📚 A-Level Physics Unit 5 Jan 19: Experimental Investigation into Spring-Mass Oscillations | A-Level物理2019年1月Unit 5实验探究：弹簧振子振动研究
The January 2019 Edexcel IAL Physics Unit 5 paper featured a comprehensive experimental investigation that tested students’ ability to plan, execute, and analyse a physics experiment. This question centred on investigating the relationship between the period of oscillation of a spring-mass system and the attached mass, aiming to determine the spring constant and assess uncertainties.
2019年1月的爱德思国际A-Level物理Unit 5试卷中包含了一道综合性实验探究题，考查学生设计、执行和分析物理实验的能力。该题围绕弹簧振子的周期与悬挂质量之间的关系进行探究，目标是测定弹簧的劲度系数并评估不确定度。
1. Experiment Overview | 实验概述
This experiment uses a vertically suspended spring with a mass hanger. By adding slotted masses and setting the system into small-amplitude oscillation, the period T is measured as a function of the total hanging mass m. For small displacements, the motion approximates simple harmonic motion, and the period is given by T = 2π √( (m + ms) / k ), where k is the spring constant and ms is the effective mass of the spring itself.
本实验使用一根垂直悬挂的弹簧和载物架。通过添加槽码并让系统进行小幅度振动，测量周期T随悬挂总质量m的变化。在小位移条件下，运动近似为简谐运动，周期公式为 T = 2π √( (m + ms) / k )，其中k为弹簧劲度系数，ms为弹簧本身的有效质量。
Squaring both sides yields T² = (4π²/k)·m + (4π²/k)·ms. Plotting T² against m allows determination of k from the gradient, and ms from the intercept.
将公式两边平方得到 T² = (4π²/k)·m + (4π²/k)·ms。通过绘制 T²-m 图像，可从斜率求出k，并从截距求出ms。
2. Apparatus and Setup | 实验器材与装置
The following equipment was used: a helical spring with pointer, a rigid clamp and stand, a metre rule, a set of slotted masses (50 g each) on a 50 g hanger, a stopwatch capable of ±0.01 s, a fiducial marker, and a set square to ensure vertical alignment.
使用的器材包括：带指针的螺旋弹簧、刚性支架和铁架台、米尺、一套50 g槽码和一个50 g的载物架、精度±0.01 s的秒表、基准标记以及用于确保垂直对齐的三角尺。
The spring was suspended from the clamp so that the bottom of the hanger could oscillate freely without touching the bench. A fiducial marker was placed at the equilibrium point to aid consistent timing.
弹簧悬挂于铁架台上，使载物架底部能自由振动而不触碰桌面。在平衡位置放置一个基准标记，以便在一致的位置进行计时。
The metre rule was aligned vertically behind the spring to measure the static extension for an alternative determination of k and to set the oscillation amplitude.
米尺垂直安装在弹簧后方，用于测量静态伸长量（作为测定k的替代方案）并设定振动幅度。
3. Procedure | 实验步骤
1. Attach the hanger to the spring and record its mass. Add slotted masses to obtain total masses of 0.100 kg, 0.150 kg, 0.200 kg, 0.250 kg, 0.300 kg and 0.350 kg.
1. 将载物架挂在弹簧上并记录其质量。添加槽码以获得0.100 kg、0.150 kg、0.200 kg、0.250 kg、0.300 kg和0.350 kg的总悬挂质量。
2. For each mass, pull the hanger down by a small distance (about 2–3 cm) and release gently to avoid sideways swing. Allow a few oscillations to stabilise.
2. 对每一质量，将载物架向下拉动一小段距离（约2–3 cm）后轻放，避免左右晃动。先让系统振动几次以进入稳定状态。
3. Using the fiducial marker as reference, measure the time for 20 complete oscillations using the stopwatch. Repeat twice for each mass to obtain three readings.
3. 以基准标记为参考，用秒表测量20个完整振动的时间。每个质量重复测量两次，共获得三组读数。
4. Calculate the mean time t for 20 oscillations, then determine the period T = t / 20. Also calculate T².
4. 计算20次振动的平均时间t，进而求得周期 T = t / 20，并计算 T²。
5. Record all data in a suitable table, including uncertainty estimates for m and T.
5. 将所有数据记录在合适的表格中，包括质量m和周期T的不确定度估算值。
4. Data Collection and Raw Data Table | 数据收集与原始数据表
Typical data from a student’s investigation are shown below. The mass values have an uncertainty of ±0.002 kg due to the balance used. The stopwatch reading uncertainty for each single 20-oscillation timing is ±0.2 s (reflecting human reaction time), giving an absolute uncertainty in T of ±0.01 s for each measurement of T.
以下展示了一份典型的学生实验数据。质量值因所用天平存在±0.002 kg的不确定度。单次20个振动时间的秒表读数不确定度为±0.2 s（反映人为反应时间），这导致每个T的绝对不确定度为±0.01 s。
| m / kg | t1 / s | t2 / s | t3 / s | Mean t / s | T / s | T² / s² |
|---|---|---|---|---|---|---|
| 0.100 | 12.04 | 12.22 | 12.39 | 12.22 | 0.611 | 0.373 |
| 0.150 | 14.49 | 14.62 | 14.32 | 14.48 | 0.724 | 0.524 |
| 0.200 | 16.75 | 16.81 | 16.92 | 16.83 | 0.842 | 0.709 |
| 0.250 | 18.91 | 19.08 | 19.18 | 19.06 | 0.953 | 0.908 |
| 0.300 | 20.93 | 21.02 | 20.85 | 20.93 | 1.047 | 1.096 |
| 0.350 | 22.64 | 22.71 | 22.58 | 22.64 | 1.132 | 1.281 |
The maximum percentage uncertainty in T² can be estimated as twice that in T, which is approximately 2 × (0.01 / 0.611) ≈ 3.3% for the smallest mass. This uncertainty is used to plot error bars on the T² axis.
T²的最大百分不确定度可估算为T百分不确定度的两倍，对于最小质量约为 2 × (0.01 / 0.611) ≈ 3.3%。该不确定度用于在T²轴上绘制误差棒。
5. Graph Plotting | 绘制图像
Plot a graph of T² (y-axis) against m (x-axis). Use a suitable scale that spreads the data points over at least half the graph grid in each direction. Label the axes as “T² / s²” and “m / kg”. Draw error bars for T² using the calculated uncertainty, and include a small horizontal uncertainty bar for m (±0.002 kg).
绘制 T² (y轴) 随 m (x轴) 变化的图像。选择合适的坐标轴分度，使数据点占据每方向至少一半的图纸空间。将坐标轴标注为“T² / s²”和“m / kg”。使用计算出的不确定度绘制T²的误差棒，同时为m绘制水平误差棒（±0.002 kg）。
The data should show a linear trend. Draw a best-fit straight line that passes as close as possible to the error bars. Then draw a worst-acceptable line (steepest or shallowest line that still reasonably fits the error bars) to determine the uncertainty in the gradient.
数据应呈现线性趋势。画一条尽可能穿过所有误差棒的最佳拟合直线。然后画一条最差可接受线（最陡或最平缓但仍合理符合误差棒的直线），用于确定斜率的不确定度。
6. Gradient and Determination of k | 斜率与k值的确定
From the best-fit line, select two points far apart on the line (not data points) and calculate the gradient G = Δ(T²) / Δm. For the sample data, a gradient of about 3.60 s²/kg is typical. Since G = 4π²/k, we find k = 4π² / G.
从最佳拟合直线上选取两个相距较远的点（非原始数据点），计算斜率 G = Δ(T²) / Δm。对于示例数据，典型斜率约为3.60 s²/kg。由 G = 4π²/k 可得 k = 4π² / G。
k = 4π² / G
Substituting G = 3.60 s²/kg gives k = 4π² / 3.60 ≈ 10.97 N/m. This should be rounded to an appropriate number of significant figures, e.g. 11.0 N/m, considering the uncertainties.
代入 G = 3.60 s²/kg 得到 k = 4π² / 3.60 ≈ 10.97 N/m。考虑到不确定度，应修约为适当的有效数字位数，例如11.0 N/m。
Determine the gradient uncertainty ΔG from the difference between the best-fit and worst-acceptable line gradients. Then the percentage uncertainty in k equals the percentage uncertainty in G, because k ∝ 1/G.
由最佳拟合线与最差可接受线斜率之差求出斜率不确定度ΔG。因为 k ∝ 1/G，所以k的百分不确定度等于G的百分不确定度。
7. Determining the Effective Mass of the Spring | 弹簧有效质量的测定
The y-intercept c of the T² vs. m graph equals (4π²/k)·ms. Once k is known, ms = c·k / (4π²). For the sample data, the intercept might be approximately 0.015 s², giving ms = (0.015 × 11.0) / (4π²) ≈ 0.0042 kg, i.e. about 4 g.
T²–m图像的纵轴截距c等于 (4π²/k)·ms。一旦k已知，ms = c·k / (4π²)。对于示例数据，截距约为0.015 s²，得出 ms = (0.015 × 11.0) / (4π²) ≈ 0.0042 kg，即约4 g。
This effective mass is typically around one-third of the actual spring mass. The small value obtained here indicates a lightweight spring, which is consistent with the assumption of SHM.
这一有效质量通常约为弹簧实际质量的三分之一。这里获得的较小数值表明弹簧很轻，这与简谐运动假设相符。
8. Uncertainty Analysis in Detail | 不确定度的详细分析
The absolute uncertainty in T is derived from the spread of the three timings. For the smallest mass, the range of T values is (12.39 – 12.04)/20 = 0.0175 s. Half of this range gives about ±0.009 s, confirming the ±0.01 s estimate used for error bars.
T的绝对不确定度来源于三次计时的分散程度。对于最小质量，T值范围是 (12.39 – 12.04)/20 = 0.0175 s。该范围的一半约为±0.009 s，验证了误差棒所用的±0.01 s估算值。
Percentage uncertainty in T² is twice that in T. For the largest mass, the percentage uncertainty in T is smaller, leading to slightly smaller error bars. Students are expected to plot error bars accordingly and comment on whether the line fits within them.
T²的百分不确定度是T的两倍。对于最大质量，T的百分不确定度更小，因此误差棒略小。学生应据此绘制误差棒，并评论直线是否符合误差棒范围。
Combining uncertainties in gradient gives a final value of k = 11.0 ± 0.5 N/m, once the worst-fit line gradient is determined.
在确定最差拟合线斜率后，结合斜率不确定度得到最终结果 k = 11.0 ± 0.5 N/m。
9. Sources of Error and Improvements | 误差来源与改进方法
• Reaction time in starting/stopping the stopwatch: This can be reduced by using a light gate connected to a data logger to measure the period automatically.
• 启动/停止秒表时的反应时间：可通过使用连接数据记录仪的光电门自动测量周期来减小。
• Parallax error in reading the metre rule and in judging the equilibrium: Use a pointer and ensure eye level alignment, or record oscillation with a high-speed camera for frame-by-frame analysis.
• 读取米尺和判断平衡位置时的视差：可使用指针并确保视线水平，或使用高速摄像机录制振动过程进行逐帧分析。
• Failure to keep amplitude small: Large amplitudes deviate from SHM due to spring non-linearity. Always start with a small, measured initial displacement using a set square.
• 未能保持小幅度振动：大幅度振动会因弹簧非线性而偏离简谐运动。始终用三角尺设定并测量一个小的起始位移。
• Mass of spring itself neglected in the theory: The linearised equation accounts for it via ms, but uncertainty in the intercept contributes to uncertainty in k if not properly handled.
• 理论中忽略了弹簧自身质量：线性化后的公式通过ms考虑了这一点，但截距的不确定度若处理不当会增大k的不确定度。
10. Exam Tips for Unit 5 Investigations | Unit 5实验探究的应考技巧
• Always derive the linear equation that relates measured quantities before plotting the graph. In this case, rearranging to T² = (4π²/k)m + (4π²/k)ms shows the gradient and intercept significance.
• 在绘图前，务必推导出被测量之间的线性关系式。在本题中，整理成 T² = (4π²/k)m + (4π²/k)ms 可以明确斜率和截距的物理意义。
• Show all workings when calculating gradient and intercept, using large triangles on the graph. Label the triangle vertices clearly.
• 计算斜率和截距时，在图像上使用大三角形，并清晰标注三角形顶点，展示所有计算过程。
• Express final answers with absolute uncertainty and to the correct number of significant figures. Compare your value with the expected one if known, and comment on the accuracy.
• 最终结果需带有绝对不确定度，并使用正确的有效数字位数。如已知理论值，应与实验值比较，并对准确度进行评述。
• In the evaluation, link every weakness to an improvement and explain how it reduces the specified uncertainty.
• 在评估部分，将每个弱点与改进措施联系起来，并解释该方法如何减少特定的不确定度。
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