📚 Experimental Investigation: Determining g using a Simple Pendulum | 实验探究：用单摆测定重力加速度
The simple pendulum experiment is a classic investigation in A-level physics, often used to determine the acceleration due to gravity, g. This experiment involves measuring the period of oscillation for a small bob suspended by a light, inextensible string, and analysing how the period depends on the length of the pendulum. A linearised graph is plotted to extract g from the gradient, allowing students to practise data handling, uncertainty analysis, and evaluation of systematic errors.
单摆实验是 A-level 物理中的经典探究，常用于测定重力加速度 g。实验中测量轻质不可伸长细线悬挂的小球的摆动周期，并分析周期如何依赖于摆长。通过绘制线性化图像，从斜率中提取 g 值，学生可以练习数据处理、不确定度分析，并评估系统误差。
1. Theory and Governing Equation | 理论与控制方程
A simple pendulum consists of a point mass m attached to a string of length L. For small angular displacements (θ < 10°), the motion approximates simple harmonic motion. The restoring force is −mg sinθ ≈ −mgθ, and using the arc length s = Lθ, we obtain the equation of motion and the period T.
单摆由长度为 L 的细绳悬挂质点质量 m 构成。当角位移很小（θ < 10°）时，运动近似为简谐运动。回复力为 −mg sinθ ≈ −mgθ，利用弧长 s = Lθ，可以推导出运动方程和周期 T。
T = 2π √(L/g)
Squaring both sides gives a linear relationship between T² and L:
将两边平方，得到 T² 与 L 之间的线性关系：
T² = (4π²/g) L
Thus, plotting T² on the y‑axis against L on the x‑axis yields a straight line passing through the origin, with gradient = 4π²/g. From the gradient, g can be calculated.
因此，以 T² 为 y 轴、L 为 x 轴作图，将得到一条通过原点的直线，其斜率 = 4π²/g。通过斜率可以计算出 g。
2. Apparatus and Setup | 仪器与装置
The following equipment is required: a small metal bob, light inextensible string (≈1.5 m), a split cork or clamp to hold the string, a metre ruler (precision ±1 mm), a digital stopwatch (precision ±0.01 s), a protractor or angle measurer, a retort stand with boss and clamp, and a fiducial marker (e.g. a small upturned pin) to define the equilibrium position accurately.
所需器材：一个小金属摆球、轻质不可伸长的细线（约 1.5 m）、用于固定细线的分体软木塞或夹子、米尺（精度 ±1 mm）、数字秒表（精度 ±0.01 s）、量角器或角度测量器、带铁圈和夹子的铁架台，以及一个用来精确定义平衡位置的参考标记（如一个倒置的小钉）。
The setup must minimise air currents and ensure the pendulum swings in a vertical plane. The length L is measured from the point of suspension to the centre of the bob. This length must be measured carefully, as it is the independent variable.
装置必须避免气流干扰，并确保摆在一个竖直平面内摆动。摆长 L 是从悬挂点到摆球中心的距离。这个长度必须仔细测量，因为它是自变量。
3. Experimental Procedure | 实验步骤
1. Secure the string at the top using a split cork so that the effective suspension point is fixed. Measure the initial length L (approximately 1.00 m) using a metre ruler. Record the uncertainty in L as half the smallest division (±0.5 mm) but typically ±1 mm due to alignment issues.
1. 用分体软木塞在上端固定细线，使有效悬挂点固定。用米尺测量初始摆长 L（约 1.00 m）。记录 L 的不确定度，通常由于对齐问题取最小分度值的一半（±0.5 mm）但一般为 ±1 mm。
2. Displace the bob by a small angle (less than 10° from the vertical) and release it smoothly. Start the stopwatch as the bob passes the fiducial marker at the equilibrium position. Count 20 complete oscillations and stop the stopwatch at the conclusion of the 20th pass through the marker.
2. 将摆球偏离一个小角度（与竖直方向的夹角小于 10°），并平稳释放。当摆球经过平衡位置的参考标记时启动秒表。计数 20 个完整的摆动，并在第 20 次通过标记的结束时停表。
3. Repeat the timing for a total of three trials at this length to evaluate random error. Calculate the average time for 20 oscillations, then divide by 20 to obtain the period T. Record the mean period and its uncertainty (e.g., half the range of the three values).
3. 在这个摆长下重复计时三次，以评估随机误差。计算 20 次摆动的平均时间，然后除以 20 得到周期 T。记录平均周期及其不确定度（例如，三次值极差的一半）。
4. Shorten the string to obtain a new length (e.g., 0.90 m, 0.80 m, …, down to about 0.30 m) and repeat steps 1–3. Obtain at least six different lengths well spread across the range.
4. 缩短摆线以获得新的摆长（如 0.90 m、0.80 m……直至约 0.30 m），并重复步骤 1–3。至少获得六个均匀分布在不同范围内的摆长数据。
4. Data Collection and Table | 数据收集与表格
Record all measurements in a structured table. Below is an example with realistic data (g ≈ 9.81 m s⁻²). Uncertainty in length is assumed ±0.001 m, and period uncertainty evaluated from repeated timings.
在结构化的表格中记录所有测量值。以下是使用真实数据的示例（g ≈ 9.81 m s⁻²）。假设长度不确定度为 ±0.001 m，周期不确定度由重复计时计算得出。
| L / m (±0.001) | 20Tₐᵥ / s | T / s | T² / s² | δT / s |
|---|---|---|---|---|
| 1.000 | 40.12 | 2.006 | 4.024 | 0.005 |
| 0.900 | 38.02 | 1.901 | 3.614 | 0.005 |
| 0.800 | 35.84 | 1.792 | 3.211 | 0.004 |
| 0.700 | 33.52 | 1.676 | 2.809 | 0.004 |
| 0.600 | 31.02 | 1.551 | 2.406 | 0.004 |
| 0.400 | 25.36 | 1.268 | 1.608 | 0.003 |
The column δT represents the absolute uncertainty in T, derived from timing repeats. The T² column is calculated directly.
δT 列表示周期 T 的绝对不确定度，由重复计时推导得出。T² 列直接计算得出。
5. Graphical Analysis | 图像分析
Plot a graph of T² (vertical axis) against L (horizontal axis). Use an appropriate scale so that the points fill at least half the graph paper in both directions. Label axes with quantity and unit: ‘T² / s²’ and ‘L / m’. Mark all data points with small crosses, and draw error bars for T² using the propagation of uncertainty: Δ(T²) = 2T ΔT.
绘制 T²（纵轴）对 L（横轴）的图像。选择合适的比例尺，使数据点在两个方向上至少占据坐标纸的一半。标注轴名和单位：‘T² / s²’ 和 ‘L / m’。用小十字标记所有数据点，并利用不确定度传播：Δ(T²) = 2T ΔT 绘制 T² 的误差棒。
Draw the best-fit straight line through the points. If the line does not pass exactly through the origin, a small intercept may indicate a systematic error, such as an inaccurate measurement of L (e.g., the effective length differs from the measured string length). The gradient m of the line is found using a large triangle on the graph.
通过数据点绘制最佳拟合直线。如果直线不严格过原点，小小的截距可能表明存在系统误差，比如摆长测量不准确（有效摆长与测得的线长有所差异）。利用图像上的大三角形求出直线斜率 m。
6. Calculating g from the Gradient | 由斜率计算 g
From the linearised equation, gradient m = 4π² / g. Therefore, the experimental value of g is determined by:
由线性化方程，斜率 m = 4π² / g。因此，实验值 g 由下式确定：
g = 4π² / m
Using the graph, suppose the gradient is found to be m = 4.00 s² m⁻¹ (as a typical value from the above table: ΔT²/ΔL ≈ (4.024−1.608)/(1.000−0.400) = 2.416/0.600 ≈ 4.03 s² m⁻¹). Then g = 4 × (3.142)² / 4.03 ≈ 9.80 m s⁻².
利用图像，假设求得斜率 m = 4.00 s² m⁻¹（上表典型值：ΔT²/ΔL ≈ (4.024−1.608)/(1.000−0.400) = 2.416/0.600 ≈ 4.03 s² m⁻¹）。那么 g = 4 × (3.142)² / 4.03 ≈ 9.80 m s⁻²。
Compare this value with the accepted local value, e.g., 9.81 m s⁻². Calculate the percentage difference to evaluate accuracy.
将此值与当地公认值（如 9.81 m s⁻²）进行比较，计算百分差以评估准确度。
7. Uncertainty in the Gradient and g | 斜率与 g 的不确定度
To find the uncertainty in g, first determine the uncertainty in the gradient. Draw the steepest and shallowest acceptable straight lines that pass through the error bars (the worst-fit lines). Obtain gradients m₁ and m₂. The absolute uncertainty in the gradient is Δm = |m₁ − m₂|/2.
要求 g 的不确定度，首先确定斜率的不确定度。绘制通过误差棒的“最陡”和“最缓”可接受的直线（最差拟合线）。得到斜率 m₁ 和 m₂。斜率的绝对不确定度为 Δm = |m₁ − m₂|/2。
Then propagate to g using the relationship g = 4π²/m. Since it is an inverse relationship, the percentage uncertainty in g equals the percentage uncertainty in m: (%Δg = %Δm). Therefore, Δg = g × (Δm/m).
然后通过关系式 g = 4π²/m 传播到 g。由于是反比关系，g 的百分不确定度等于 m 的百分不确定度：(%Δg = %Δm)。因此 Δg = g × (Δm/m)。
Express the final result as: g = gₑₓₚ ± Δg (in m s⁻²). Ensure the number of significant figures in the uncertainty and the value are consistent.
最终结果表示为：g = gₑₓₚ ± Δg（单位 m s⁻²）。确保不确定度和测量值的有效数字位数一致。
8. Sources of Error and Their Effects | 误差来源与影响
Systematic errors: The measured length may not be exactly the distance to the centre of mass of the bob; the string might stretch slightly; the angle might exceed 10° causing the motion to deviate from simple harmonic, making the period larger than the formula predicts. Each of these biases the result either systematically high or low.
系统误差：所测长度可能不是到摆球质心的精确距离；细线可能轻微伸长；摆角超过 10° 导致运动偏离简谐，使得周期大于公式预测值。这些都会使结果系统性地偏高或偏低。
Random errors: Reaction time in stopwatch readings introduces scatter in the period values. Using a fiducial marker and timing 20 oscillations helps reduce this effect. Parallax when measuring L adds uncertainty.
随机误差：秒表读数时的反应时间给周期值带来离散。使用参考标记并计时 20 次摆动有助于减小这一影响。测量摆长时的视差增加了不确定度。
Mitigation strategies include: using a heavier, small bob to approximate a point mass; using a clamp stand firmly fixed to avoid wobble; timing from the fiducial marker at the centre where speed is greatest; and repeating measurements for multiple lengths.
缓解策略包括：使用较大质量的小球以近似质点；使用稳固的铁架台避免晃动；从速度最大的中心参考标记处开始计时；对多个摆长重复测量。
9. Modifications and Deeper Investigation | 实验改进与深入探究
A more accurate value of g can be obtained using a compound pendulum (bar pendulum) where the radius of gyration is known. Alternatively, vary the amplitude systematically and investigate the breakdown of the small-angle approximation, plotting T against θ₀ to examine the deviation.
通过使用已知回转半径的复摆（棒摆），可以获得更准确的 g 值。或者，系统地改变振幅，研究小角近似的失效，绘制 T 对 θ₀ 的图像以考察偏离。
Another extension is to investigate the effect of the mass of the bob: though theory predicts independence, a light string and a very light bob may introduce air resistance and damping. Measuring the decay of amplitude over time can link to damping concepts in Unit 4.
另一项拓展是研究摆球质量的影响：尽管理论预测周期与质量无关，但轻质细线和极轻的摆球可能引入空气阻力和阻尼。测量振幅随时间衰减可以联系到 Unit 4 中的阻尼概念。
10. Conclusion and Report Writing | 结论与实验报告撰写
This investigation provides a reliable determination of g, typically within 2% of the accepted value if careful technique is applied. The linearisation method demonstrates the power of transforming variables to test physical laws. A full report should include an introduction, theory, method, results with uncertainty analysis, discussion of errors, and a concise conclusion that evaluates the success of the experiment against the objective.
如果操作得当，本探究能够可靠地测定 g，通常与公认值的偏差在 2% 以内。线性化方法展示了转换变量以检验物理定律的威力。一份完整的报告应包含引言、理论、方法、带不确定度分析的结果、误差讨论，以及对照目标评估实验成功与否的简明结论。
When answering examination questions on experimental inserts like PH04, students must be able to identify variables, suggest improvements, calculate uncertainties from given data, and critically assess limitations. Familiarity with the pendulum investigation is thus essential for the International A-level Physics practical assessment.
在回答如 PH04 这样的实验插入页考题时，学生必须能够识别变量、提出改进、根据给定数据计算不确定度，并批判性地评估局限性。因此，熟悉单摆探究对于 International A-level 物理实验评估至关重要。
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