Measuring g by Free Fall | 利用自由落体法测量重力加速度

📚 Measuring g by Free Fall | 利用自由落体法测量重力加速度

In AS Physics, one of the most important required practicals is to determine the acceleration due to gravity, g, using a free-fall method. This experiment involves dropping a dense object from rest and accurately measuring the time it takes to fall through a known distance. By applying kinematic equations, a value for g can be calculated and compared with the accepted standard of approximately 9.81 m s⁻². The June 2019 style experimental investigation focuses on precision, repeated readings, graphical analysis and uncertainty evaluation — all key skills for any aspiring physicist.

在 AS 物理中，最重要的必做实验之一是利用自由落体法测定重力加速度 g。该实验涉及让一个密度较大的物体从静止下落，并精确测量它通过已知距离所需的时间。通过应用运动学方程，可以计算出 g 的数值，并与公认的标准值约 9.81 m s⁻² 进行比较。2019 年 6 月风格的实验探究侧重于精确度、重复读数、图像分析以及不确定度评估，这些对任何未来的物理学家而言都是关键技能。

1. Background Theory | 背景原理

When an object is released from rest and falls freely under gravity (ignoring air resistance), its motion is uniformly accelerated. The kinematic equation that relates displacement h, initial velocity u, acceleration a and time t is s = ut + ½at². With initial velocity u = 0 and acceleration a = g, the equation simplifies to h = ½gt². Rearranging gives g = 2h/t². This means that if we measure the height fallen, h, and the corresponding time of fall, t, we can calculate g. To reduce random error, we perform multiple trials at various heights and use a linear graph to extract a best-fit value.

当物体从静止释放并在重力作用下自由下落（忽略空气阻力）时，其运动是匀加速的。描述位移 h、初速度 u、加速度 a 和时间 t 的运动学方程为 s = ut + ½at²。当初速度 u = 0、加速度 a = g 时，该方程简化为 h = ½gt²。整理后得到 g = 2h/t²。这意味着如果我们测量下落的高度 h 和相应的下落时间 t，就可以计算出 g。为减少随机误差，我们在不同高度进行多次试验，并利用线性图像提取最佳拟合值。

h = ½gt²    →    g = 2h/t²

2. Apparatus | 实验器材

A list of typical apparatus includes: a metre rule or measuring tape (with millimetre precision), an electromagnet and trapdoor arrangement (or a light gate connected to a data logger), a steel ball bearing or a dense iron sphere, a switch or release mechanism, a timer or stopwatch (resolution 0.01 s), a plumb line, clamps and stands, and a soft landing pad to catch the ball. Some setups replace the trapdoor with a light gate and a pin or card to trigger timing more precisely.

典型的实验器材清单包括：一把有毫米精度的米尺或卷尺、电磁铁与活板门装置（或连接到数据记录器的光闸）、一个钢球或密度较大的铁球、开关或释放机构、秒表或计时器（分辨率 0.01 s）、铅垂线、铁架台和夹子，以及一个用于接住球的软着陆垫。一些装置用光闸和触发小片或卡片代替活板门，以实现更精确的计时。

3. Experimental Setup and Procedure | 实验装置与步骤

The electromagnet is clamped vertically above the bench, and a trapdoor or light gate is placed directly beneath it. Using a plumb line, ensure the ball’s trajectory is vertical and the ball falls through the centre of the detecting device. The distance h is measured from the bottom of the ball while held by the electromagnet to the top of the trapdoor or the light beam. Record the release height accurately. The ball is released by cutting the current to the electromagnet, and the timer records the fall time. Repeat the drop at least three times for each height, and then change h to a new value (e.g., 0.200 m, 0.400 m, 0.600 m, 0.800 m, 1.000 m). Record all raw data in a table.

将电磁铁垂直夹紧在工作台上方，并将活板门或光闸直接放置在其下方。使用铅垂线确保球的轨迹是竖直的，并且球能穿过检测装置的中心。距离 h 是从电磁铁吸住球时球的底部到活板门顶部或光束的距离。准确记录释放高度。通过切断电磁铁的电流来释放球，计时器记录下落时间。在每个高度至少重复三次下落，然后改变 h 到新的数值（例如 0.200 m、0.400 m、0.600 m、0.800 m、1.000 m）。将所有原始数据记录在表格中。

4. Data Collection and Table | 数据收集与表格

A well-designed data table is essential. For each height h, record the individual fall times t₁, t₂, t₃, calculate the mean time t_mean, and then compute t². Also include columns for h and the absolute uncertainty in h (usually ±0.001 m or ±½ of the smallest division) and in t (estimated from the spread of timing data or instrument precision). An example table is shown below.

设计良好的数据表至关重要。对于每个高度 h，记录各次下落时间 t₁、t₂、t₃，计算平均时间 t_mean，然后计算 t²。还应包含 h 的绝对不确定度（通常为 ±0.001 m 或最小刻度的一半）以及 t 的绝对不确定度（根据计时数据的离散度或仪器精度估算）的列。下面展示了一个示例表格。

5. Graphical Analysis | 图像分析

Plot a graph of h (vertical axis) against t² (horizontal axis). According to h = ½gt², the relationship should be a straight line through the origin with gradient m = ½g. Draw a best-fit straight line that passes as close as possible to all plotted points. Calculate the gradient using a large triangle: m = Δh / Δ(t²). Then g = 2 × m. The expected value is about 9.81 m s⁻². If the line does not go through the origin, it may suggest a systematic error such as an incorrect zero for height or a delay in the timing system.

绘制 h（纵轴）对 t²（横轴）的图像。根据 h = ½gt²，这种关系应该是一条通过原点的直线，斜率 m = ½g。画一条尽可能接近所有数据点的最佳拟合直线。使用一个大三角形计算斜率：m = Δh / Δ(t²)。然后 g = 2 × m。预期值约为 9.81 m s⁻²。如果直线不经过原点，这可能暗示存在系统误差，例如高度零位不正确或计时系统有延迟。

gradient m = (h₂ − h₁) / (t₂² − t₁²)    and    g = 2m

6. Uncertainty Estimation | 不确定度估算

To find the percentage uncertainty in g, we combine the uncertainties in the gradient. One method is to draw a worst-acceptable line (steepest or shallowest) that still passes through the error bars of the points. The gradient of the worst fit, m_worst, gives g_worst = 2m_worst. The absolute uncertainty in g is |g_best − g_worst| and percentage uncertainty = (Δg / g_best) × 100%. Alternatively, for a directly calculated g = 2h/t² from a single pair, the percentage uncertainty can be approximated by %u(g) = %u(h) + 2 × %u(t), where %u(t) = (½ range / mean) × 100% if multiple readings are taken.

为求得 g 的百分不确定度，我们要合并斜率中的不确定度。一种方法是画一条仍然穿过各点误差棒的“最不可接受的线”（最陡或最平缓的那条）。最差拟合的斜率 m_worst 给出 g_worst = 2m_worst。g 的绝对不确定度为 |g_best − g_worst|，百分不确定度 = (Δg / g_best) × 100%。或者，对于直接从单组数据计算的 g = 2h/t²，若取多次读数，其百分不确定度可近似为 %u(g) = %u(h) + 2 × %u(t)，其中 %u(t) = (½极差 / 平均值) × 100%。

7. Sources of Error | 误差来源

• Reaction time: If a manual stopwatch is used, human reaction time (≈0.2 s) can introduce large random errors, especially for short falls. Using an electronic timer with electromagnet release and trapdoor or light gate drastically reduces this.
• 反应时间：如果使用手动秒表，人的反应时间（约 0.2 s）会引入很大的随机误差，特别是对于短距离下落。使用带电磁铁释放和活板门或光闸的电子计时器可大幅减少这种误差。
• Parallax error in height measurement: The rule must be placed vertically and eyes level with the scale to avoid parallax. A set square and plumb line help minimise this.
• 高度测量中的视差：米尺必须竖直放置，眼睛应与刻度齐平以避免视差。使用三角板和铅垂线有助于尽量减少这种误差。
• Air resistance: A dense, smooth sphere (like a steel ball) is chosen to minimise air drag. Lighter or irregularly shaped objects would fall with a lower effective acceleration.
• 空气阻力：选择密度大且光滑的球体（如钢球）以尽量减少空气阻力。较轻或形状不规则的物体会以较小的有效加速度下落。
• Zero error in height: The reference point for h (e.g., bottom of ball to trapdoor surface) must be consistent. A misreading of a few millimetres can shift the graph intercept.
• 高度的零点误差：h 的参考点（例如从球底到活板门表面）必须一致。几毫米的误读可能使图像截距偏移。
• Electromagnet residual magnetism: Sometimes the ball is slightly delayed after current is cut, introducing a systematic delay. Using a thin non-magnetic separator or releasing from a small distance can help, but must be noted.
• 电磁铁剩磁：有时断电后球会因剩磁而略有延迟，这引入了系统延迟。使用薄的非磁性隔板或从微小距离释放可能有帮助，但须注意注明。

8. Improvements and Refinements | 改进与精化

To improve accuracy, a light gate connected to a data logger eliminates reaction time completely and allows measurement to 0.001 s or better. Using two light gates at a fixed separation and measuring the time interval between them allows a method based on v² = u² + 2as, which can be more robust. An alternative is the ‘picket fence’ method using a card with evenly spaced black bands passing through a single light gate; a computer calculates g from the acceleration directly. Repeating the whole experiment with the ball drop reversed (drop from a higher initial point but measuring the same height intervals) can check for consistency.

为了提高准确度，连接到数据记录器的光闸完全消除了反应时间，并可以实现 0.001 s 或更好的测量精度。使用间距固定的两个光闸并测量它们之间通过的时间间隔，可以采用基于 v² = u² + 2as 的方法，这可能更加稳健。另一种方案是“栅栏式”方法，使用带有均匀黑色条带的卡片通过单个光闸；计算机直接从加速度计算出 g。将整个实验在球下落方向反转的情况下重复进行（从更高初始点落下但测量相同的高度间隔）可以检查一致性。

9. Safety Precautions | 安全注意事项

Although this is a low-risk experiment, the steel ball can cause injury if it hits a person. Always use a soft landing pad (e.g., sand tray or thick cloth) and ensure the area beneath the electromagnet is clear. Keep feet away from the drop zone. When using an electromagnet, avoid overheating the coil by not leaving the current on for too long. If an elevated setup is used, secure clamps firmly to prevent toppling.

虽然这是一个低风险实验，但钢球如果击中人员可能造成伤害。始终使用软着陆垫（例如沙盘或厚布），并确保电磁铁下方区域畅通。双脚远离下落区域。使用电磁铁时，不要长时间通电以免线圈过热。如果使用升高装置，要牢固地夹紧夹子以防倾倒。

10. Summary and Exam Tips | 总结与考试技巧

The free-fall method for determining g is a core practical that elegantly combines kinematics, data handling and error analysis. Remember these key points: plot h vs t², not t; the gradient is ½g; always comment on whether the line passes through the origin; distinguish between systematic and random errors. In the June 2019 style exam questions, you may be asked to suggest improvements, calculate percentage difference from the accepted value, or evaluate the reliability of a student’s data. Practise drawing lines of best and worst fit, and clearly state your answer with the correct number of significant figures.

自由落体法测定 g 是一个核心实验，它优雅地结合了运动学、数据处理和误差分析。请记住这些关键点：绘制 h 对 t² 的关系图，而非对 t；斜率是 ½g；总要评论直线是否通过原点；要区分系统误差和随机误差。在 2019 年 6 月风格的考题中，你可能会被要求提出改进建议、计算与公认值的百分差异，或者评估某位学生数据的可靠性。练习绘制最佳和最差拟合线，并用正确的有效数字位数清晰地写出你的答案。

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