📚 Measurements and their errors | 测量及其误差
In AQA A-level Physics, the opening topic 'Measurements and their errors' underpins every practical skill and written question across the specification. It sets out how physicists quantify the physical world, judge the quality of data, and express confidence in results.
在 AQA A-level 物理中，开篇主题 '测量及其误差' 是整个课程所有实验技能和笔试问题的基础。它说明了物理学家如何量化物理世界、判断数据质量，以及表达对结果的信心。
1. SI Units and Base Quantities | 国际单位制与基本量
The International System of Units (SI) defines seven base quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance and luminous intensity. Their base units are the metre (m), kilogram (kg), second (s), ampere (A), kelvin (K), mole (mol) and candela (cd).
国际单位制 (SI) 定义了七个基本量：长度、质量、时间、电流、热力学温度、物质的量和发光强度。它们的基本单位分别是米 (m)、千克 (kg)、秒 (s)、安培 (A)、开尔文 (K)、摩尔 (mol) 和坎德拉 (cd)。
All other physical quantities are expressed as derived units formed from combinations of these base units. For example, the newton is equivalent to kg m s⁻² and the joule is equivalent to kg m² s⁻².
所有其他物理量都用这些基本单位组合而成的导出单位表示。例如，牛顿等价于 kg m s⁻²，焦耳等价于 kg m² s⁻²。
In AQA examination questions, you may need to express a unit in base units, especially for quantities such as force, energy, power, potential difference and resistance.
在 AQA 考试题中，你可能需要将某个单位写成基本单位，尤其是力、能量、功率、电势差和电阻等量。
2. Derived Units and Homogeneity | 导出单位与量纲一致性
Derived units are obtained by multiplying or dividing base units. Common examples include speed (m s⁻¹), acceleration (m s⁻²), force (kg m s⁻²), pressure (kg m⁻¹ s⁻²), charge (A s), potential difference (kg m² s⁻³ A⁻¹) and resistance (kg m² s⁻³ A⁻²).
导出单位是由基本单位相乘或相除得到的。常见的例子包括速度 (m s⁻¹)、加速度 (m s⁻²)、力 (kg m s⁻²)、压强 (kg m⁻¹ s⁻²)、电荷量 (A s)、电势差 (kg m² s⁻³ A⁻¹) 和电阻 (kg m² s⁻³ A⁻²)。
for errors in a formula. For instance, in s = ut + ½ at², both terms have the unit metre.
当一个方程两边的基准单位相同时，该方程就是量纲一致的。这可以快速检查公式是否有错误。例如，在 s = ut + ½ at² 中，两项的单位都是米。
s = ut + ½ at²
Homogeneity cannot guarantee that an equation is physically correct, because dimensionless constants such as 2 or π carry no units, but it can rule out many incorrect relationships.
量纲一致性并不能保证一个方程在物理上一定正确，因为像 2 或 π 这样的无量纲常数不携带单位，但它可以排除许多错误的关系式。
3. Prefixes and Standard Form | 单位前缀与标准形式
Very large and very small quantities are written using prefixes or standard form. AQA expects you to know the common prefixes tera (T, 10¹²), giga (G, 10⁹), mega (M, 10⁶), kilo (k, 10³), centi (c, 10⁻²), milli (m, 10⁻³), micro (μ, 10⁻⁶), nano (n, 10⁻⁹), pico (p, 10⁻¹²) and femto (f, 10⁻¹⁵).
非常大和非常小的量会使用前缀或标准形式来表示。AQA 要求你掌握常见前缀：太拉 (T, 10¹²)、吉咖 (G, 10⁹)、兆 (M, 10⁶)、千 (k, 10³)、厘 (c, 10⁻²)、毫 (m, 10⁻³)、微 (μ, 10⁻⁶)、纳 (n, 10⁻⁹)、皮 (p, 10⁻¹²) 和飞 (f, 10⁻¹⁵)。
nd accurate. For example, 3.2 mm = 3.2 × 10⁻³ m and 450 nm = 450 × 10⁻⁹ m = 4.50 × 10⁻⁷ m.
带前缀单位与基本单位之间的换算必须快速准确。例如，3.2 mm = 3.2 × 10⁻³ m，450 nm = 450 × 10⁻⁹ m = 4.50 × 10⁻⁷ m。
In calculations, convert everything to base units before substituting into an equation unless the question specifically asks for an answer in prefixed units.
在计算中，除非题目特别要求使用前缀单位作答，否则应先将所有量换算成基本单位，再代入公式。
4. Precision and Accuracy | 精密度与准确度
Precision describes how closely repeated readings agree with one another. A set of measurements clustered tightly around a mean value is precise, even if that mean is far from the true value.
精密度描述的是多次重复读数之间相互接近的程度。一组紧密围绕平均值分布的数据是精密的，即使该平均值远离真实值。
Accuracy describes how close a measured value is to the true or accepted value. A measurement can be precise but inaccurate if a systematic error is shifting all readings by the same amount.
准确度描述的是测量值与真实值或公认值之间的接近程度。如果系统误差使所有读数向同一方向偏移，那么测量可以精密但不准确。
In practical work, use the terms precision and accuracy separately, and link them to random and systematic errors respectively. Do not treat them as interchangeable.
在实验工作中，应分别使用精密度和准确度这两个术语，并分别将它们与随机误差和系统误差联系起来。不要把它们混用。
5. Random and Systematic Errors | 随机误差与系统误差
Random errors cause fluctuations in repeated readings. They arise from unpredictable changes such as reaction time, vibrations or reading a scale from slightly different positions. Repeating measurements and taking the mean reduces their effect.
随机误差会引起重复读数的波动。它们来源于不可预测的变化，例如反应时间、振动或从略有不同的位置读取刻度。重复测量并取平均值可以减小随机误差的影响。
Systematic errors shift all readings in the same direction. They may be caused by a zero error on a meter, wrong calibration, or a consistent parallax error. Repeats do not reveal or reduce a systematic error.
系统误差会使所有读数向同一方向偏移。它们可能由仪表零误差、错误校准或持续的视差引起。重复测量不能发现或减小系统误差。
The mean of many readings improves reliability against random errors, but a systematic error can only be reduced by calibration, correction or a better experimental procedure.
多次读数的平均值可以提高数据对随机误差的可靠性，但系统误差只能通过校准、修正或改进实验方法来减小。
6. Absolute and Percentage Uncertainty | 绝对不确定度与百分不确定度
the measurement. It is written as measured value ± uncertainty, for example 24.5 ± 0.1 cm.
绝对不确定度是用与测量值相同单位表示的不确定度。它写作测量值 ± 不确定度，例如 24.5 ± 0.1 cm。
Percentage uncertainty is the absolute uncertainty divided by the measured value, multiplied by 100.
百分不确定度是绝对不确定度除以测量值，再乘以 100。
percentage uncertainty = (absolute uncertainty ÷ measured value) × 100%
le, a metre rule may give ±0.5 mm, while a digital voltmeter reading 1.56 V gives ±0.01 V.
对于模拟仪器，分辨率通常是刻度最小分度的一半。对于数字仪器，分辨率是最小显示位数。例如，米尺可能给出 ±0.5 mm，而读数为 1.56 V 的数字电压表给出 ±0.01 V。
7. Combining Uncertainties | 不确定度的合成
When quantities are added or subtracted, absolute uncertainties add in quadrature or simply add. For AQA, the standard approach is to add absolute uncertainties directly.
当量进行加减运算时，绝对不确定度直接相加。对于 AQA，标准做法是直接将绝对不确定度相加。
If Q = a + b or Q = a - b, then ΔQ = Δa + Δb
tainties add. This rule is essential when processing data such as speed = distance ÷ time.
当量进行乘除运算时，百分不确定度相加。在处理如速度 = 距离 ÷ 时间等数据时，这一规则必不可少。
If Q = ab or Q = a ÷ b, then %ΔQ = %Δa + %Δb
e kinetic energy expression ½ mv² has a percentage uncertainty from v squared equal to 2 × %Δv.
当一个量被乘方时，百分不确定度要乘以该幂次。例如，动能表达式 ½ mv² 中，由 v 的平方带来的百分不确定度等于 2 × %Δv。
If Q = aⁿ, then %ΔQ = n × %Δa
8. Graphs and Error Bars | 图像与误差棒
Plotting a graph is a key skill in AQA Physics. The independent variable goes on the x-axis, the dependent variable on the y-axis. Axes must be labelled with quantity and unit, and the scale should use at least half the graph paper.
绘制图像是 AQA 物理的一项关键技能。自变量放在 x 轴，因变量放在 y 轴。坐标轴必须标出量和单位，并且坐标分度应至少使用图表纸的一半。
gradient = Δy / Δx
Error bars show the uncertainty in each plotted point. A best-fit line should pass through the error bars. If the line is curved, draw a smooth curve rather than joining points with straight lines.
误差棒表示每个数据点的不确定度。最佳拟合线应穿过误差棒。如果线是弯曲的，应画出平滑曲线，而不是用直线连接各点。
To estimate uncertainty in a gradient, draw the steepest and shallowest reasonable lines through the data points. The uncertainty is half the difference between these two gradients.
要估计斜率的不确定度，可画出穿过数据点的最陡和最浅两条合理直线。不确定度是这两个斜率之差的一半。
uncertainty in gradient = (max gradient - min gradient) ÷ 2
9. Significant Figures and Rounding | 有效数字与修约
Significant figures show the precision of a measured or calculated value. Leading zeros are not significant, but zeros between non-zero digits and trailing zeros after a decimal point are significant.
有效数字反映测量值或计算值的精密度。前导零不算有效数字，但非零数字之间的零以及小数点后的尾随零都是有效数字。
Your final answer should usually be quoted to the same number of significant figures as the least precise piece of data used in the calculation. For example, if a distance is given as 12.3 m and time as 2.1 s, the speed should not be given as 5.857 m s⁻¹ but as 5.9 m s⁻¹.
最终答案通常应与计算中使用的最不精确数据保持相同的有效数字位数。例如，如果距离为 12.3 m，时间为 2.1 s，那么速度不应写作 5.857 m s⁻¹，而应写作 5.9 m s⁻¹。
Uncertainties are usually quoted to one significant figure, or at most two. The measured value should then be rounded to the same decimal place as the absolute uncertainty.
不确定度通常保留一位有效数字，最多两位。测量值应修约到与绝对不确定度相同的小数位。
10. Planning and Reducing Errors | 实验设计与减小误差
A well-planned experiment reduces uncertainty and removes systematic bias. In timing experiments such as a pendulum, time 20 complete oscillations rather than one, and divide the total time by 20 to obtain the period.
经过良好设计的实验可以减小不确定度并消除系统偏差。在如单摆等计时实验中，应计时 20 个完整周期而不是一个，再用总时间除以 20 得到周期。
Use a fiducial marker at the centre of oscillation, not at the end, so the timing point is clearly defined. Repeat readings and use a digital instrument where possible to reduce random uncertainty.
在振动中心使用基准标记，而不是在端点，这样计时点更明确。重复读数，并尽可能使用数字仪器，以减小随机不确定度。
To reduce systematic error, check zero settings before use, calibrate instruments against known standards, and read scales perpendicular to avoid parallax.
为了减小系统误差，使用前检查零位，根据已知标准校准仪器，并垂直于刻度读取以避免视差。
Finally, evaluate the largest source of uncertainty in your procedure and suggest a practical improvement. This is a common AQA exam skill in practical-based questions.
最后，评估实验中最大的不确定度来源，并提出切实可行的改进方法。这是 AQA 实验类题目中常见的考试技能。
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