Uncertainty in A-Level Physics | 物理测量中的不确定度

📚 Uncertainty in A-Level Physics | 物理测量中的不确定度

In physics, every measurement has some doubt. The range of values within which the true value is likely to lie is called the uncertainty. Understanding uncertainty allows you to judge how reliable your results are and to compare them with theoretical predictions.

在物理学中，每一次测量都存在一定的疑问。真实值可能落入的数值范围称为不确定度。理解不确定度可以帮助你判断结果有多可靠，并可将实验结果与理论预测进行比较。

1. What is Uncertainty? | 什么是不确定度？

Uncertainty is the interval around a measured value that expresses the lack of exact knowledge of the true value. It is not a mistake; it is an unavoidable feature of measurement.

不确定度是围绕测量值的一个区间，表示对真实值缺乏精确了解。它不是错误，而是测量中不可避免的特征。

For example, if you measure a length as 25.0 cm with an uncertainty of ±0.2 cm, you are saying that the true length is likely between 24.8 cm and 25.2 cm.

例如，如果你测得某长度为 25.0 cm，不确定度为 ±0.2 cm，那就是说真实长度很可能在 24.8 cm 到 25.2 cm 之间。

Uncertainty = ± Δx

不确定度 = ± Δx

In AQA A-Level Physics, uncertainties are usually stated to one significant figure unless they are a leading digit.
在 AQA A-Level 物理中，不确定度通常保留一位有效数字，除非首位是 1 或 2 时可保留两位。

2. Random vs Systematic Errors | 随机误差与系统误差

Random errors cause readings to scatter above and below the true value. They can be reduced by repeating measurements and calculating the mean.

随机误差使读数在真实值上下分散。它们可以通过重复测量并计算平均值来减小。

Systematic errors affect all readings in the same way, often by a fixed amount. They cannot be reduced by repetition, but they can be identified by calibrating the instrument or using a different method.

系统误差以相同方式影响所有读数，通常是一个固定量。它们不能通过重复测量来减小，但可以通过校准仪器或采用不同方法来识别。

Type	Effect	Reduction
Random	Scatter of readings	Repeat and average
Systematic	Shift in same direction	Calibrate or zero the instrument

In exams, you may be asked to identify which type of error a zero error represents. A zero error is systematic because it shifts every reading by the same amount.

在考试中，你可能会被要求识别零误差属于哪种误差。零误差属于系统误差，因为它使每个读数偏移相同的量。

3. Precision, Accuracy and Resolution | 精密度、准确度和分辨率

Precision describes how closely repeated readings agree with each other. Accuracy describes how close a measurement is to the true value. Resolution is the smallest change a measuring instrument can detect.

精密度描述重复读数之间的一致性。准确度描述测量值与真实值的接近程度。分辨率是测量仪器能够检测到的最小变化。

A measurement can be precise but inaccurate if a systematic error is present. For example, a digital balance that is zeroed incorrectly can give precise but inaccurate readings.

如果存在系统误差，测量可能精密但不准确。例如，未正确调零的电子天平可能给出精密但不准确的读数。

The uncertainty due to the resolution of an analogue instrument is often taken as half the smallest division. For a digital instrument, the uncertainty is usually ±1 in the last digit.

模拟仪器由分辨率引起的不确定度通常取最小刻度的一半。数字仪器的不确定度通常为最末位数字的 ±1。

Analogue: Δx = ½ × smallest division

模拟仪器：Δx = ½ × 最小刻度

4. Absolute, Fractional and Percentage Uncertainties | 绝对、分数和百分比不确定度

Absolute uncertainty has the same unit as the measurement, for example ±0.5 cm. Fractional uncertainty is the absolute uncertainty divided by the measured value. Percentage uncertainty is the fractional uncertainty multiplied by 100%.

绝对不确定度与测量值具有相同单位，例如 ±0.5 cm。分数不确定度是绝对不确定度除以测量值。百分比不确定度是分数不确定度乘以 100%。

If a length is measured as 15.0 cm with an absolute uncertainty of ±0.3 cm, then:

如果测得长度为 15.0 cm，绝对不确定度为 ±0.3 cm，则：

Fractional uncertainty = 0.3 / 15.0 = 0.02

分数不确定度 = 0.3 / 15.0 = 0.02

Percentage uncertainty = 0.02 × 100% = 2%

百分比不确定度 = 0.02 × 100% = 2%

When representing a value with its uncertainty, use the notation x ± Δx. Always include units.

当表示带有不确定度的数值时，使用记号 x ± Δx。始终包含单位。

5. Combining Uncertainties: Addition and Subtraction | 组合不确定度：加减法

When quantities are added or subtracted, the absolute uncertainties are added together.

当量相加或相减时，绝对不确定度相加。

Suppose p = a + b or p = a − b. Then:

设 p = a + b 或 p = a − b，则：

Δp = Δa + Δb

For example, if a = 5.0 ± 0.1 m and b = 2.0 ± 0.2 m, then a + b = 7.0 ± 0.3 m and a − b = 3.0 ± 0.3 m.

例如，若 a = 5.0 ± 0.1 m，b = 2.0 ± 0.2 m，则 a + b = 7.0 ± 0.3 m，a − b = 3.0 ± 0.3 m。

Note that even when subtracting, uncertainties add, which can make a small difference very unreliable.

注意，即使做减法，不确定度也相加，这可能使一个很小的差值变得非常不可靠。

6. Combining Uncertainties: Multiplication and Division | 组合不确定度：乘除法

When quantities are multiplied or divided, the percentage uncertainties are added.

当量相乘或相除时，百分比不确定度相加。

If p = a × b or p = a / b, then:

若 p = a × b 或 p = a / b，则：

(Δp / p) × 100% = (Δa / a) × 100% + (Δb / b) × 100%

For example, the area of a rectangle measured as length = 4.0 ± 0.1 cm and width = 2.0 ± 0.2 cm:

例如，测得矩形长度 = 4.0 ± 0.1 cm，宽度 = 2.0 ± 0.2 cm，求面积：

Percentage uncertainty in length = 2.5%, in width = 10%. Total percentage uncertainty = 12.5%. Area = 8.0 cm², so absolute uncertainty = 8.0 × 0.125 = 1.0 cm².

长度的百分比不确定度 = 2.5%，宽度的百分比不确定度 = 10%。总百分比不确定度 = 12.5%。面积 = 8.0 cm²，因此绝对不确定度 = 8.0 × 0.125 = 1.0 cm²。

Area = 8.0 ± 1.0 cm²

面积 = 8.0 ± 1.0 cm²

7. Combining Uncertainties: Powers and Constants | 组合不确定度：幂与常数

When a quantity is raised to a power, the percentage uncertainty is multiplied by that power.

当一个量被提升到某次幂时，其百分比不确定度乘以该次幂。

If p = aⁿ, then:

若 p = aⁿ，则：

Δp / p = n × (Δa / a)

For example, the volume of a cube with side length s = 2.0 ± 0.1 cm. The percentage uncertainty in s is 5%. For V = s³, the percentage uncertainty in V is 3 × 5% = 15%.

例如，正方体边长 s = 2.0 ± 0.1 cm。边长的不确定度为 5%。对于 V = s³，体积的百分比不确定度为 3 × 5% = 15%。

Constants such as π have no uncertainty and do not affect the percentage uncertainty when multiplied. If a quantity is multiplied by a constant, the absolute uncertainty is multiplied by that constant, but the percentage uncertainty remains the same.

像 π 这样的常数没有不确定度，相乘时不影响百分比不确定度。如果某个量乘以常数，绝对不确定度乘以该常数，但百分比不确定度保持不变。

V = 8.0 cm³, ΔV = 0.15 × 8.0 = 1.2 cm³

V = 8.0 cm³，ΔV = 0.15 × 8.0 = 1.2 cm³

8. Graphical Methods and Error Bars | 图形方法与误差线

In AQA practical work, you should plot a graph with error bars to represent the uncertainty in each data point. The length of an error bar is 2 × Δy (or 2 × Δx) centred on the measured value.

在 AQA 实验操作中，你应该绘制带有误差线的图表来表示每个数据点的不确定度。误差线的长度是 2 × Δy（或 2 × Δx），以测量值为中心。

The best-fit line should pass through all error bars if possible. Since each data point has a range of possible locations, a range of lines can be drawn that are consistent with the error bars. The steepest and shallowest possible lines give the uncertainty in the gradient and intercept.

最佳拟合直线应尽可能穿过所有误差线。由于每个数据点可能处于一个范围内，可以画出多条与误差线一致的直线。最陡和最缓的直线给出了斜率和截距的不确定度。

Plot the data as points with vertical error bars if the uncertainty is in y.
如果不确定度在 y 方向，则绘制带垂直误差线的数据点。
Draw the line of best fit, then draw two additional lines: maximum gradient and minimum gradient.
绘制最佳拟合线，然后再画两条附加线：最大斜率和最小斜率。

For a straight line through the origin, the gradient can be found from one distant point rather than from a best fit, but using a best fit is more reliable.

对于过原点的直线，可以从一个较远的点求斜率，而不必用最佳拟合线，但使用最佳拟合线更可靠。

9. Uncertainty in Gradient and Intercept | 斜率和截距的不确定度

To find the uncertainty in a gradient, calculate the gradient of the best-fit line, then calculate the gradients of the steepest and shallowest acceptable lines.

要求斜率的不确定度，先计算最佳拟合线的斜率，再计算可接受的最陡线和最缓线的斜率。

Δm = (m_max − m_min) / 2

Similarly, for the intercept c:

类似地，对于截距 c：

Δc = (c_max − c_min) / 2

These should be stated in the form m ± Δm with suitable units. A common AQA question asks you to use this method to determine whether a line passes through the origin within uncertainty.

这些应以 m ± Δm 的形式与适当单位一起表示。AQA 常见问题会要求你用这种方法判断直线是否在不确定度范围内过原点。

10. Reducing Uncertainty | 减小不确定度

You should know practical strategies to reduce uncertainties and improve reliability.

你应该了解减小不确定度和提高可靠性的实用策略。

Repeat measurements and calculate the mean. This reduces the effect of random errors.
重复测量并计算平均值。这可以减小随机误差的影响。
Use an instrument with a smaller resolution or a more precise technique.
使用分辨率更小的仪器或更精密的技术。
Check for zero errors and calibrate equipment before use.
使用前检查零误差并校准设备。
Measure a larger quantity where possible. For example, measure the time for 20 oscillations instead of one, then divide by 20. This reduces the percentage uncertainty in the timer reading.
尽可能测量更大的量。例如，测量 20 次振荡的时间而不是一次，然后除以 20。这样可以减小计时器读数的百分比不确定度。
Use a fiducial marker or parallax-free reading to reduce reading errors.
使用基准标记或避免视差来减小读数误差。
Control environmental variables such as temperature and draughts during an experiment.
在实验期间控制环境变量，如温度和气流。

Remember that reducing random error improves precision, but systematic errors must be eliminated by checking the experimental design.

记住，减小随机误差可以提高精密度，但系统误差必须通过检查实验设计来消除。

11. Exam Tips for AQA | AQA 考试提示

In AQA A-Level Physics exams, uncertainty questions often appear in the practical section and in data analysis questions. Here are key strategies.

在 AQA A-Level 物理考试中，不确定度问题常出现在实验部分和数据分析题中。以下是一些关键策略。

Always quote the absolute uncertainty to one significant figure, e.g. 2.5 ± 0.2 N.
始终将绝对不确定度保留一位有效数字，例如 2.5 ± 0.2 N。
When giving an answer, the last significant figure of the measured value should match the decimal place of the uncertainty.
给出答案时，测量值的最后一位有效数字应与不确定度的小数位对齐。
If you are asked to find the uncertainty in a calculated quantity, first identify whether the quantities are added/subtracted (use absolute) or multiplied/divided (use percentage).
如果要求计算量的不确定度，首先判断量是加减（用绝对）还是乘除（用百分比）。
For a power relation such as g = 4π²l/T², the percentage uncertainty in g is (Δl/l × 100%) + 2 × (ΔT/T × 100%).
对于幂函数关系如 g = 4π²l/T²，g 的百分比不确定度是 (Δl/l × 100%) + 2 × (ΔT/T × 100%)。
Use the gradient uncertainty formula with actual numerical values from the graph, not just written symbols.
使用斜率不确定度公式时，应代入图中具体的数值，而不只是写出符号。
Do not forget units on every final answer.
不要忘记在每个最终答案上写单位。

Final answer format: value ± uncertainty (unit)

最终答案格式：数值 ± 不确定度（单位）

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Uncertainty in A-Level Physics | 物理测量中的不确定度