📚 IB Physics: Measurements and Uncertainties (Student Cards) | IB物理学：测量与不确定度（学生卡片）
Mastering measurements and uncertainties is the first critical step in IB Physics. These ideas underpin every practical investigation, data analysis question, and Internal Assessment. This set of student cards condenses the essential definitions, rules, and calculation methods you need for Papers 1, 2, and 3.
掌握测量与不确定度是学习IB物理的第一个关键步骤。这些概念支撑着每一份实验探究、数据分析题和内部评估。本套学生卡片浓缩了你应对试卷一、试卷二和试卷三所需的基本定义、规则与计算方法。
1. SI Units and Prefixes | 国际单位制与词头
The IB Physics course uses the International System of Units (SI). There are seven base units: kilogram (kg) for mass, metre (m) for length, second (s) for time, ampere (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity. All other units are derived from these base units.
IB物理课程采用国际单位制（SI）。共有七个基本单位：千克（kg）表示质量，米（m）表示长度，秒（s）表示时间，安培（A）表示电流，开尔文（K）表示温度，摩尔（mol）表示物质的量，坎德拉（cd）表示发光强度。所有其他单位均由这些基本单位导出。
Prefixes are used to express very large or very small quantities. You must know the full list from 10¹² (tera, T) down to 10⁻¹² (pico, p). The most commonly tested prefixes are kilo (k, 10³), centi (c, 10⁻²), milli (m, 10⁻³), micro (μ, 10⁻⁶), and nano (n, 10⁻⁹).
词头用于表示非常大或非常小的量。你必须掌握从10¹²（太，T）到10⁻¹²（皮，p）的完整列表。最常考查的词头是千（k，10³）、厘（c，10⁻²）、毫（m，10⁻³）、微（μ，10⁻⁶）和纳（n，10⁻⁹）。
- Example: 5.4 km = 5.4 × 10³ m
- 示例：5.4 km = 5.4 × 10³ m
- Example: 20 μs = 20 × 10⁻⁶ s = 2.0 × 10⁻⁵ s
- 示例：20 μs = 20 × 10⁻⁶ s = 2.0 × 10⁻⁵ s
revents errors in derived units such as newtons (N = kg m s⁻²) and joules (J = kg m² s⁻²).
在把数值代入方程之前，一定要先换算词头。这样可以避免在导出单位（如牛顿 N = kg m s⁻² 和焦耳 J = kg m² s⁻²）中出现错误。
2. Scientific Notation and Significant Figures | 科学记数法与有效数字
nteger. It is the standard way to avoid ambiguity about trailing zeros and to handle extreme values in IB Physics.
科学记数法将数字表示为 a × 10ⁿ，其中1 ≤ a < 10，n为整数。这是IB物理中避免末尾零产生歧义并处理极端数值的标准方法。
Significant figures (s.f.) include all digits known with certainty plus one uncertain digit. Zeros between non-zero digits are significant, but leading zeros are never significant. Trailing zeros are significant only if a decimal point is shown.
有效数字（s.f.）包括所有确定的数字再加上一位不确定的数字。非零数字之间的零是有效的，但前导零永远无效。末尾零只有在有小数点时才有效。
| Value | Significant figures | 数值 | 有效数字 |
|---|---|---|---|
| 0.00350 | 3 | 0.00350 | 3位 |
| 10500 | 3 (or ambiguous) | 10500 | 3位（或存在歧义） |
| 1.0500 × 10⁴ | 5 | 1.0500 × 10⁴ | 5位 |
In calculations, the final answer should be quoted to the same number of significant figures as the least precise input quantity. For addition and subtraction, use the least number of decimal places.
在计算中，最终答案的有效数字位数应与最不精确的输入量相同。对于加法和减法，则使用最少的小数位数。
3. Random and Systematic Errors | 随机误差与系统误差
Random errors cause readings to scatter around the true value. They can be reduced by taking repeated measurements and finding the mean. Random errors affect precision but not necessarily accuracy.
随机误差使读数在真实值附近分散。可以通过多次测量并求平均值来减小随机误差。随机误差影响精密度，但不一定影响准确度。
Systematic errors shift all readings in the same direction. They arise from faulty equipment, zero error, or an incorrect calibration. Systematic errors affect accuracy but not precision, and repeated measurements do not reduce them.
系统误差使所有读数向同一方向偏移。它们来源于仪器故障、零误差或校准不当。系统误差影响准确度，但不影响精密度，多次测量无法减小系统误差。
- Random error example: reaction time when starting a stopwatch.
- 随机误差示例：启动秒表时的反应时间。
- Systematic error example: a voltmeter reading 0.2 V too high because of a zero offset.
- 系统误差示例：由于零点偏移，电压表读数偏高0.2 V。
Always state one realistic source of random error and one source of systematic error in a practical write-up. Be specific to the experiment, not generic.
在实验报告中，始终指出一个现实的随机误差来源和一个系统误差来源。要针对具体实验，而不是使用笼统的描述。
4. Accuracy and Precision | 准确度与精密度
Accuracy describes how close a measured value is to the accepted or true value. Precision describes how close repeated measurements are to each other. The two terms are independent: results can be precise but inaccurate, or accurate but imprecise.
准确度描述测量值与公认值或真实值的接近程度。精密度描述重复测量值彼此之间的接近程度。这两个术语相互独立：结果可以精密度高但准确度低，也可以准确度高但精密度低。
A common illustration uses a target: precise results cluster tightly even if far from the bullseye; accurate results average on the bullseye even if scattered. In IB exam questions, you must not confuse these definitions.
一个常见的比喻是靶子：精密的结果即使远离靶心也聚集得很紧；准确的结果即使分散也平均落在靶心上。在IB考试题中，你不能混淆这些定义。
In a data table, precision is indicated by the number of decimal places consistent across repeats. Accuracy is judged by comparing the mean to a literature value or theoretical prediction.
在数据表中，精密度由重复数据的小数位数一致性来表示。准确度则通过将平均值与文献值或理论预测值进行比较来判断。
5. Absolute, Fractional and Percentage Uncertainties | 绝对、相对与百分比不确定度
Every measurement carries an uncertainty. The absolute uncertainty Δx has the same units as the measured quantity x. It is usually estimated as half the smallest scale division on an analogue instrument, or as the smallest digit on a digital display.
每次测量都带有不确定度。绝对不确定度Δx与测量量x具有相同的单位。通常对于模拟仪器，估计为最小刻度值的一半；对于数字显示器，则取最小的显示位数。
Fractional uncertainty is Δx / x, and percentage uncertainty is (Δx / x) × 100%. Fractional uncertainty has no units; percentage uncertainty is expressed in %.
相对不确定度为Δx / x，百分比不确定度为(Δx / x) × 100%。相对不确定度没有单位；百分比不确定度以%表示。
Percentage uncertainty = (absolute uncertainty / measured value) × 100%
百分比不确定度 =（绝对不确定度 / 测量值）× 100%
: A length measured as 12.0 cm ± 0.1 cm has a percentage uncertainty of (0.1 / 12.0) × 100% = 0.83%.
示例：测得长度为12.0 cm ± 0.1 cm，百分比不确定度为(0.1 / 12.0) × 100% = 0.83%。
or a stopwatch reading 25.31 s, the uncertainty is ± 0.01 s, ignoring human reaction time.
对于数字仪器，绝对不确定度通常是最小读数，而不是一半。对于读数为25.31 s的秒表，不确定度为± 0.01 s，这里忽略了人的反应时间。
6. Combining Uncertainties | 不确定度的合成
B level, simply add. For y = a + b or y = a − b, the absolute uncertainty is Δy = Δa + Δb.
当量进行加法或减法时，绝对不确定度按平方和开方合成，或在IB水平中简单相加。对于 y = a + b 或 y = a − b，绝对不确定度为 Δy = Δa + Δb。
ivided, percentage uncertainties add. For y = ab or y = a/b, the percentage uncertainty is %Δy = %Δa + %Δb.
当量进行乘法或除法时，百分比不确定度相加。对于 y = ab 或 y = a/b，百分比不确定度为 %Δy = %Δa + %Δb。
Addition/Subtraction: Δy = Δa + Δb
加法/减法：Δy = Δa + Δb
Multiplication/Division: %Δy = %Δa + %Δb
乘法/除法：%Δy = %Δa + %Δb
For a power, if y = aⁿ, the percentage uncertainty is %Δy = |n| × %a. For a constant multiplier, the absolute uncertainty is multiplied by the same constant.
对于幂运算，若 y = aⁿ，则百分比不确定度为 %Δy = |n| × %Δa。对于常数倍数，绝对不确定度乘以相同的常数。
- If a = 4.0 ± 0.1 cm and b = 3.0 ± 0.1 cm, then a + b = 7.0 ± 0.2 cm.
- 若 a = 4.0 ± 0.1 cm，b = 3.0 ± 0.1 cm，则 a + b = 7.0 ± 0.2 cm。
- , add the percentage uncertainties of l, w, and h to find the percentage uncertainty in V.
- 若 V = l × w × h，将 l、w 和 h 的百分比不确定度相加，即可求出 V 的百分比不确定度。
7. Graphing and Error Bars | 绘图与误差棒
All graphs in IB Physics must be drawn on graph paper or using software, with labelled axes, units, and a clear scale. Data points should occupy more than half of each axis, and the line of best fit must have a balanced spread of points.
IB物理中的所有图表必须绘制在坐标纸上或使用软件绘制，并标注坐标轴、单位和清晰的比例。数据点应占据每条坐标轴的一半以上，最佳拟合线必须使点均匀分布在线两侧。
Error bars show the absolute uncertainty in each plotted quantity. A vertical error bar extends one Δy above and below the point; a horizontal error bar extends one Δx left and right.
误差棒表示每个作图量的绝对不确定度。垂直误差棒在点上方和下方各延伸一个Δy；水平误差棒在点左侧和右侧各延伸一个Δx。
ass through the error bars. The gradient uncertainty is (gradient_max − gradient_min) / 2.
斜率的绝对不确定度通过能穿过误差棒的最陡和最浅两条可能直线求得。斜率不确定度为（最大斜率 − 最小斜率）÷ 2。
Always refer to 'line of best fit', never 'line of best fit by eye' in a final report. The line should be thin and drawn with a sharp pencil.
在最终报告中，始终使用“最佳拟合线”这一表述，而不要说“目测最佳拟合线”。线条应细且用削尖的铅笔绘制。
8. Linearising Data | 数据线性化
-line form y = mx + c. This allows the gradient and intercept to give physical quantities.
许多IB物理关系是非线性的。为了分析这些关系，你必须将数据变换为直线形式 y = mx + c。这样斜率与截距就能给出物理量。
Examples of linearisation: for a simple pendulum T = 2π√L/g, plot T² against L to get gradient = 4π²/g. For a capacitor discharge V = V₀e−t/RC, plot ln V against t to get gradient = −1/RC.
线性化的例子：对于单摆 T = 2π√L/g，作 T² 对 L 的图，斜率为 4π²/g。对于电容器放电 V = V₀e−t/RC，作 ln V 对 t 的图，斜率为 −1/RC。
In Internal Assessment, linearisation is expected for any non-linear proposal. You should state the theoretical equation, identify what to plot on each axis, and explain how the gradient or intercept yields the target constant.
在内部评估中，任何非线性设计方案都要求进行线性化。你应写出理论方程，确定每条坐标轴上要绘制什么量，并解释斜率或截距如何得出目标常数。
9. Vectors and Scalars | 矢量与标量
Scalars have magnitude only: distance, speed, mass, energy, temperature, time. Vectors have both magnitude and direction: displacement, velocity, acceleration, force, momentum, electric field strength.
标量只有大小：距离、速率、质量、能量、温度、时间。矢量既有大小又有方向：位移、速度、加速度、力、动量、电场强度。
llelogram or tip-to-tail method. For perpendicular vectors, the resultant magnitude is R = √A² + B², and the direction is found from tan θ = B/A.
矢量相加时，使用平行四边形法或首尾相接法。对于相互垂直的矢量，合矢量的大小为 R = √A² + B²，方向由 tan θ = B/A 求出。
R = √A² + B² for perpendicular vectors
R = √A² + B²（适用于垂直矢量）
When resolving a vector into components, the horizontal component is A cos θ and the vertical component is A sin θ, where θ is the angle to the horizontal. This is fundamental for projectile motion and inclined plane problems.
当把矢量分解为分量时，水平分量为 A cos θ，垂直分量为 A sin θ，其中θ是与水平方向的夹角。这是抛体运动和斜面问题的基础。
10. Worked Examples and Common Pitfalls | 典型例题与常见误区
nd a resistance of 200 ± 10 Ω. Calculate the power P = I²R and its percentage uncertainty.
典型例题：一学生测得电流为0.50 ± 0.01 A，电阻为200 ± 10 Ω。计算功率 P = I²R 及其百分比不确定度。
Solution: P = (0.50)² × 200 = 50 W. Percentage uncertainties: %ΔI = (0.01/0.50) × 100% = 2%; %ΔR = (10/200) × 100% = 5%. For I², %Δ(I²) = 2 × 2% = 4%. Total %ΔP = 4% + 5% = 9%. Therefore P = 50 W ± 9%.
解答：P = (0.50)² × 200 = 50 W。百分比不确定度：%ΔI = (0.01/0.50) × 100% = 2%；%ΔR = (10/200) × 100% = 5%。对于 I²，%Δ(I²) = 2 × 2% = 4%。总 %ΔP = 4% + 5% = 9%。因此 P = 50 W ± 9%。
Common pitfalls: forgetting to convert units before calculating uncertainties, quoting too many significant figures in a final answer, and confusing absolute with percentage uncertainty. Always round uncertainties to one or two significant figures and match the decimal place of the quantity.
常见误区：在计算不确定度之前忘记换算单位，最终答案中有效数字位数过多，以及混淆绝对不确定度与百分比不确定度。始终将不确定度保留一位或两位有效数字，并与测量量的小数位保持一致。
In multiple-choice questions, look for answers that correctly apply the rules of propagation rather than simply adding all uncertainties regardless of operation. In written responses, show every step and quote uncertainties explicitly.
在选择题中，要选择正确应用传播规则的选项，而不是无论运算类型都简单地把所有不确定度相加。在书面作答时，展示每一步，并明确写出不确定度。
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