📚 A-Level Physics Insert 4 Jan22: Experimental Investigation of a Spring | A-Level 物理：2022年1月第四套附加材料 弹簧实验探究
In the AQA A-Level Physics examination, insert booklets often supply experimental data or apparatus diagrams for Paper 3 Section B. Insert 4 from the January 2022 series presents a student’s investigation into the behaviour of a helical spring, including raw measurements of extension under increasing load. This article unpacks that insert, guiding you through the process of designing an experiment, recording data, plotting a graph, calculating the spring constant, and evaluating uncertainties — all essential skills for any aspiring physicist.
在AQA A-Level物理考试中，附加材料（insert）通常为Paper 3 Section B提供实验数据或装置示意图。2022年1月系列的第四套附加材料展示了一名学生对螺旋弹簧行为的探究，包括随载荷增加的伸长量原始测量数据。本文将解析这份附加材料，带你完成实验设计、数据记录、绘图、计算劲度系数以及评估不确定度的全过程——这些都是每一位物理学习者必备的技能。
1. Understanding the Insert 4 Scenario | 理解Insert 4情景
Insert 4 describes a laboratory setup where a spring is suspended vertically from a rigid clamp. A slotted mass hanger is attached to the free end, and masses are added in increments. The student records the total mass, converts it to force, and measures the resulting extension using a metre rule and a set square. A second metre rule is placed close to the spring to avoid parallax errors. The raw data table lists mass in grams, force in newtons, and extension in millimetres for five different loads.
附加材料4描述了一个实验室装置：一个弹簧竖直悬挂在坚固的夹具上，自由端挂有槽码吊架，并逐步增加质量。学生记录总质量，将其转换为力，并使用米尺和三角尺测量相应的伸长量。第二把米尺放置在弹簧附近以避免视差。原始数据表格列出了五种不同负载下的质量（克）、力（牛顿）和伸长量（毫米）。
2. Experimental Aim and Variables | 实验目的与变量
The primary aim is to determine the spring constant k of the spring and to verify Hooke’s Law, which states that the extension Δx is directly proportional to the applied force F, provided the elastic limit is not exceeded. In this investigation, the independent variable is the force applied (F), the dependent variable is the extension (Δx), and the control variables include the spring’s original length, room temperature, and the point of suspension to ensure consistency.
主要目的是测定弹簧的劲度系数 k，并验证胡克定律——即在不超过弹性限度的前提下，伸长量 Δx 与施加的力 F 成正比。本实验中，自变量是施加的力（F），因变量是伸长量（Δx），而控制变量包括弹簧原始长度、室温以及悬挂点位置，以保证一致性。
3. Apparatus and Setup | 仪器与装置
According to Insert 4, the apparatus consists of a helical spring, a clamp and stand, a slotted mass hanger with 100 g masses, a metre rule, a set square, and a second metre rule as a fiducial marker. The spring is hung so that its bottom end is level with the top of the second metre rule. A set square is used to align the bottom of the spring with the scale, reducing parallax error. The masses are added carefully to avoid oscillations before taking readings.
根据附加材料4，实验装置包括一个螺旋弹簧、夹具和支架、一个带有100 g槽码的吊架、一把米尺、一个三角尺以及用作参考标记的第二把米尺。弹簧悬挂时使其底端与第二把米尺的顶端齐平。使用三角尺将弹簧底部与刻度对齐，以减少视差误差。加载槽码时需轻放，以免在读数前引起振动。
4. Method and Procedure | 方法与步骤
The student first measures the unloaded length L₀ of the spring and records it. A 100 g mass is placed on the hanger, giving a total mass of 150 g (including the hanger). The new length L is measured, and the extension is calculated as Δx = L − L₀. The force is calculated using F = mg, where g = 9.81 m s⁻². This process is repeated for four additional masses, up to a total of 500 g. Care is taken to ensure the spring is stationary before each reading.
学生首先测量弹簧的无负载长度 L₀ 并记录。吊架上放置一个100 g的槽码，总质量（含吊架）为150 g。测量新长度 L，并按 Δx = L − L₀ 计算伸长量。力用 F = mg 计算，其中 g = 9.81 m s⁻²。重复此过程，增加另外四个质量，直至总质量达到500 g。每次读数前确保弹簧静止。
5. Raw Data from Insert 4 | 附加材料4的原始数据
The insert provides the following table of results, where the student has already converted mass to force and measured extensions. Note that all extensions are given to the nearest millimetre.
附加材料提供了以下结果表格，学生已经将质量转换为力并测量了伸长量。所有伸长量均精确到毫米。
| Force F / N | Extension Δx / mm |
|---|---|
| 0.00 | 0 |
| 1.47 | 42 |
| 2.94 | 84 |
| 4.41 | 125 |
| 5.89 | 167 |
| 7.36 | 210 |
You will immediately notice that the extension does not double exactly when the force doubles; this is due to measurement uncertainties and the real behaviour of the spring. Your task is to analyse this data to find the spring constant and assess the validity of Hooke’s Law.
你会立即注意到，当力加倍时伸长量并非精确加倍；这源于测量不确定度以及弹簧的实际行为。你的任务是分析这些数据，求出劲度系数，并评估胡克定律的有效性。
6. Plotting the Graph | 绘制图表
The standard approach is to plot a graph of force F (y-axis) against extension Δx (x-axis). According to Hooke’s Law, F = kΔx, so the gradient of the best-fit straight line will be equal to the spring constant k. Insert 4 instructs you to use the grid provided, choose appropriate scales, and label axes with units. Plot all six data points, and draw a line of best fit that passes through the origin as nearly as possible, since there should be zero extension at zero force.
标准方法是以力 F（y轴）对伸长量 Δx（x轴）作图。根据胡克定律 F = kΔx，因此最佳拟合直线的斜率等于劲度系数 k。附加材料4要求你使用提供的方格纸，选择合适的比例，并给坐标轴标上单位。标出所有六个数据点，画一条尽可能通过原点的最佳拟合线，因为当力为零时伸长量应为零。
7. Calculating the Spring Constant | 计算劲度系数
To find k from the graph, select two well-spaced points on the best-fit line — do not use the data points themselves unless they lie perfectly on the line. For instance, choose (Δx₁, F₁) = (50 mm, 1.78 N) and (Δx₂, F₂) = (180 mm, 6.40 N). Convert extension to metres: 50 mm = 0.050 m and 180 mm = 0.180 m. The gradient is calculated as
k = (F₂ − F₁) / (Δx₂ − Δx₁) = (6.40 − 1.78) / (0.180 − 0.050) = 4.62 / 0.130 ≈ 35.5 N m⁻¹
This value represents the spring constant. Depending on the quality of the line, you might obtain a slightly different value, but it should be around 35 to 36 N m⁻¹. Note that using points near the ends of the line maximises precision.
要从图中求出 k，需在最佳拟合线上选择两个间隔较大的点——不要直接使用数据点，除非它们恰好完全落在线。例如选择 (Δx₁, F₁) = (50 mm, 1.78 N) 和 (Δx₂, F₂) = (180 mm, 6.40 N)。将伸长量转换为米：50 mm = 0.050 m，180 mm = 0.180 m。斜率计算如上所示，得到 k ≈ 35.5 N m⁻¹。此值即劲度系数。根据拟合线的质量，可能得到略有不同的值，但应在 35 至 36 N m⁻¹ 范围内。注意使用线两端的点可最大限度提高精度。
8. Uncertainties and Error Analysis | 不确定度与误差分析
The metre rule has a precision of ±1 mm, so each length measurement carries an absolute uncertainty of ±1 mm. Since extension is the difference between two lengths, the uncertainty in Δx is ±2 mm. The mass values are given to the nearest 1 g, leading to a negligible force uncertainty (~0.01 N). The largest source of uncertainty is likely the judgement of the spring’s end point, which can be reduced by using a fiducial marker and a set square. To quote the spring constant with its uncertainty, you could draw worst-fit lines (steepest and shallowest) and calculate the gradient difference: e.g., k_max = 37.0 N m⁻¹, k_min = 34.2 N m⁻¹. The uncertainty Δk is then (k_max − k_min)/2 = 1.4 N m⁻¹. Hence, the final result is k = 35.5 ± 1.4 N m⁻¹.
米尺的精度为 ±1 mm，因此每次长度测量带有 ±1 mm 的绝对不确定度。由于伸长量是两个长度之差，Δx 的不确定度为 ±2 mm。质量值精确到 1 g，导致力的不确定度可忽略（约 0.01 N）。最大的不确定度来源可能是对弹簧端点位置的判断，可通过使用参考标记和三角尺来减小。为了给出带有不确定度的劲度系数，可以画出最陡和最浅的包络线，计算斜率差值：例如 k_max = 37.0 N m⁻¹，k_min = 34.2 N m⁻¹。不确定度 Δk = (k_max − k_min)/2 = 1.4 N m⁻¹。因此最终结果为 k = 35.5 ± 1.4 N m⁻¹。
9. Assessing Validity of Hooke’s Law | 评估胡克定律的适用性
A straight line through the origin confirms that the spring obeys Hooke’s Law within the tested range. However, if the data points show a clear curve or if the line does not pass close to the origin, this might indicate permanent deformation or exceeding the elastic limit. In the given data, the point (4.41 N, 125 mm) lies slightly above a straight line fit, possibly due to random error. It is essential to comment on any anomalous points and suggest improvements, such as taking repeat readings and averaging, or using a pointer attached to the spring for better alignment.
一条通过原点的直线表明弹簧在测试范围内遵循胡克定律。但如果数据点呈现出明显曲线，或直线并不接近原点，则可能表明发生了永久形变或超出了弹性限度。在所给数据中，点 (4.41 N, 125 mm) 略高于直线拟合，可能由随机误差引起。重要的是要针对异常点进行评论，并提出改进建议，比如重复读数取平均值，或在弹簧上附加指针以改善对齐。
10. Common Pitfalls and Examiner Tips | 常见错误与考官建议
Many candidates lose marks by forgetting to convert millimetres to metres in gradient calculations, or by using the data points instead of the best-fit line. Always draw a large graph occupying more than half the grid, label axes with quantities and units, and indicate the points used for gradient with a triangle symbol. When quoting final results, give the spring constant in N m⁻¹ and to an appropriate number of significant figures, usually 2 or 3. Also, be prepared to discuss whether the extension is directly proportional to force if the line does not pass through the origin.
许多考生因在斜率计算中忘记将毫米转换为米，或直接使用数据点而非最佳拟合线而失分。务必绘制占据方格纸一半以上的大图，用物理量和单位标记坐标轴，并用三角符号标明用于计算斜率的点。给出最终结果时，劲度系数以 N m⁻¹ 为单位，并使用适当的有效数字位数，通常为 2 或 3 位。此外，准备好讨论如果直线不通过原点，伸长量是否与力成正比。
11. Extending the Investigation | 实验拓展
Beyond the simple determination of k, you might investigate the energy stored in the spring, using the area under the F–Δx graph. The elastic potential energy E = ½kΔx² can be verified by comparing the work done by the falling masses. Alternatively, you could study the oscillation of the spring-mass system to determine the period T, and use T = 2π√(m/k) to find an independent value of k, comparing it with the static method.
除了简单测定 k 值，你还可以探究弹簧中储存的能量，利用 F–Δx 图线下的面积。弹性势能 E = ½kΔx² 可通过比较下落质量所做的功来验证。或者，你可以研究弹簧-质量系统的振动，测量周期 T，利用 T = 2π√(m/k) 求出独立的 k 值，并与静态法的结果进行比较。
12. Summary and Key Takeaways | 总结与核心要点
Insert 4 Jan22 serves as an excellent model for how A-Level Physics practical skills are assessed. By carefully analysing the supplied data, you develop competence in graphing, error analysis, and critical evaluation. Remember that a successful answer demonstrates not only correct numerical manipulation but also an understanding of experimental limitations and the logical justification of conclusions. Practising with past inserts is one of the most effective revision strategies for Paper 3.
2022年1月的第四套附加材料是A-Level物理实验技能评估的绝佳范例。通过仔细分析所提供的数据，你能提升绘图、误差分析和批判性评估的能力。请记住，一份满分的答案不仅要展示正确的数值处理能力，还要体现对实验局限性的理解以及对结论的逻辑论证。利用以往的附加材料进行练习是准备Paper 3最有效的复习策略之一。
Published by TutorHao | Physics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导Cancel reply