📚 Exploring the Relationship between Period and Mass in a Spring-Mass System | 探究弹簧振子周期与质量的关系
of vertical oscillations of a loaded spring, in order to determine the spring constant and to verify the relationship T = 2π√m / k. This article provides a comprehensive walkthrough of the experimental procedure, data handling, graphical analysis and uncertainty evaluation, mirroring the style of the June 2023 PH05 Insert.
在爱德思国际 A Level 物理单元 5（PH05）考试中，实验数据分析是一项关键技能。一个经典的探究实验是测量弹簧振子竖直振动的周期，以确定弹簧的劲度系数并验证关系式 T = 2π√m / k。本文全面介绍实验步骤、数据处理、图像分析与不确定度评定，风格与 2023 年 6 月 PH05 插入文件中的探究任务类似。
1. Introduction to Simple Harmonic Motion and the Mass-Spring System | 简谐运动与弹簧振子简介
Simple harmonic motion (SHM) occurs when the restoring force on a displaced object is directly proportional to the displacement and acts towards the equilibrium position. A mass hung on a light spring, when displaced vertically and released, executes SHM, provided the spring's extension remains within its elastic limit.
当物体所受的回复力与位移成正比且始终指向平衡位置时，物体做简谐运动（SHM）。把质量块挂在轻弹簧下方，竖直拉离平衡位置后释放，只要弹簧的伸长量仍在弹性限度内，系统就会做简谐运动。
ing force, x is the displacement from equilibrium and k is the spring constant. The minus sign shows that the force always opposes the displacement, so the acceleration is always directed back towards the centre.
简谐运动的判定条件可写作 F = −k x，其中 F 为回复力，x 为相对平衡位置的位移，k 为劲度系数。负号表示力始终与位移方向相反，因此加速度总是指向平衡位置。
For a mass-spring oscillator, the period T is independent of the amplitude (for small amplitudes) and depends only on the suspended mass and the stiffness of the spring. This is what makes the system such a clean vehicle for a relationship investigation in the practical examination.
对于弹簧振子，在小振幅条件下周期 T 与振幅无关，只取决于悬挂的质量和弹簧的软硬程度。这正是该系统适合在实验考试中用于探究变量关系的原因。
2. Theoretical Background: The Period Formula | 理论背景：周期公式
For an ideal massless spring, the period of oscillation T (in seconds) is related to the mass m (in kg) suspended from it and the spring constant k (in N/m) by the standard equation below.
对于理想的无质量弹簧，振动周期 T（单位：秒）与悬挂质量 m（单位：千克）和劲度系数 k（单位：牛每米）之间的关系由下面的标准公式给出。
T = 2π√m / k
Squaring both sides gives a form that is far more useful for graph work, because it converts a curved relationship into a straight line.
两边平方后得到一个对作图分析极为有用的形式，因为它把曲线关系转化为直线关系。
T² = (4π² / k) × m
From this equation, a graph of T² on the y-axis against m on the x-axis should be a straight line through the origin, with gradient 4π² / k. The constant 4π² is a pure number (approximately 39.48), so the gradient carries the unit s² kg⁻¹.
由该式可知，以 T² 为纵轴、m 为横轴作图应得到一条过原点的直线，斜率为 4π² / k。常数 4π² 是无量纲的纯数（约 39.48），因此斜率的单位为 s² kg⁻¹。
Since the relationship between T and m is of the form T ∝ √m, plotting T against m directly would give a curve. Examiners expect candidates to justify squaring: linearisation is what allows a reliable gradient to be extracted.
由于 T 与 m 满足 T ∝ √m 的关系，直接以 T 对 m 作图会得到一条曲线。考官期望考生能说明为什么要平方：线性化才能可靠地求出斜率。
3. The Effective Mass Correction | 有效质量修正
The simple formula T = 2π√m / k assumes that all the oscillating mass is concentrated at the end of the spring. In reality the spring itself has mass mₛ, and the coils near the fixed end move very little while those near the bottom move almost as much as the load.
简单公式 T = 2π√m / k 假设全部振动质量都集中在弹簧末端。但实际上弹簧本身也有质量 mₛ，靠近固定端的簧圈几乎不动，而靠近下端的簧圈位移接近负载。
A more accurate treatment replaces m by (m + meff), where meff is the effective mass of the spring. Integration over a uniformly extending spring gives meff ≈ mₛ / 3.
更精确的处理是用 (m + meff) 代替 m，其中 meff 为弹簧的有效质量。对均匀伸长的弹簧进行积分可得 meff ≈ mₛ / 3。
T² = (4π² / k) × (m + meff)
linear with gradient 4π² / k, and with a small positive y-intercept equal to (4π² / k) × meff.
这个扩展公式才与实际实验数据吻合。它表明 T² 与 m 的图像应为直线，斜率为 4π² / k，并有一个小的正截距，等于 (4π² / k) × meff。
The presence of a non-zero intercept is not a failure of the experiment; it is direct experimental evidence for the finite mass of the spring. A candidate who identifies and quantifies this earns credit for critical thinking.
截距不为零并不是实验失败，而是弹簧具有有限质量的直接实验证据。能够识别并定量处理该截距的考生，可以获得批判性思维方面的分数。
4. Experimental Set-up and Equipment | 实验装置与器材
A typical set-up for this investigation includes the following items. The spring must be clamped rigidly from above so that the support does not vibrate.
该探究的典型装置包括以下器材。弹簧必须从上方被牢固夹持，以免支架发生振动。
- Helical spring of known mass mₛ (measured on a balance). | 已知质量的螺旋弹簧 mₛ（用天平称量）。
- Rigid retort stand, boss and clamp. | 刚性铁架台、十字夹与夹头。
- Mass hanger plus a set of slotted masses (e.g. 10 g to 500 g). | 砝码挂盘与一套开槽砝码（例如 10 g 至 500 g）。
- Stopwatch (or a video camera / slow-motion phone app for better timing). | 秒表（或摄像机、手机慢动作应用，以改进计时）。
- Metre rule with a fiducial marker attached to the bottom of the spring. | 米尺，并在弹簧下端固定一个参考标记。
- Balance for measuring the mass of the spring and the slotted masses. | 天平，用于称量弹簧与砝码质量。
In a PH05-style question, the mark scheme usually rewards a clear statement that the masses are known or measured, that the spring is clamped vertically, and that the timing is repeated and averaged.
在 PH05 类题目中，评分标准通常给分于：明确说明质量已知或已测量、弹簧竖直夹持、计时重复取平均。
5. Experimental Procedure | 实验步骤
The investigation proceeds in a defined sequence so that each independent variable setting is tested under the same conditions.
实验应按照固定顺序进行，使每一个自变量取值都在相同条件下被测试。
- Measure the mass of the spring mₛ on a balance and record it. | 用天平测量弹簧质量 mₛ 并记录。
- Clamp the spring vertically and hang the mass hanger from its lower end. | 竖直夹好弹簧，在其下端挂上砝码挂盘。
- Add slotted masses one at a time, recording the total suspended mass m. | 逐个添加砝码，记录总悬挂质量 m。
- Displace the mass slightly downwards (small amplitude) and release it without a sideways push. | 将质量块向下轻拉一小段（小振幅），不加侧向推力地释放。
- Allow one or two oscillations to settle, then time N complete oscillations (N = 10 or 20). | 先让它摆动一两个周期稳定，然后计时 N 个完整周期（N = 10 或 20）。
- Repeat the timing three times and calculate the mean time, then T = mean time / N. | 重复计时三次取平均值，再由 T = 平均时间 / N 求周期。
- Repeat for five to eight different values of m, spanning as wide a range as the spring allows. | 对 5 至 8 个不同的 m 值重复实验，范围尽量宽，但不超出弹簧弹性限度。
Each set of readings should be taken at the same small amplitude, because the theoretical relationship assumes SHM, and large amplitudes introduce measurable deviations from the ideal model.
每组数据都应在相同的小振幅下测得，因为理论关系式假设简谐运动，而大振幅会带来可测量的偏差。
6. Measuring the Period Accurately | 准确测量周期
Timing is the dominant source of random uncertainty in this experiment. Human reaction time is typically 0.2 s to 0.3 s, so timing a single oscillation would give a percentage uncertainty of tens of percent.
计时是本实验中随机不确定度的主要来源。人的反应时间通常为 0.2 s 至 0.3 s，若只测一个周期，百分不确定度可达数十个百分点。
The standard remedy is to time many oscillations. If 20 oscillations are timed, the reaction-time uncertainty is divided by 20 when T is calculated.
标准做法是测量多个周期。若计时 20 个周期，则计算 T 时反应时间带来的不确定度被除以 20。
A fiducial marker (a small pointer or a piece of tape) fixed at the bottom of the spring, level with a reference mark on the stand, gives a repeatable start and stop point. Counting should begin from "zero" and the stopwatch started and stopped as the marker passes the reference in the same direction each time.
在弹簧下端固定一个小指针或一条胶带作为参考标记，与支架上的参考线对齐，可得到可重复的起止点。计数从"零"开始，秒表在标记每次沿同一方向经过参考线时启动和停止。
Alternatively, a slow-motion video with a known frame rate allows the period to be found frame by frame, removing the reaction-time problem almost entirely.
另一种方法是用已知帧率的慢动作视频逐帧测量周期，几乎完全消除反应时间的影响。
7. Data Handling and Tabulation | 数据处理与表格
Raw data should be recorded in a clear table with correct units in the column headings and consistent significant figures. The processed column T² is then derived from the mean period.
原始数据应记录在清晰的表格中，列标题写明单位，有效数字保持一致，再根据平均周期计算 T² 列。
| m / kg | t₁ / s | t₂ / s | t₃ / s | mean t / s | T = mean t / 20 / s | T² / s² |
|---|---|---|---|---|---|---|
| 0.100 | 12.62 | 12.70 | 12.66 | 12.66 | 0.633 | 0.401 |
| 0.200 | 17.86 | 17.92 | 17.88 | 17.89 | 0.894 | 0.800 |
| 0.300 | 21.92 | 21.96 | 21.90 | 21.93 | 1.096 | 1.202 |
Note that T² values are quoted to three significant figures, consistent with the timing precision. Working in SI units throughout avoids unit-conversion slips at the graph stage.
注意 T² 保留三位有效数字，与计时精度相符。全程使用国际单位制，可避免作图阶段的单位换算错误。
8. Graphical Analysis: T² against m | 图像分析：T² 与 m
The processed data are plotted as T² (y-axis) against m (x-axis). Error bars in T² are usually small and often omitted, but a well-presented graph includes them and a line of best fit drawn by eye or by least squares.
将处理后的数据以 T² 为纵轴、m 为横轴作图。T² 方向的误差棒通常较小，可以省略，但规范的图像应画出误差棒，并用目测或最小二乘法画出最佳拟合直线。
The gradient of this line is 4π² / k, and the y-intercept is (4π² / k) × meff g was over-stretched, that the amplitude was too large, or that the timing method drifted.
该直线的斜率为 4π² / k，纵截距为 (4π² / k) × meff。图像应为直线；若出现系统性弯曲，说明弹簧被拉伸过度、振幅过大，或计时方法出现漂移。
A useful check is to compare the intercept with the gradient: meff lue is close to mₛ / 3, the data are consistent with the theoretical effective-mass model.
一个有用的检验是用截距除以斜率：meff = 截距 / 斜率。若该值接近 mₛ / 3，说明数据与理论有效质量模型相符。
9. Determining the Spring Constant and Effective Mass | 求劲度系数与有效质量
Once the gradient G is found, the spring constant follows directly from rearranging the gradient expression.
求出斜率 G 后，通过改写斜率表达式即可直接得到劲度系数。
k = 4π² / G
9.75 N m⁻¹. The unit follows automatically from s² kg⁻¹ inverted, since 1 N = 1 kg m s⁻².
例如，若斜率为 4.05 s² kg⁻¹，则 k = 39.48 / 4.05 ≈ 9.75 N m⁻¹。单位可由 s² kg⁻¹ 取倒数直接得到，因为 1 N = 1 kg m s⁻²。
The intercept c gives the effective mass, meff radient cancels both 4π² and k, which is why this ratio is a neat way to extract the effective mass without knowing k first.
截距 c 给出有效质量 meff = c / G。截距除以斜率可同时消去 4π² 与 k，因此这个比值是在未知 k 的情况下求有效质量的简便方法。
The investigation can be cross-checked with a static method: hanging known weights and measuring the extension, then plotting force against extension to find k from the gradient. Agreement between the dynamic and static values of k within their uncertainties strengthens the conclusion considerably.
该探究可用静态法交叉验证：悬挂已知重物测量伸长量，作力与伸长量图像，由斜率求 k。若动态法与静态法所得的 k 在各自不确定度范围内一致，结论的可信度将大大提高。
10. Uncertainty Evaluation | 不确定度评定
Uncertainties in this experiment come from several sources, and they should be combined to give a final uncertainty in k. The most important are the timing uncertainty and the mass uncertainty.
本实验的不确定度来自多个方面，应合成后给出 k 的最终不确定度。其中最重要的是计时不确定度与质量不确定度。
- Timing: ±0.2 s per reading, reduced by a factor of N when T is calculated. | 计时：每次读数 ±0.2 s，计算 T 时被 N 除而减小。
- : limited by the balance resolution, typically ±0.1 g or ±1 g depending on the instrument. | 质量：受天平分辨率限制，视仪器而定，通常为 ±0.1 g 或 ±1 g。
- Amplitude and spring mass: systematic, not reduced by averaging. | 振幅与弹簧质量：属于系统误差，取平均不能减小。
For the period, the percentage uncertainty is ΔT / T × 100%. Because T² is used on the graph, the percentage uncertainty in T² is twice that in T.
对于周期，百分不确定度为 ΔT / T × 100%。由于图像中使用的是 T²，T² 的百分不确定度是 T 的两倍。
tainty in k equals the percentage uncertainty in the gradient, since k = 4π² / G and 4π² is exact.
斜率的不确定度可通过穿过误差棒的"最大斜率"与"最小斜率"两条直线求得。由于 k = 4π² / G 且 4π² 为精确常数，k 的百分不确定度等于斜率的百分不确定度。
11. Sources of Error and Improvements | 误差来源与改进
A strong answer for a PH05 practical question identifies errors and proposes realistic improvements, rather than simply listing generic 'human error'.
在 PH05 实验题中，优秀答案会指出具体误差并提出可行的改进措施，而不是笼统地写"人为误差"。
- Swinging sideways rather than vertically: release the mass without a lateral push and let it settle before timing. | 摆动方向倾斜而非竖直：释放时不要侧推，等其稳定后再计时。
- Parallax when reading the marker: view the pointer and reference line at eye level, perpendicular to the scale. | 读数时视差：视线与指针、参考线等高，垂直于刻度方向观察。
- Over-stretching the spring beyond its elastic limit: keep the total load within the manufacturer's range. | 弹簧超出弹性限度：总负载须控制在允许范围内。
- Amplitude too large: restrict the displacement so the small-angle and SHM approximations hold. | 振幅过大：限制位移，使小角度与简谐近似成立。
- Counting errors in the oscillation number: count aloud, or use a video recording to verify. | 周期计数错误：出声计数，或用录像核对。
Temperature changes can also alter the spring constant slightly, so readings should be taken reasonably quickly and under the same laboratory conditions.
温度变化也会轻微改变劲度系数，因此读数应在较短时间内、在相同的实验室条件下完成。
12. Exam-style Conclusions and Mark Scheme Tips | 考试型结论与得分要点
A complete conclusion states the value of k with its uncertainty and comments on whether the data support the theoretical relationship. The word 'verify' requires a judgement, not merely a calculation.
完整的结论应给出 k 的值及其不确定度，并评论数据是否支持理论关系。题目中的"验证"要求作出判断，而不仅仅是计算。
Typical earning statements in a mark scheme include: squaring to linearise the relationship; identifying gradient as 4π² / k; recognising a non-zero intercept as evidence of spring mass; and comparing the experimental value of k with a static determination.
评分标准中常见的得分点包括：通过平方实现线性化；指出斜率等于 4π² / k；认识到非零截距是弹簧质量的证据；将实验所得的 k 与静态法结果进行比较。
Finally, check units, significant figures and the direction of each axis. A clear, labelled graph with a stated gradient and its percentage uncertainty will secure the analysis marks far more reliably than a long paragraph of prose.
最后，检查单位、有效数字与坐标轴方向。一张标注清晰、给出斜率及其百分不确定度的图像，比一大段文字描述更能稳妥地拿到分析分。
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