📚 AS Physics Unit 2 Insert Jan20: Spring Constant Investigation | AS物理单元2 插入页2020年1月：弹簧劲度系数探究
The AS Physics Unit 2 data-sheet insert from January 2020 provides a set of experimental readings designed to test your understanding of practical skills, data analysis, and uncertainty evaluation. In this article, we will walk through a typical spring constant investigation that could be presented using such an insert. You will learn how to process raw results, construct a force–extension graph, determine the spring constant k, assess uncertainties, and suggest improvements—all crucial skills for your AS practical assessment.
2020年1月的AS物理单元2数据表插入页提供了一组实验读数，旨在考查你对实验技能、数据分析和不确定度评估的理解。本文将带你走过一个可能出现在该插入页中的典型弹簧劲度系数探究实验。你将学会如何处理原始结果、绘制力–伸长量图像、确定弹簧的劲度系数k、评估不确定度并提出改进建议——这些都是AS实验考核中至关重要的技能。
1. Overview of the Insert | 插入页概述
The insert typically contains a labelled diagram of the apparatus, a list of quantities to measure, and a table of raw data recorded by a student. For a spring constant investigation, the setup usually includes a spring hanging vertically with a pointer, a ruler or Vernier scale to measure the spring length, and a set of slotted masses. The student records the unstretched length l₀ and then adds masses, measuring the new length l each time. The extension x is found by x = l − l₀.
插入页通常包含仪器的标注图、要测量的物理量列表以及学生记录的原始数据表格。在弹簧劲度系数的探究中，装置通常包括竖直悬挂的弹簧（带指针）、用于测量弹簧长度的刻度尺或游标尺，以及一组槽码。学生记录未拉伸时的原始长度l₀，然后增加质量，每次测量新的长度l。伸长量x由x = l − l₀得出。
The insert may also provide the value of gravitational field strength g = 9.81 N kg⁻¹ and remind candidates to convert mass from grams to kilograms. Your task is to use the processed data to find the spring constant and to discuss the reliability of the result.
插入页还可能提供重力场强度g = 9.81 N kg⁻¹，并提醒考生将质量的单位从克转换为千克。你的任务是利用处理后的数据求出弹簧的劲度系数，并讨论结果的可靠性。
2. Raw Data: Mass and Length | 原始数据：质量与长度
A typical set of raw data from the insert might look like the table below. The unstretched spring length l₀ has been measured as 22.5 mm. Additional masses are attached and the new lengths recorded.
插入页中一组典型的原始数据如下表所示。未拉伸时的弹簧长度l₀已测得为22.5 mm。添加额外质量后，记录下新的长度。
| Mass m / g | Spring length l / mm |
|---|---|
| 40.0 | 25.0 |
| 80.0 | 27.5 |
| 120.0 | 30.2 |
| 160.0 | 32.8 |
| 200.0 | 35.5 |
Before using these numbers, convert mass to kilograms and length to metres. Remember that precise unit conversion is a key practical skill. The extension x in metres is (l − 22.5) × 10⁻³ m. For the first row, x = (25.0 − 22.5) × 10⁻³ = 2.5 × 10⁻³ m.
使用这些数据前，需将质量转换为千克，长度转换为米。请牢记，准确的单位换算是一项关键的实验技能。以米为单位的伸长量x为 (l − 22.5) × 10⁻³ m。对第一行数据，x = (25.0 − 22.5) × 10⁻³ = 2.5 × 10⁻³ m。
3. Processing Data: Force vs Extension | 数据处理：力与伸长量
The force applied is the weight of the masses: F = mg, with g = 9.81 m s⁻². Calculate F for each mass and then create a new table of F against x. This step transforms the raw data into the quantities that Hooke’s law relates.
施加的力是质量所受的重力：F = mg，其中g = 9.81 m s⁻²。计算出每个质量对应的力，然后建立一个力F对伸长量x的新表格。这一步将原始数据转化为胡克定律所关联的物理量。
For example, when m = 0.040 kg, F = 0.040 × 9.81 = 0.3924 N, and x = 0.0025 m. Repeat for all five data points. The complete processed table is shown below.
例如，当m = 0.040 kg时，F = 0.040 × 9.81 = 0.3924 N，x = 0.0025 m。对所有五个数据点重复此操作。完整的处理后表格如下所示。
| Mass m / kg | Force F / N | Extension x / 10⁻³ m |
|---|---|---|
| 0.040 | 0.392 | 2.5 |
| 0.080 | 0.785 | 5.0 |
| 0.120 | 1.177 | 7.7 |
| 0.160 | 1.570 | 10.3 |
| 0.200 | 1.962 | 13.0 |
Notice that the extension values have been given to an appropriate number of significant figures based on the raw length measurements. Keeping consistent significant figures throughout the analysis is essential for grading.
请注意，伸长量的数值已根据原始长度测量的有效数字位数适当给出。整个分析过程中保持有效数字的一致性对于评分至关重要。
4. Plotting the Graph | 绘制图形
Plot a graph of force F on the y‑axis against extension x on the x‑axis. Use a sharp pencil and choose scales that make the plotted points occupy at least half of the grid in both directions. The graph should have a sensible origin—not necessarily (0,0) if the data lie far from zero, but here the spring starts unstretched so the origin is meaningful.
绘制以力F为纵轴、伸长量x为横轴的图像。使用削尖的铅笔，并选择使所描点至少占据网格纸两个方向中一半空间的坐标尺度。原点的选择要合理——如果数据点远离零点，则不一定从(0,0)开始；但在本实验中弹簧从自然长度开始，因此原点是有意义的。
Draw a line of best fit. Since Hooke’s law predicts F = kx, the points should lie on a straight line passing through the origin. Do not force the line through the origin; instead, judge by eye whether the intercept is close to zero. Include error bars if the insert provides uncertainty estimates for length or mass.
画出最佳拟合直线。由于胡克定律预测F = kx，数据点应落在一条通过原点的直线上。不要勉强让直线恰好经过原点；而是通过目测判断截距是否接近零。如果插入页给出了长度或质量的不确定度估计，可在图中加入误差棒。
5. Determining the Spring Constant | 确定劲度系数
The spring constant k is the gradient of the F–x graph. Select two widely spaced points on the line of best fit—not raw data points—and calculate:
弹簧劲度系数k是F–x图像的斜率。在最佳拟合直线上选择两个相距较远的点（非原始数据点），然后计算：
k = ΔF / Δx
Using the best-fit line, suppose one point reads (x₁ = 0.0010 m, F₁ = 0.15 N) and another reads (x₂ = 0.0120 m, F₂ = 1.80 N). Then ΔF = 1.80 − 0.15 = 1.65 N, Δx = 0.0120 − 0.0010 = 0.0110 m, giving k = 1.65 / 0.0110 = 150 N m⁻¹. Always quote the unit and round to the appropriate number of significant figures (here 150 N m⁻¹ or 1.50 × 10² N m⁻¹).
利用最佳拟合直线，假设取点 (x₁ = 0.0010 m, F₁ = 0.15 N) 和 (x₂ = 0.0120 m, F₂ = 1.80 N)。则 ΔF = 1.80 − 0.15 = 1.65 N，Δx = 0.0120 − 0.0010 = 0.0110 m，由此得出 k = 1.65 / 0.0110 = 150 N m⁻¹。务必注明单位，并根据有效数字适当修约（此处可写作150 N m⁻¹ 或 1.50 × 10² N m⁻¹）。
Check that the line passes close to the origin. The intercept from your chosen points can be estimated using the straight‑line equation F = kx + c. If c is small compared to the force values, the spring obeys Hooke’s law within experimental error.
检查直线是否接近原点。利用所选点可依据直线方程F = kx + c估算截距。若c相对于力的数值很小，则说明在实验误差范围内弹簧遵循胡克定律。
6. Uncertainty Analysis | 不确定度分析
The insert may ask you to determine the absolute uncertainty in k. A common method is to draw the steepest and shallowest lines of best fit that are still consistent with the error bars. The gradient of the steepest line (k_max) and the gradient of the shallowest line (k_min) define a range. The absolute uncertainty in k is approximately (k_max − k_min)/2.
插入页可能要求你确定k的绝对不确定度。常用的方法是画出仍与误差棒相符的最陡和最浅的最佳拟合直线。最陡直线的斜率（k_max）与最浅直线的斜率（k_min）给出一个范围。k的绝对不确定度约为 (k_max − k_min)/2。
For example, if the data scatter gives k_max = 160 N m⁻¹ and k_min = 142 N m⁻¹, then Δk = (160 − 142)/2 = 9 N m⁻¹. The result is reported as k = 150 ± 9 N m⁻¹. You can also calculate the percentage uncertainty: (9/150) × 100% = 6%.
例如，若由数据点的离散情况得出 k_max = 160 N m⁻¹，k_min = 142 N m⁻¹，则 Δk = (160 − 142)/2 = 9 N m⁻¹。结果可表示为 k = 150 ± 9 N m⁻¹。你还可以计算百分比不确定度：(9/150) × 100% = 6%。
If the insert provides explicit instrumental uncertainties, propagate them through the calculations. For instance, if the length measurement has an uncertainty of ±0.1 mm and mass ±0.5 g, convert these to standard uncertainties and combine them appropriately when computing extension and force.
如果插入页明确给出了仪器不确定度，则需在计算中传播这些不确定度。例如，若长度测量的不确定度为 ±0.1 mm，质量的不确定度为 ±0.5 g，则需将这些转化为标准不确定度，并在计算伸长量和力时进行适当合成。
7. Experimental Improvements | 实验改进
Questions on the insert often ask for suggestions to reduce uncertainty or improve reliability. For a spring experiment, you might suggest using a Vernier scale or a digital camera with a reference scale to measure length more precisely. Clamping the ruler vertically and aligning it carefully avoids parallax errors.
插入页上的问题常要求你提出减少不确定度或提高可靠性的建议。对于弹簧实验，你可以建议使用游标尺或带有参考标尺的数码相机来更精确地测量长度。将刻度尺竖直夹持并仔细对准，可避免视差误差。
Ensuring the spring is not overloaded beyond its elastic limit is crucial; otherwise permanent deformation leads to non‑linear behaviour. Repeating the readings for each mass multiple times and averaging reduces random errors. Also, lightly tapping the apparatus to overcome static friction at the pulley or pointer can improve data quality.
确保弹簧加载不超出其弹性极限至关重要；否则永久性形变会导致非线性行为。对每个质量下的读数进行多次重复测量并取平均值，可减小随机误差。此外，轻敲装置以克服滑轮或指针处的静摩擦，也能改善数据质量。
Using a longer spring or a set of springs in series can increase the extension for a given mass, thereby reducing the percentage uncertainty in length measurements. These points are commonly credited in practical‑based exam questions.
使用更长的弹簧或将弹簧串联可以增加给定质量下的伸长量，从而减小长度测量的百分比不确定度。这些要点在以实验为基础的考题中常能得到给分。
8. Application: Measuring g | 应用：测量重力加速度
Once the spring constant k has been determined, the same setup can be used to measure the local gravitational field strength g. By oscillating the spring vertically and measuring the period T for small oscillations, we can use the formula:
一旦确定了劲度系数k，同一装置还可用于测量当地重力场强度g。通过让弹簧在竖直方向上作小幅振动并测量其周期T，我们可以利用公式：
T = 2π √(m/k)
However, this expression ignores the mass of the spring. A more accurate treatment includes an effective mass m_eff = m + m_s/3, where m_s is the spring’s mass. By plotting T² against m and finding the gradient, g can be extracted if k is known, or k can be checked independently.
然而，该表达式忽略了弹簧本身的质量。更精确的处理需引入有效质量 m_eff = m + m_s/3，其中 m_s 为弹簧的质量。通过绘制 T²–m 图像并求其斜率，若已知k便可得出g，或可独立检验k。
The insert might provide a set of period data alongside the static extension data, allowing you to compare static and dynamic methods. Agreement within experimental uncertainty supports the validity of Hooke’s law and the simple harmonic motion model.
插入页可能在静态伸长数据之外还提供一组周期数据，让你能够对静态法和动态法进行比较。若结果在实验不确定度范围内一致，则支持胡克定律和简谐运动模型的有效性。
9. Common Pitfalls and How to Avoid Them | 常见错误与规避方法
Forgetting to convert units is a frequent mistake. Always record mass in kg and length in metres before substituting into F = mg or k = F/x. Another pitfall is using the stretched length l instead of the extension x. Always subtract the original length l₀.
忘记换算单位是常见错误。务必在代入 F = mg 或 k = F/x 之前将质量换算为kg、长度换算为m。另一个陷阱是使用拉伸后的长度l而非伸长量x。务必减去原始长度l₀。
Misinterpreting the intercept on the F–x graph can lead to incorrect conclusions. A small positive intercept might indicate that a small initial force was needed before measurable extension began—often due to a tightly coiled spring or zero error in the length scale. Annotate the graph clearly to show that this is an experimental systematic error.
错误解读F–x图像上的截距可能导致错误结论。一个小的正截距可能表明在可测得的伸长量出现之前需要一小点的力——通常是由于弹簧紧密缠绕或长度尺的零点误差所致。在图上清晰标注，说明这是一个实验的系统误差。
Drawing error bars incorrectly can mask the uncertainty. If the length uncertainty is ±1 mm, the error bar on x should extend ±1 mm in the horizontal direction. Similarly, if the mass uncertainty is ±1 g, the vertical error bar corresponds to ±mg. Showing error bars allows you to justify whether a straight line is a good fit.
错误地绘制误差棒可能掩盖不确定度。若长度不确定度为 ±1 mm，则x的误差棒应在水平方向延伸 ±1 mm。类似地，若质量不确定度为 ±1 g，则竖直误差棒对应 ±mg。画出误差棒可使你有理由判断直线拟合是否合理。
10. Linking to Exam Questions | 与考题的关联
In the exam, you might be asked to complete the table of processed data, plot the graph, and determine k. You could also be given a second spring and asked to predict the behaviour of springs in series or parallel. Recall that for springs in parallel, k_total = k₁ + k₂, and for springs in series, 1/k_total = 1/k₁ + 1/k₂.
在考试中，你可能会被要求完成处理后的数据表格、绘制图像并确定k。你也可能被给定另一根弹簧，并被要求预测弹簧串联或并联时的行为。请记住，对于并联弹簧，k_total = k₁ + k₂；对于串联弹簧，1/k_total = 1/k₁ + 1/k₂。
Questions on significant figures and uncertainty calculations are very common. Practise stating final results with the correct absolute or percentage uncertainty and rounding to match the least precise input. Mark schemes reward clear working and correct units.
与有效数字和不确定度计算相关的问题非常常见。请练习在陈述最终结果时给出正确的绝对或百分比不确定度，并根据精度最低的输入进行修约。评分标准奖励清晰的解题步骤和正确的单位。
Additionally, you may be required to evaluate the method. Comment on whether repeating readings, using a digital balance, or a motion sensor would improve accuracy. Always support your suggestions with physics reasoning.
此外，你还可能被要求评价实验方法。评述重复读数、使用数字天平或运动传感器是否能提高准确性。请始终用物理学原理来支持你的建议。
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