📚 A-Level Physics: Experiment Investigations Using the Data and Formula Booklet (Paper 1, Jan 2018) | A-Level物理：利用数据与公式册进行实验探究（2018年1月试卷1）
Mastering experimental analysis in A-Level Physics requires more than just memorising equations – it demands a deep understanding of how to use the Data and Formula Booklet provided in Paper 1. The January 2018 booklet is a powerful tool that supplies essential constants, formulae, and conversion factors. This article explores key experiments, shows how to link them to the booklet’s resources, and develops the practical skills needed to excel in examination questions on investigations.
掌握A-Level物理的实验分析，不仅仅需要记忆公式——它要求深刻理解如何在试卷1提供的《数据与公式册》中获取信息。2018年1月的这本册子是强有力的工具，提供了基本常数、公式和换算因子。本文探究关键实验，展示如何将它们与册子中的资源关联起来，并培养在考试中应对实验探究题目所需的实用技能。
1. The Role of the Data and Formula Booklet in Experiments | 数据与公式册在实验中的角色
The January 2018 booklet contains fundamental quantities such as the acceleration of free fall (g = 9.81 m s⁻²), the Planck constant, and the permittivity of free space. In an experiment question, you are expected to identify which given constant or formula is relevant, and then apply it with your measured data to calculate a target quantity like resistivity or the Young modulus.
2018年1月的册子包含基本量，如自由落体加速度（g = 9.81 m s⁻²）、普朗克常数和真空介电常数。在实验题中，你需要识别哪个给定的常数或公式与之相关，然后将它与测量数据一起应用，计算出目标量，如电阻率或杨氏模量。
Often, the booklet will list alternative forms of an equation. For instance, the time period of a simple pendulum appears as T = 2π√(l/g). You might need to rearrange it to plot a straight-line graph, a skill heavily tested in Paper 1 practical analysis.
册子常常列出一个方程的多种形式。例如，单摆的周期公式为 T = 2π√(l/g)。你可能需要将其变形以绘制直线图，这是试卷1中实验分析重点考查的技能。
2. Key Constants You Must Locate Instantly | 必须能立刻定位的关键常数
During the exam, wasting time searching for a constant is costly. The January 2018 booklet typically groups constants near the front. Common ones include: gravitational field strength g = 9.81 N kg⁻¹, electron charge e = 1.60 × 10⁻¹⁹ C, and the speed of light in vacuum c = 3.00 × 10⁸ m s⁻¹. In an experiment, if you are measuring the force on a wire in a magnetic field, you will need e or the permeability of free space μ₀ = 4π × 10⁻⁷ H m⁻¹, all found in the booklet.
考试期间，花时间搜寻常数是得不偿失的。2018年1月的册子通常将常数集中在前部。常见的包括：重力场强 g = 9.81 N kg⁻¹、电子电荷 e = 1.60 × 10⁻¹⁹ C 和真空中光速 c = 3.00 × 10⁸ m s⁻¹。在实验中，若测量磁场中通电导线所受的力，你会需要 e 或真空磁导率 μ₀ = 4π × 10⁻⁷ H m⁻¹，这些均在册子中。
Practise highlighting or mentally bookmarking the page that lists these values so that you can cite them accurately in your experimental write-up or when performing calculations.
练习对列出这些值的页面进行高亮或心理标记，以便在实验报告或进行计算时准确引用。
3. Experiment 1: Determining g Using a Free Fall Method | 实验1：用自由落体法测定g
A classic investigation involves dropping a steel ball from rest and recording the time taken to fall through a measured distance. The relevant equation from the booklet is s = ut + ½at². With u = 0, we obtain s = ½gt². By plotting a graph of s against t², the gradient equals ½g. The booklet provides the accepted value of g as 9.81 m s⁻², so you can calculate a percentage difference.
一个经典探究是让钢球从静止下落，记录下落一段测量距离所用的时间。册子中的相关公式是 s = ut + ½at²。因 u = 0，可得 s = ½gt²。通过绘制 s 对 t² 的图像，斜率等于 ½g。册子给出了 g 的标准值为 9.81 m s⁻²，因此你可以计算百分误差。
s = ½ g t² → gradient = ½ g
To reduce uncertainty, an electromagnet and electronic timer or light gates are used. You must also measure the distance s with a metre ruler, correcting for parallax error. The final experimental value of g should be compared with the booklet’s constant, and possible sources of systematic error, such as air resistance or timer delay, must be discussed.
为减小不确定度，会用到电磁铁和电子计时器或光闸。你还必须用米尺测量距离 s，并纠正视差。最终实验测得的 g 值应与册子中的常数比较，并需讨论可能的系统误差来源，如空气阻力或计时器延迟。
4. Experiment 2: Resistivity of a Wire | 实验2：导线的电阻率
The resistivity ρ of a metal wire can be found using the formula from the booklet: R = ρL/A, where A is the cross-sectional area. By measuring the potential difference V and current I for a known length L, you can calculate R = V/I. The area A is determined from the diameter d using A = πd²/4. The booklet often lists the formula for area of a circle, but if not, you need to recall it.
金属导线的电阻率 ρ 可用册子中的公式求得：R = ρL/A，其中 A 是横截面积。通过测量已知长度 L 两端的电势差 V 和电流 I，可算出 R = V/I。面积 A 由直径 d 用 A = πd²/4 求出。册子可能列出圆面积公式，但若未列出，则需自行回忆。
Rearrange to ρ = RA/L. A graph of R against L yields a straight line through the origin with gradient ρ/A. By measuring d with a micrometer screw gauge, you can thus determine ρ. The booklet may provide the resistivity of copper (1.72 × 10⁻⁸ Ω m) at a specific temperature, which serves as a reference for your result.
变形得 ρ = RA/L。绘制 R 对 L 的图像，将得到一条过原点的直线，斜率为 ρ/A。通过用千分尺测量 d，即可求出 ρ。册子可能提供特定温度下铜的电阻率（1.72 × 10⁻⁸ Ω m），作为结果的参考值。
5. Experiment 3: Investigating Newton’s Second Law | 实验3：探究牛顿第二定律
Newton’s second law states F = ma, which is prominent in the booklet. A common experiment uses a trolley on a friction-compensated ramp, pulled by a falling mass m. The accelerating force is mg, and the total mass of the system (M + m) is accelerated. The equation becomes mg = (M + m)a. By plotting a against 1/(M+m) for constant force, a straight line confirms the law.
牛顿第二定律 F = ma 在册子中十分醒目。一个常见实验使用在补偿摩擦力斜面上的小车，由下落质量 m 牵引。加速力为 mg，整个系统的质量（M + m）被加速。方程变为 mg = (M + m)a。通过保持力恒定，绘制 a 对 1/(M+m) 的图像，直线关系可验证该定律。
Alternatively, keeping total mass constant and varying the force allows you to plot a against F, giving gradient 1/(total mass). The booklet’s value of g is used to compute force. Care must be taken to ensure the ramp is correctly adjusted so that the trolley moves with constant velocity when pushed gently, eliminating friction.
另一种方法是保持总质量不变、改变力，绘制 a 对 F 的图像，斜率为 1/(总质量)。册子中 g 的数值用于计算力。必须注意正确调节斜面，使得轻推小车后它能匀速运动，从而消除摩擦力。
6. Experiment 4: Young Modulus of a Wire | 实验4：金属丝的杨氏模量
The Young modulus E is defined in the booklet as tensile stress / tensile strain = (F/A) / (ΔL/L). A long, thin wire is loaded with known weights F, and the extension ΔL is measured using a marker on the wire and a travelling microscope or ruler. The original length L and diameter d must be measured precisely to find area A = πd²/4.
杨氏模量 E 在册子中定义为拉伸应力/拉伸应变 = (F/A) / (ΔL/L)。用已知重物 F 加载一根长细丝，通过丝上的标记和读数显微镜或尺子测量伸长量 ΔL。原始长度 L 和直径 d 需精确测量，以求得面积 A = πd²/4。
A graph of F against ΔL is plotted. From the linear region, the gradient = EA/L. Thus E = gradient × (L/A). The booklet might list typical values for steel (2.0 × 10¹¹ Pa) or copper, allowing a comparison. One must ensure the elastic limit is not exceeded; otherwise, the graph will curve, and the linear portion must be used.
绘制 F 对 ΔL 的图像。在线性区域内，斜率 = EA/L。因此 E = 斜率 × (L/A)。册子可能列出钢（2.0 × 10¹¹ Pa）或铜的典型值，以作比较。必须确保不超过弹性极限；否则图像会弯曲，此时应使用线性部分。
7. Graphical Analysis and Linearisation | 图像分析与线性化处理
Many Paper 1 experimental questions ask you to show how a given formula can be rearranged into a straight-line form y = mx + c. For instance, the discharge of a capacitor, V = V₀e-t/RC, becomes ln V = ln V₀ – t/RC. A plot of ln V against t yields a straight line with gradient -1/RC. The booklet provides the exponential decay equation, but you must perform the linearisation.
试卷1的许多实验题要求你展示如何将给定公式变形为直线形式 y = mx + c。例如，电容器的放电方程 V = V₀e-t/RC 可变为 ln V = ln V₀ – t/RC。绘制 ln V 对 t 的图像，得到一条直线，其斜率为 -1/RC。册子提供了指数衰减方程，但你必须自行进行线性化处理。
Accurate graphs require sensible scales, labelled axes with units, and careful plotting. The gradient and intercept should be calculated using a large triangle. Your data analysis must reference the formula booklet’s equations to link the gradient to the physical quantity sought.
绘制精确图像需要合理的坐标比例、标注了单位的坐标轴，以及细致的描点。应使用大三角形来计算斜率和截距。数据分析必须引用公式册中的方程，以将斜率与所求的物理量联系起来。
8. Uncertainty Calculations and the Booklet | 不确定度计算与公式册
The booklet often includes statistical formulas for percentage uncertainty, combining uncertainties, and absolute/relative forms. For a measurement like diameter d taken with a micrometer (resolution 0.01 mm), the absolute uncertainty might be ±0.01 mm, leading to a percentage uncertainty in area of 2 × (% uncertainty in d). When you finally calculate resistivity, you must combine uncertainties using the rules, such as adding percentage uncertainties for multiplication.
册子常包含统计公式，如百分不确定度、不确定度的合成，以及绝对/相对形式。对于用千分尺（分辨力 0.01 mm）测量的直径 d，绝对不确定度可能为 ±0.01 mm，导致面积的百分不确定度为 2 ×（d 的百分不确定度）。当你最终计算电阻率时，须用合成规则，例如乘法运算时百分不确定度相加。
Where explicit rules are not in the booklet, the principle of half the smallest division for analogue instruments and the given resolution for digital instruments are standard. Your answer should always be compared with the data booklet’s accepted value, and the percentage difference quoted.
若册子中没有明确给出规则，模拟仪器采用最小分度值的一半、数字仪器采用显示分辨力，这是标准做法。你的答案始终应与数据册中的标准值进行比较，并列出百分差异。
9. Using the Formula List for Experimental Design | 利用公式表进行实验设计
A less obvious but vital skill is reverse-engineering an experiment from a formula. Suppose you are given E = ½ kx² for elastic potential energy. You could be asked to design an experiment to determine the spring constant k. The formula suggests measuring the energy stored indirectly by allowing the spring to launch a mass and measuring its kinetic energy (½mv²) using a motion sensor. The booklet’s kinematic formulas may then be needed.
一项不那么明显但至关重要的技能是，从某个公式出发逆向推导一个实验。假设给你弹性势能公式 E = ½ kx²。可能会要求设计一个实验来测定弹簧的劲度系数 k。公式提示，可以通过让弹簧弹射一个质量，并用运动传感器测量其动能（½mv²）来间接测量储存的能量。此时可能就需要册子中的运动学公式。
Being able to spot the direct or indirect measurement possibilities in each equation in the booklet is what separates high achievers. The January 2018 booklet might also include aspirational constants like the unified atomic mass unit (u = 1.66 × 10⁻²⁷ kg), which opens up nuclear experiments.
能够发现册子中每个方程所蕴含的直接或间接测量可能性，正是高分学生脱颖而出的原因。2018年1月的册子还可能包含一些并非每个实验都会用到的常数，如原子质量单位 u = 1.66 × 10⁻²⁷ kg，这就为核物理实验提供了可能。
10. Common Pitfalls and Examiner Advice | 常见陷阱与考官建议
Examiners frequently report that candidates fail to link their calculated value with the booklet’s constant, or they misquote the constant. Another pitfall is using the wrong number of significant figures – the data booklet often provides constants to 2 or 3 significant figures, so your final answer should match this. Also, when the booklet gives a formula in terms of specific symbols, maintain those symbols exactly in your working; do not invent new variables.
考官经常报告称，考生未能将计算值与册子中的常数关联起来，或者错误地引用了常数。另一个陷阱是使用了错误的有效数字位数——数据册中的常数通常给出2或3位有效数字，因此你的最终答案应与之匹配。此外，当册子用特定符号给出公式时，在解题过程中必须严格使用这些符号；不要自创变量。
A final tip: always annotate the booklet gently with a pencil if allowed; mark the constants you have used and note the page reference in your answer. This demonstrates good experimental practice and makes your reasoning clear to the examiner.
最后一条建议：若允许，用铅笔轻轻在册子上作注释；标记出你用过的常数，并在答案中注明页码出处。这展示了良好的实验规范，并使你的推理对考官一目了然。
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