重复并取 平均值 来克服随机误差
- 合成不确定度:$+\,-$ 时 加绝对值,$\times\,\div$ 时 加百分比,$a^{n}$ 时 乘以幂