📚 AS Physics Experimental Investigation: June 2019 Unit 1 Spring Systems | AS物理实验探究：2019年6月单元1弹簧系统
In the June 2019 Edexcel AS Physics Unit 1 examination, one of the core experimental investigation questions focused on the behaviour of springs under load, including series and parallel combinations. This article revisits that experiment, covering the underlying physics, practical techniques, data analysis, and common pitfalls. By studying this investigation in detail, students will strengthen their understanding of Hooke’s law, spring constants, and the principles of effective stiffness in spring systems.
在 2019 年 6 月 Edexcel AS 物理单元 1 考试中，其中一道核心实验探究题考察了弹簧在负载下的行为，包括串联和并联组合。本文重新审视该实验，涵盖其基本原理、实验技巧、数据分析及常见错误。通过深入学习这一探究，学生可以加深对胡克定律、劲度系数以及弹簧系统等效刚度原理的理解。
1. Introduction to the Experiment | 实验简介
The investigation required students to examine the extension of a single spring when different masses were attached, then to explore how the effective spring constant changes when two identical springs are arranged in series and in parallel. This classic experiment allows learners to verify Hooke’s law (F = kΔx) and to compare theoretical predictions with measured results for combined springs.
该实验要求学生研究单个弹簧悬挂不同质量时的伸长量，进而探索两个相同弹簧串联和并联时有效劲度系数的变化。这一经典实验让学习者得以验证胡克定律（F = kΔx），并比较组合弹簧的理论预测与实测结果。
2. Apparatus and Setup | 器材与装置
A typical setup includes a retort stand, a clamp, a boss, two identical light springs, a metre rule, a set of slotted masses (e.g. 100 g each), a mass hanger, a pointer, and a set square or vernier calipers for accurate length measurements. The spring is suspended vertically, and a pointer attached to its lower end helps read the position on the metre rule.
典型装置包括一个铁架台、一个夹子、一个卡头、两个相同的轻质弹簧、一把米尺、一组槽码（例如每个 100 g）、一个挂钩、一个指针以及用于精确测量长度的直角规或游标卡尺。弹簧竖直悬挂，指针固定在其下端，便于在米尺上读数。
3. Procedure for Single Spring | 单个弹簧实验步骤
Hang the spring freely and record the initial pointer position x₀ without any load. Add known masses m one by one, recording the new pointer position x each time. The extension is calculated as Δx = x – x₀. Ensure the spring oscillates briefly and comes to rest before taking readings to avoid friction-induced errors. Repeat the unloading process to check for hysteresis.
将弹簧自由悬挂，记录无负载时的初始指针位置 x₀。逐一增加已知质量 m，每次记录新的指针位置 x。伸长量计算为 Δx = x – x₀。确保弹簧短暂振荡后静止再读数，以避免摩擦引起的误差。重复卸载过程以检查是否存在迟滞现象。
4. Data Collection and Tabulation | 数据收集与表格
Design a table with columns for mass m (kg), weight F = mg (N), position x (m), and extension Δx (m). Typical data for a soft spring might look like this:
设计一个表格，包含质量 m（kg）、重力 F = mg（N）、位置 x（m）和伸长量 Δx（m）等列。一个软弹簧的典型数据可能如下：
| m / kg | F / N | x / m | Δx / m |
|---|---|---|---|
| 0.100 | 0.981 | 0.155 | 0.045 |
| 0.200 | 1.962 | 0.188 | 0.078 |
| 0.300 | 2.943 | 0.221 | 0.111 |
| 0.400 | 3.924 | 0.254 | 0.144 |
5. Graph Plotting and Analysis | 作图与分析
Plot a graph of force F (y‑axis) against extension Δx (x‑axis). According to Hooke’s law, the graph should be a straight line through the origin. The gradient of the line gives the spring constant k. If the line does not pass through the origin, check for systematic errors such as incorrect zero reading or a pre‑stretched spring.
绘制力 F（y 轴）对伸长量 Δx（x 轴）的图形。根据胡克定律，图形应为一条过原点的直线。直线斜率即为劲度系数 k。如果直线不过原点，需检查是否存在系统误差，例如零点读数不正确或弹簧已预先拉伸。
6. Determining Spring Constant k | 测定劲度系数 k
Using the gradient, k = ΔF / Δ(Δx). For the sample data, taking two points (0,0) and (0.144 m, 3.924 N) gives k ≈ 27.3 N m⁻¹. Express the final value with its absolute uncertainty, estimated from the scatter of points or the resolution of instruments. In the exam, you may be asked to calculate the percentage uncertainty and state the result as k ± Δk.
利用斜率计算 k = ΔF / Δ(Δx)。对于示例数据，选取（0,0）和（0.144 m, 3.924 N）两点，得出 k ≈ 27.3 N m⁻¹。应根据数据点的离散程度或仪器分辨率估算绝对不确定度，并最终表示为 k ± Δk。考试中可能要求计算百分不确定度并给出结果。
7. Series Combination of Springs | 弹簧串联组合
When two identical springs (each with constant k) are connected end‑to‑end, the effective spring constant kₛ is given by 1/kₛ = 1/k + 1/k = 2/k, so kₛ = k/2. This means the combination is softer, and the same force produces twice the extension of a single spring. The experimental setup must ensure the springs are linked without twisting, and the pointer must be repositioned.
当两个相同的弹簧（劲度系数均为 k）首尾相连时，等效劲度系数 kₛ 由 1/kₛ = 1/k + 1/k = 2/k 给出，即 kₛ = k/2。这意味着弹簧组更软，相同力产生的伸长量是单个弹簧的两倍。实验装置必须确保弹簧无扭结连接，且指针需重新定位。
8. Parallel Combination of Springs | 弹簧并联组合
For two identical springs arranged side‑by‑side supporting the same load, the effective spring constant kₚ = k + k = 2k. Here the system is stiffer, and for the same force the extension is halved. Care must be taken to keep both springs vertical and to distribute the mass equally, perhaps using a light bar.
对于两个相同弹簧并列支撑同一负载，等效劲度系数 kₚ = k + k = 2k。此时系统更硬，相同力作用下伸长量减半。需注意保持两弹簧竖直，并借助轻质横杆使质量均匀分配。
9. Experimental Verification of Effective Constants | 等效劲度系数的实验验证
Repeat the mass‑extension procedure for series and parallel setups. Plot F–Δx graphs for each case and find the gradients to obtain kₛ and kₚ experimentally. Compare these values with k/2 and 2k from the single‑spring measurement. A good agreement within experimental uncertainties confirms the theoretical models, while significant discrepancies may indicate misalignment or slipping of springs.
对串联和并联装置重复质量‑伸长量实验步骤。分别绘制 F–Δx 图并求斜率，以获得实验值 kₛ 和 kₚ。将这些值与单个弹簧测得的 k/2 和 2k 进行比较。如果在实验不确定度范围内吻合良好，则验证了理论模型；若存在显著偏差，可能表明弹簧未对齐或发生了滑移。
10. Sources of Error and Uncertainties | 误差与不确定度来源
Common errors include parallax when reading the metre rule, zero error in the ruler or pointer alignment, oscillation of the mass not fully ceased, and non‑linearity if the spring is overstretched beyond its elastic limit. The finite mass of the spring itself can cause a small offset. Ensure that g = 9.81 m s⁻² is used consistently and that repeated readings are taken for each load.
常见误差包括读取米尺时的视差、尺子或指针未对齐的零误差、重物未完全静止便开始读数，以及弹簧过度拉伸超出弹性极限导致的非线性。弹簧自身的质量也会引入微小偏移。应始终使用 g = 9.81 m s⁻²，并对每个负载进行重复读数。
11. Improvements and Further Investigation | 改进与深入探究
Use a digital vernier calliper or a motion sensor to record extension more precisely. A set square aligned with the pointer can reduce parallax. To investigate the transition to plastic deformation, add masses until the spring does not return to its original length. You can also study springs of different materials or coil diameters to see how k varies with geometric factors, linking to Young modulus concepts.
可以使用数字游标卡尺或运动传感器更精确地记录伸长量。用直角规对准指针可减少视差。为研究向塑性变形的过渡，可持续增加质量直至弹簧无法恢复原长。还可研究不同材料或不同圈径的弹簧，观察 k 如何随几何因素变化，这一探究与杨氏模量概念相关。
12. Exam-Style Tips | 考试技巧
In the Unit 1 paper, always label graph axes with quantities and units, draw a best‑fit line, and show the gradient triangle clearly. When explaining the series/parallel theory, derive from fundamentals: in series, each spring experiences the same force so extension adds; in parallel, forces add while extension is common. Mention assumptions like springs are identical and massless. Be prepared to calculate percentage difference between theoretical and experimental effective constants.
在单元 1 试卷中，务必为图轴标上物理量和单位，绘制最佳拟合线，并清晰地画出斜率三角形。解释串联/并联原理时，要从基础推导：串联时每根弹簧受力相同，伸长量相加；并联时力相加而伸长量相同。应提及弹簧相同且不计质量的假设。准备好计算理论等效劲度系数与实验值的百分差异。
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