📚 A-Level Physics Unit 5 Insert Jan 20 Experiment: Spring-Mass Oscillator Investigation | A-Level 物理 Unit 5 2020年1月实验探究：弹簧振子研究
In the January 2020 A-Level Physics Unit 5 examination, the insert booklet provided the foundation for a practical investigation into the behaviour of a spring-mass oscillator. Students were presented with a laboratory scenario, raw timing data, and a series of prompts to process results, linearise relationships, calculate the spring constant, and evaluate uncertainties. This article reconstructs that experimental inquiry, guiding you through the complete analytical journey from raw measurements to a fully evaluated conclusion.
在2020年1月的A-Level物理Unit 5考试中，插页材料为一个弹簧振子实验探究提供了背景信息。试卷呈现了一个实验室场景、原始计时数据以及一系列处理数据、线性化关系、计算劲度系数和评估不确定度的要求。本文重构了这一实验探究过程，带你完整走过从原始测量到充分评估结论的分析之旅。
1. Experiment Overview and Insert Context | 实验概述与插页背景
The investigation centred on a helical spring suspended from a clamp stand with a slotted mass hanger. By adding known masses to the hanger, the system was set into vertical oscillations, and the time for multiple complete cycles was recorded. The main objective was to determine the spring constant k of the spring by analysing how the period T depends on the total suspended mass m. The insert supplied a diagram of the setup, a table of mass values, and three repeat time measurements for each mass.
该探究的核心是一个用夹具悬吊的螺旋弹簧，下方悬挂槽码盘。通过在码盘上添加已知质量，让系统作竖直振荡，并记录多个完整周期所需的时间。主要目标是通过分析周期T与悬挂总质量m的关系来确定弹簧的劲度系数k。插页提供了装置示意图、质量值表格以及每个质量对应的三次重复时间测量数据。
2. Underlying Physical Model | 基础物理模型
For an ideal mass-spring system, the period of vertical oscillations is given by T = 2π √(m/k) when the spring’s own mass is negligible. In practice, the spring has an effective mass ms that contributes to inertia. A more accurate model treats the system as T2 = (4π2/k)(m + meff), where meff is a fraction of the spring’s mass. This linear relationship between T2 and m allows k to be found from the gradient of a straight-line graph.
对于理想的弹簧振子系统，当忽略弹簧自身质量时，竖直振荡的周期由公式T = 2π √(m/k)给出。实际中，弹簧具有有效质量ms，会对惯性产生贡献。更精确的模型将系统处理为T2 = (4π2/k)(m + meff)，其中meff是弹簧质量的一部分。T2与m之间的线性关系使我们能够通过直线图的斜率求出k。
3. Raw Data from the Insert | 插页中的原始数据
A typical extract from the insert is shown below. Three trials of the time for 10 oscillations (t10) were recorded for each mass, reducing the impact of human reaction time on the period determination.
下方展示了插页中一组典型的数据。每个质量下记录了10次振荡所需时间(t10)的三次测量值，以减小人为反应时间对周期测定的影响。
| Mass m / kg | t10 Trial 1 / s | t10 Trial 2 / s | t10 Trial 3 / s |
|---|---|---|---|
| 0.100 | 4.82 | 4.79 | 4.84 |
| 0.200 | 6.75 | 6.72 | 6.78 |
| 0.300 | 8.15 | 8.10 | 8.20 |
| 0.400 | 9.30 | 9.25 | 9.35 |
| 0.500 | 10.30 | 10.25 | 10.35 |
4. Processing the Raw Data | 处理原始数据
For each mass, calculate the mean t10 and the absolute uncertainty Δt10 from the half-range (max − min)/2. The period for one oscillation is then T = t10/10, and its uncertainty ΔT = Δt10/10. Finally, compute T2 and the percentage uncertainty in T2 for plotting.
对每一质量，计算平均t10和绝对不确定度Δt10（由半区间宽(max − min)/2得出）。单次振荡周期为T = t10/10，其不确定度ΔT = Δt10/10。最后计算T2及其百分比不确定度，以便绘图。
The processed table illustrates these steps:
处理后的表格直观展示了这些步骤：
| m/kg | Mean t10/s | Δ t10/s | T/s | ΔT/s | T2/s2 |
|---|---|---|---|---|---|
| 0.100 | 4.817 | 0.025 | 0.4817 | 0.0025 | 0.232 |
| 0.200 | 6.750 | 0.030 | 0.6750 | 0.0030 | 0.456 |
| 0.300 | 8.150 | 0.050 | 0.8150 | 0.0050 | 0.664 |
| 0.400 | 9.300 | 0.050 | 0.9300 | 0.0050 | 0.865 |
| 0.500 | 10.300 | 0.050 | 1.0300 | 0.0050 | 1.061 |
5. Linearising the Relationship and Graphing | 线性化关系与绘图
According to T2 = (4π2/k)m + (4π2/k)meff, plotting T2 on the y-axis against m on the x-axis should yield a straight line. Both variables are plotted with error bars: vertical error bars represent ±Δ(T2), and horizontal error bars represent ±Δm (typically ±0.001 kg for standard slotted masses).
依据T2 = (4π2/k)m + (4π2/k)meff，以T2为y轴、m为x轴作图，应得到一条直线。两变量均用误差棒表示：纵向误差棒为±Δ(T2)，横向误差棒为±Δm（标准槽码通常为±0.001 kg）。
Students were expected to draw a line of best fit and, where appropriate, a worst-acceptable line to determine the gradient and intercept uncertainties. A graph of the processed data gives a gradient close to 1.90 s2 kg−1 and a positive intercept on the T2 axis, confirming the effective mass contribution.
要求学生绘制最佳拟合线，并在适当时绘制最差可接受线，以确定斜率和截距的不确定度。所处理数据的图形显示斜率接近1.90 s2 kg−1，且T2轴上具有正截距，证实了有效质量的贡献。
6. Determining the Spring Constant from the Gradient | 由斜率确定劲度系数
From the linear equation, gradient = 4π2/k. Therefore, k = 4π2 / gradient. Using the measured gradient of 1.90 s2 kg−1, we obtain k = (4 × 9.87) / 1.90 ≈ 20.8 N m−1. To estimate the uncertainty in k, the difference between the best-fit and worst-fit gradients is used: Δgradient ≈ ±0.10 s2 kg−1 leads to Δk ≈ 1.1 N m−1. The result is expressed as k = 20.8 ± 1.1 N m−1.
由线性方程可知，斜率 = 4π2/k。因此k = 4π2 / 斜率。用测得的斜率1.90 s2 kg−1计算，得k = (4 × 9.87) / 1.90 ≈ 20.8 N m−1。估计k的不确定度时，使用最佳拟合线与最差拟合线的斜率差值：Δ斜率 ≈ ±0.10 s2 kg−1，得Δk ≈ 1.1 N m−1。结果表示为k = 20.8 ± 1.1 N m−1。
7. Intercept and Effective Mass | 截距与有效质量
The y-intercept is given by (4π2/k)meff, so meff = intercept / gradient. With intercept ≈ 0.040 s2 and gradient ≈ 1.90 s2 kg−1, we find meff ≈ 0.021 kg. This value can be compared with the actual spring mass (measured separately as 0.065 kg) to check consistency; typically meff is about one-third of the spring’s mass, which agrees with theoretical expectations for a uniform spring.
y轴截距由(4π2/k)meff给出，因此meff = 截距 / 斜率。已知截距 ≈ 0.040 s2，斜率 ≈ 1.90 s2 kg−1，得meff ≈ 0.021 kg。该值可与单独测量的弹簧实际质量（0.065 kg）进行比较，以检查一致性；通常meff约为弹簧质量的1/3，与均匀弹簧的理论预期一致。
8. Uncertainty Analysis in Detail | 详细的不确定度分析
Percentage uncertainties combine individual contributions. For a typical point, T relies on t10; the dominant uncertainty arises from the reaction time variation in starting and stopping the stopwatch (±0.05 s on t10). Instruments like the digital balance (Δm = ±0.001 kg) contribute much less. The percentage uncertainty in T2 is twice that in T, doubling the influence.
百分比不确定度由各分量合成。对于某一数据点，T依赖于t10；主要不确定度来自启动和停止秒表时的反应时间差异（t10的±0.05 s）。电子天平（Δm = ±0.001 kg）等仪器的贡献要小得多。T2的百分比不确定度是T百分比不确定度的两倍，使影响加倍。
- For m = 0.100 kg: T = 0.4817 s, ΔT = 0.0025 s → %U(T) = 0.52%, %U(T2) = 1.04%
- For m = 0.500 kg: T = 1.0300 s, ΔT = 0.0050 s → %U(T) = 0.49%, %U(T2) = 0.98%
- 当m = 0.100 kg: T = 0.4817 s, ΔT = 0.0025 s → 百分比不确定度(T) = 0.52%, 百分比不确定度(T2) = 1.04%
- 当m = 0.500 kg: T = 1.0300 s, ΔT = 0.0050 s → 百分比不确定度(T) = 0.49%, 百分比不确定度(T2) = 0.98%
Error bars on the graph reflect these values and remain roughly constant in absolute size for larger m, indicating that increasing the number of oscillations (e.g. timing 20 swings) could improve precision.
图形上的误差棒反映了这些值，且在大m时绝对大小大致保持不变，表明增加振荡次数（如计时20次摆动）可提高精度。
9. Error Sources and Suggested Improvements | 误差来源与改进建议
Several systematic and random errors are present in this investigation. The most significant random error is human reaction time when using a manual stopwatch. A light gate or motion sensor connected to a data logger would drastically reduce this uncertainty. Systematic errors may include the non-linearity of the spring at large amplitudes or the mass of the hanger being overlooked. Ensuring oscillations remain small (amplitude < 5 cm) and zeroing the balance before weighing hanger plus masses addresses these.
该探究中存在若干系统误差和随机误差。最显著的随机误差是手动使用秒表时的人为反应时间。连接到数据采集器的光门或运动传感器将大幅降低这一不确定度。系统误差可能包括大振幅时弹簧的非线性或忽略了码盘的质量。确保振荡保持小幅度（振幅< 5 cm）并在称量码盘和槽码前对天平调零，可以解决这些问题。
- Use a fiducial marker at equilibrium to start and stop timing at the same point in the cycle.
- Repeat each measurement at least 5 times and use a larger oscillation count (20T) to reduce fractional uncertainty.
- Clamp the spring firmly and avoid sideways motion by guiding oscillations with a vertical rod.
- 在平衡位置设置基准标记，以便在周期内同一点开始和停止计时。
- 每项测量至少重复5次，并使用更大的振荡次数（20T）以降低相对不确定度。
- 稳固夹持弹簧，并通过竖直导杆限制摆动，避免横向运动。
10. Conclusion and Link to Unit 5 Assessment Objectives | 结论与Unit
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