📚 Case Study in Action: Investigating a Damped Pendulum | 案例分析实战演练：阻尼单摆的研究
A case study approach is a powerful way to bring together experimental skills, data analysis, and physical understanding. In this walkthrough, you will follow a typical Year 13 investigation into the damping of a simple pendulum due to air resistance. The focus is on how to process data, interpret graphs, estimate uncertainties, and evaluate the method critically – all essential for WJEC A Level Physics practical assessments.
案例研究是融合实验技能、数据分析与物理理解的绝佳方式。本文将通过一个典型的13年级探究项目——空气阻力对单摆的阻尼作用——带你完整走一遍实战流程。重点在于如何处理数据、解读图像、估算不确定度并批判性地评估方法，这些对WJEC A Level 物理的实践考核至关重要。
1. Introduction to the Investigation | 调查简介
A student aims to determine the damping constant of a simple pendulum oscillating in air. The pendulum is assumed to experience a resistive force proportional to its velocity, giving an exponentially decaying amplitude. By measuring the angular amplitude at known times, the student can test the exponential model and evaluate the damping constant from the logarithmic decrement.
一名学生旨在测定空气中小摆幅单摆的阻尼常数。假设摆受到与速度成正比的阻力，振幅将呈指数衰减。通过测量不同时刻的角振幅，该学生可以检验指数模型，并利用对数减量求出阻尼常数。
The governing equation for light damping is:
适用的光阻尼方程为：
A(t) = A₀ e−λt
where A₀ is the initial amplitude, λ is the damping constant, and t is time. Taking natural logarithms gives a linear relation:
其中 A₀ 为初始振幅，λ 为阻尼常数，t 为时间。取自然对数后得到线性关系：
ln A = ln A₀ − λt
Thus, a graph of ln A against t should yield a straight line with gradient −λ.
因此，绘制 ln A 对 t 的图像应得到斜率为 −λ 的直线。
2. Experimental Setup and Procedure | 实验装置与步骤
The pendulum consists of a small, dense bob attached to a light, inextensible string of length about 1.20 m. A protractor with a resolution of 1° is fixed behind the point of suspension to measure the angular amplitude from the vertical. A stopwatch (resolution 0.01 s) is used to record the total time since release.
单摆装置：将一个小而密的摆球悬挂在一根轻质、不可伸长的弦上，弦长约1.20 m。在悬点后方固定一个分度值为1°的量角器，用于测量摆离开竖直方向的角度振幅。使用分度值为0.01 s的秒表记录从释放开始的总时间。
The bob is pulled to an initial amplitude of 20° and released without push. The amplitude is read at the moment when the bob reaches the maximum displacement on the same side, every 10 complete oscillations. The stopwatch reading is noted at that instant. The measurement is repeated for decreasing amplitudes until the amplitude falls below 5°.
将摆球拉至振幅20°处无初速释放。每经过10个完整周期，摆球回到同侧最大位移时读取振幅，并记录秒表读数。重复此过程，直至振幅降至5°以下。
3. Data Collection: Amplitude vs Time | 数据收集：振幅随时间的变化
Below is a typical set of data obtained in such an experiment. The period T of the pendulum is independently measured as 2.20 s by timing 20 oscillations and averaging.
以下是该实验获得的典型数据集。通过计时20次振荡取平均，测得单摆周期 T = 2.20 s。
| Number of oscillations, N | Time t / s | Amplitude A / ° |
|---|---|---|
| 0 | 0.0 | 20.0 |
| 10 | 22.0 | 17.5 |
| 20 | 44.0 | 15.0 |
| 30 | 66.0 | 13.0 |
| 40 | 88.0 | 11.2 |
| 50 | 110.0 | 9.5 |
| 60 | 132.0 | 8.0 |
| 70 | 154.0 | 6.8 |
| 80 | 176.0 | 5.7 |
| 90 | 198.0 | 4.9 |
4. Initial Data Processing | 初步数据处理
tionship is independent of units, this is acceptable. The uncertainty in A is ±0.5° (half the protractor resolution). The uncertainty in t is taken as ±0.2 s to account for reaction time.
为将指数衰减线性化，计算每个振幅值的 ln(A)。振幅以度为单位，但对数关系与单位无关，故可直接使用。A 的不确定度为 ±0.5°（量角器分度值的一半）。时间的不确定度取 ±0.2 s，以计入反应时间。
| t / s | A / ° | ln(A/°) |
|---|---|---|
| 0.0 | 20.0 | 2.996 |
| 22.0 | 17.5 | 2.862 |
| 44.0 | 15.0 | 2.708 |
| 66.0 | 13.0 | 2.565 |
| 88.0 | 11.2 | 2.416 |
| 110.0 | 9.5 | 2.251 |
| 132.0 | 8.0 | 2.079 |
| 154.0 | 6.8 | 1.917 |
| 176.0 | 5.7 | 1.740 |
| 198.0 | 4.9 | 1.589 |
A = 20.0°, Δ(ln A) ≈ 0.5/20 = 0.025. This value will be used for error bars on the graph.
ln(A) 的不确定度通过相对不确定度估算：Δ(ln A) ≈ ΔA/A。例如 A = 20.0° 时，Δ(ln A) ≈ 0.5/20 = 0.025。此数值将用于图像上的误差棒。
5. Plotting a Suitable Graph | 绘制合适的图像
Plot a graph of ln(A) on the vertical axis against time t on the horizontal axis. Use sensible scales so that the data points occupy more than half the graph paper. Label axes with quantities and units. Plot error bars for ln(A); the time uncertainties are negligible at this scale and can be ignored.
绘制 ln(A) 为纵轴、时间 t 为横轴的图像。选用合理的尺度，使数据点占据图纸的大半。坐标轴标注物理量和单位。绘出 ln(A) 的误差棒；该尺度下时间不确定度可忽略不计。
The plotted points should show a clear downward trend, suggesting a linear relationship consistent with exponential decay. Draw a best-fit straight line through the points, ensuring equal numbers of points lie above and below the line.
数据点应呈现清晰的下降趋势，表明存在与指数衰减一致的线性关系。画一条最佳拟合直线，确保线上方和下方的点数量大致相等。
6. Determining the Damping Constant from Graph | 从图像确定阻尼常数
the gradient. For example, use (t₁, ln A₁) = (0 s, 3.00) and (t₂, ln A₂) = (200 s, 1.55).
在最佳拟合线上选取两个相距较远的非数据点来计算斜率。例如，取 (t₁, ln A₁) = (0 s, 3.00) 和 (t₂, ln A₂) = (200 s, 1.55)。
= (ln A₂ − ln A₁) / (t₂ − t₁) = (1.55 − 3.00) / (200 − 0) = −1.45 / 200 = −7.25 × 10⁻³ s⁻¹
Therefore, the damping constant λ = 7.25 × 10⁻³ s⁻¹. The negative sign confirms the decay.
根据线性化方程，斜率 = −λ。因此阻尼常数 λ = 7.25 × 10⁻³ s⁻¹。负号证实了衰减。
7. Estimating Uncertainties | 不确定度估算
To estimate the uncertainty in λ, draw the steepest and shallowest plausible straight lines through the error bars (worst-fit lines). Calculate their gradients and find the difference from the best gradient.
为估算 λ 的不确定度，通过误差棒画出最陡和最平缓的可行直线（最差拟合线）。计算它们的斜率，并求出与最佳斜率的差值。
.80 × 10⁻³ s⁻¹ and −7.70 × 10⁻³ s⁻¹. The uncertainty in the gradient is half the spread: Δ|gradient| = (7.70 − 6.80) × 10⁻³ / 2 = 0.45 × 10⁻³ s⁻¹.
假设最差拟合斜率分别为 −6.80 × 10⁻³ s⁻¹ 和 −7.70 × 10⁻³ s⁻¹。斜率的不确定度为跨度的一半：Δ|gradient| = (7.70 − 6.80) × 10⁻³ / 2 = 0.45 × 10⁻³ s⁻¹。
Thus, λ = (7.25 ± 0.45) × 10⁻³ s⁻¹. The percentage uncertainty is (0.45/7.25) × 100% ≈ 6.2%.
因此 λ = (7.25 ± 0.45) × 10⁻³ s⁻¹。百分比不确定度约为 6.2%。
8. Discussion: Sources of Error | 讨论：误差来源
The main sources of uncertainty are the measurement of amplitude using the protractor and the timing of the exact moment of maximum displacement. The protractor's resolution is the dominant contribution, especially at smaller amplitudes where the percentage uncertainty in A becomes larger.
主要的不确定度来源是使用量角器测量振幅以及确定最大位移时刻的计时。量角器的分度值是主导因素，尤其在较小振幅时，A 的相对不确定度变大。
There may also be systematic errors: if the pendulum is not exactly aligned with the protractor's centre, the amplitude readings could be consistently low. The assumption of a velocity-dependent resistive force may break down at larger amplitudes where quadratic drag becomes noticeable.
也可能存在系统误差：如果摆未与量角器中心精确对准，振幅读数可能持续偏低。速度相关阻力的假设在大振幅时可能失效，此时平方型阻力会变得明显。
Furthermore, the string may not be perfectly light or inextensible, and the bob's motion may include a slight sideways component, causing the motion to depart from the ideal simple pendulum model.
此外，弦不可能完全轻质且不可伸长，摆球的运动可能包含轻微的侧向分量，使运动偏离理想单摆模型。
9. Evaluation of the Method | 方法评估
The method using a protractor is simple and low-cost, but it limits precision. Using a motion sensor or a rotary encoder would give much smaller uncertainties in amplitude and eliminate reaction-time errors. Alternatively, recording a video and analysing the motion frame by frame can dramatically improve accuracy.
使用量角器的方法简单且成本低，但精度有限。采用运动传感器或旋转编码器将大幅降低振幅不确定度，并消除反应时间误差。另一种方法是录制视频并逐帧分析运动，这能显著提高精确度。
The assumption of linearity appears valid, as the ln A vs t plot is close to a straight line. However, the slight scatter at small amplitudes suggests that the exponential model is not perfect. A more rigorous analysis could use the logarithmic decrement method on successive amplitudes to check for consistency of λ.
线性假设似乎成立，因为 ln A 对 t 的图像接近直线。然而，小振幅处略微离散表明指数模型并不完美。更严谨的分析可采用连续振幅的对数减量法，以检验 λ 的一致性。
To reduce background disturbances, the experiment was performed in still air. The suspension point was firmly clamped to minimise energy loss to support vibrations. These precautions improve the reliability of the result.
为减少背景扰动，实验在无风环境中进行。悬点被牢固夹紧，以尽量减少因支架振动造成的能量损失。这些措施提高了结果的可靠性。
10. Conclusion and Further Work | 结论与进一步工作
endulum was found to be (7.25 ± 0.45) × 10⁻³ s⁻¹, with a percentage uncertainty of about 6% d oscillations and provides a practical example of log-linear graphing and error analysis.
该单摆的阻尼常数测定为 (7.25 ± 0.45) × 10⁻³ s⁻¹，百分比不确定度约为6%。该实验成功展示了光阻尼振荡的指数衰减模型，并提供了对数-线性绘图与误差分析的实践范例。
For further work, the student could investigate how the damping constant varies with bob size or with immersion in a more viscous fluid. Comparing the measured λ with a theoretical prediction based on Stokes' law would give deeper insight into the validity of the linear drag assumption.
进一步工作中，学生可以探究阻尼常数如何随摆球大小或在更黏滞流体中变化。将测得的 λ 与基于斯托克斯定律的理论预测进行比较，可深入理解线性阻力假设的有效性。
Published by TutorHao | Physics Revision Series | aleveler.com
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