📚 A-Level Physics: Unit 5 Experimental Investigation (January 2021 Paper) | A-Level物理：Unit 5 实验探究（2021年1月真题）
In A-Level Physics Unit 5, experimental investigation questions demand more than recalling facts – they test your ability to design procedures, handle uncertainties, and critically evaluate data. The January 2021 paper featured a classic investigation that required careful measurement of a simple pendulum to determine the acceleration of free fall, g. This article unpacks the essential skills behind such a question and shows how to structure a high‑scoring answer.
在A-Level物理Unit 5中，实验探究题不只考查对知识的回忆——它测试的是你设计实验步骤、处理不确定度以及批判性评价数据的能力。2021年1月的试卷中出现了一个经典实验，要求通过准确测量单摆来确定重力加速度g。本文将剖析这类题目背后的核心技能，并展示如何组织一份高分答案。
1. Understanding the Experimental Context | 理解实验背景
The aim is to determine g using the relationship between the length of a pendulum and its period. The accepted formula is T = 2π√(ℓ/g), provided the amplitude is small (less than about 10°).
实验目标是利用摆长与周期的关系来测定g。公认的公式为 T = 2π√(ℓ/g)，前提是振幅很小（约小于10°）。
To linearise the data, we square both sides: T² = (4π²/g) ℓ. Hence a graph of T² against ℓ should yield a straight line through the origin, with gradient = 4π²/g.
为使数据线性化，我们对两边平方：T² = (4π²/g) ℓ。因此 T²–ℓ 图像应是一条通过原点的直线，斜率等于 4π²/g。
In the Jan 21 question, students were given incomplete data and asked to calculate missing values, plot a graph, find the gradient, determine g, and evaluate uncertainties.
在2021年1月的考题中，学生拿到不完整的数据，需要算出缺失值、绘制图像、求斜率、计算g并评价不确定度。
2. Variables and Preliminary Setup | 变量与初步设置
Identify the independent variable (pendulum length ℓ), the dependent variable (period T), and the control variables: mass of the bob, amplitude of swing, and shape of the bob.
确定自变量（摆长 ℓ）、因变量（周期 T）以及控制变量：摆球质量、摆动振幅和摆球形状。
The length ℓ is measured from the point of suspension to the centre of the bob. A metre rule and a vernier calliper can be used for this purpose.
摆长 ℓ 从悬点测量到摆球中心，可使用米尺和游标卡尺进行测量。
For timing, measure the duration of at least 20 complete oscillations to reduce the percentage uncertainty caused by human reaction time.
计时方面，测量至少20次完整振荡的持续时间，以减小由人反应时间引起的百分不确定度。
3. Choosing Appropriate Measuring Instruments | 选择恰当的测量工具
A digital stopwatch with a resolution of 0.01 s is suitable for timing. However, the uncertainty in a manually stopped measurement is dominated by reaction time (typically ±0.2 s), not the resolution.
分辨力为0.01 s的数字秒表适合计时。但手动停表的测量不确定度主要由反应时间（通常为±0.2 s）决定，而不是由分辨力决定。
For length, a metre rule (resolution 1 mm) can be used for the string, while a vernier calliper (resolution 0.01 cm) measures the bob diameter, allowing the radius to be added to the string length.
长度方面，米尺（分辨力1 mm）测量摆线，游标卡尺（分辨力0.01 cm）测量摆球直径，从而可将半径加到线长上。
Always state the resolution and the estimated absolute uncertainty for each instrument in your plan.
在实验方案中，永远要说明每件仪器的分辨力及其估计的绝对不确定度。
4. Minimising Systematic and Random Errors | 减少系统误差与随机误差
A systematic error could arise if the ruler has a zero error or if the clamp position is not properly aligned. Check instruments before use and subtract any zero error.
若直尺存在零误差或夹持位置未对齐，则会产生系统误差。使用前应检查仪器并扣除零误差。
Random timing errors are reduced by repeating the measurement of oscillation time three times for each length and calculating the mean.
对每个摆长重复测量三次振荡时间并计算平均值，可以减小随机计时误差。
Use a small release angle (≤10°) and ensure the pendulum swings in a single vertical plane to avoid conical motion, which would make the period slightly shorter.
采用小释放角度（≤10°），并确保单摆在同一个竖直平面内摆动，以避免锥摆运动——这会使周期略微减小。
5. Recording Data with Appropriate Precision | 以适当精度记录数据
Design a clear results table with columns for length ℓ (m), time for 20 swings t₂₀ (s), period T (s), and T² (s²). All raw data should be recorded to the precision of the instrument, while calculated values should carry one more significant figure.
设计清晰的结果表格，包含摆长 ℓ (m)、20次摆动时间 t₂₀ (s)、周期 T (s) 和 T² (s²) 等栏目。所有原始数据需记录至仪器精度，而计算值应多保留一位有效数字。
| ℓ / m | t₂₀ / s | Mean t₂₀ / s | T / s | T² / s² |
|---|---|---|---|---|
| 0.500 | 28.32, 28.40, 28.28 | 28.33 | 1.417 | 2.007 |
Reject any obviously anomalous repeats (those differing by more than 0.5 s) and, if time allows, take an extra reading.
剔除明显异常的重复值（与其它值相差超过0.5 s的），如果时间允许，补测一次。
6. Calculating Uncertainty in Derived Quantities | 计算导出量的不确定度
The absolute uncertainty in length Δℓ arises from the ruler and calliper readings; for a typical metre rule it is ±1 mm, and for the calliper ±0.1 mm. Combine them using Δℓ = Δℓruler + Δℓcalliper.
长度绝对不确定度 Δℓ 来自直尺和游标卡尺读数；典型米尺为±1 mm，游标卡尺为±0.1 mm。使用 Δℓ = Δℓruler + Δℓcalliper 合并。
The period T is obtained from t₂₀/20, so its absolute uncertainty is ΔT = Δt₂₀ / 20, where Δt₂₀ is the spread of the repeat times (half the range) or the reaction time, whichever is larger.
周期 T 由 t₂₀/20 得到，因此其绝对不确定度 ΔT = Δt₂₀ / 20，其中 Δt₂₀ 取重复时间的半范围或反应时间中较大者。
The quantity T² has an absolute uncertainty given by Δ(T²) = 2T ΔT, and the percentage uncertainty in T² is twice the percentage uncertainty in T.
T² 的绝对不确定度为 Δ(T²) = 2T ΔT，且 T² 的百分不确定度是 T 的百分不确定度的两倍。
7. Plotting a Graph and Drawing a Line of Best Fit | 绘制图表并画最佳拟合线
Plot T² on the vertical axis and ℓ on the horizontal axis. Choose scales that use at least half the graph grid and label axes with quantities and units.
将 T² 标在纵轴，ℓ 标在横轴。选择能占据至少一半图纸的刻度，并在坐标轴标注物理量及单位。
Plot data points as small crosses (×) or circled dots. Draw a single straight best‑fit line that passes through the centroid of the points. Do not force the line through the origin unless there is a valid theoretical reason and the data clearly support it.
数据点用小十字（×）或带圈的圆点绘制。画一条穿过数据点形心的单一直线最佳拟合线。除非有充分的理论理由且数据明显支持，否则不要强制让直线通过原点。
Identify and clearly mark any outliers—points that deviate noticeably from the trend. Exclude them from the best‑fit judgment.
识别并清晰标出任何异常值——明显偏离趋势的点。在判断最佳拟合线时不应计入这些点。
8. Determining the Gradient and Intercept with Uncertainty | 确定梯度和截距及其不确定度
Select two points far apart on the best‑fit line (not data points) and calculate the gradient m = ΔT²/Δℓ. Use as large a triangle as possible to minimise uncertainty.
在最佳拟合线上选取相距较远的两点（非数据点），计算斜率 m = ΔT²/Δℓ。尽可能使用大三角形以减小不确定度。
m = (T²₂ − T²₁) / (ℓ₂ − ℓ₁)
From the gradient, calculate g using the relation g = 4π² / m.
由斜率通过 g = 4π² / m 计算重力加速度。
To find the uncertainty in the gradient, draw the maximum and minimum plausible gradient lines (steepest and shallowest) that still fit the error bars. The uncertainty Δm is half the difference: Δm = (mₘₐₓ − mₘᵢₙ) / 2.
要得到斜率的不确定度，画出仍符合误差棒的极大和极小梯度线（最陡和最平缓）。不确定度 Δm 取两者差值的一半：Δm = (mₘₐₓ − mₘᵢₙ) / 2。
The percentage uncertainty in g is then equal to the percentage uncertainty in m, because g ∝ 1/m. So Δg = g × (Δm / m).
g 的百分不确定度等于 m 的百分不确定度，因为 g ∝ 1/m。故 Δg = g × (Δm / m)。
9. Evaluating the Experiment and Suggesting Improvements | 评估实验并提出改进建议
Compare your experimental value of g with the accepted value, 9.81 m s⁻². A percentage difference less than 5 % indicates good agreement, but discuss whether the accepted value lies within your calculated uncertainty range.
将实验测得的 g 值与公认值 9.81 m s⁻² 比较。若百分差异小于 5 % 则认为一致性好，但需讨论公认值是否落在你计算的不确定度范围内。
Major sources of uncertainty usually include: reaction time when starting/stopping the stopwatch, difficulty in measuring the exact length to the centre of the bob, and assuming the bob is a point mass.
主要不确定度来源通常包括：启停秒表时的反应时间、难以精确测量到摆球中心的长度，以及把摆球当作质点处理。
Suggested improvements: use a light gate or motion sensor connected to a data logger to record period automatically; use a longer pendulum (e.g. ℓ > 1 m) to reduce fractional length uncertainty; and adopt a smaller spherical bob to better approximate a point mass.
改进建议：使用光闸或运动传感器连接数据采集器自动记录周期；采用更长的摆（例如 ℓ > 1 m）以降低长度的相对不确定度；并选用更小的球形摆以更接近质点模型。
10. Writing a High‑Scoring Conclusion | 写高分结论
State the final result for g in the standard form: g = (best value ± absolute uncertainty) m s⁻², e.g. g = (9.8 ± 0.3) m s⁻².
以标准形式给出 g 的最终结果：g = (最佳值 ± 绝对不确定度) m s⁻²，例如 g = (9.8 ± 0.3) m s⁻²。
Comment on the reliability of the experiment by referencing the uncertainty. If the accepted value lies within the uncertainty range, the experiment supports the theory within the limitations of the procedure.
通过指出不确定度来评述实验的可靠性。若公认值落在不确定度范围内，则实验在方案限制内支持了理论。
Reflect on whether a straight line through the origin was expected and whether your data justify that. Mention any systematic shift in the intercept that could suggest unaccounted errors, such as an inaccurate zero position on the ruler.
反思是否预期直线通过原点，以及你的数据是否支持这一点。提到截距的任何系统性偏移可能暗示未计及的误差，例如直尺零点位置不准确。
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