A-Level Physics: June 2018 Insert 5 – Measuring g by Free Fall | A-Level 物理：2018年6月插入材料5 – 自由落体法测重力加速度

📚 A-Level Physics: June 2018 Insert 5 – Measuring g by Free Fall | A-Level 物理：2018年6月插入材料5 – 自由落体法测重力加速度

In the June 2018 A-Level Physics examination, Insert 5 provided a diagram of an electromagnet and trapdoor apparatus, along with a table of measured distances and times for a steel ball falling from rest. This investigation is a classic method to determine the acceleration due to gravity, g, and the data must be analysed using graphical techniques and error analysis typical of practical examination questions.

在 2018 年 6 月的 A-Level 物理考试中，插入材料 5 提供了一幅电磁铁和触发门的装置图，以及一组钢球从静止下落的距离和时间测量数据表。本次探究是测定重力加速度 g 的经典方法，考试要求考生运用图像处理和误差分析技术来解析数据，这是实验考题的典型要求。

1. Understanding Insert 5 | 理解插入材料 5

Insert 5 contains a circuit diagram of an electromagnet holding a steel ball, a trapdoor switch connected to a timer, and a metre rule to vary the drop height h. The timer starts when the circuit is broken and stops when the trapdoor opens. A table gives values of h and the corresponding fall time t for several heights.

插入材料 5 包含一个电磁铁吸附钢球的电路图、一个连接计时器的触发门开关，以及一把改变下落高度 h 的米尺。当电路断开时计时器开始计时，触发门打开时停止计时。表格给出了多个高度对应的下落时间 t。

The data in Insert 5 allow students to plot a graph of h against t² and hence determine g from the slope. The experiment assumes negligible air resistance and that the ball is released from rest precisely when the timer starts.

插入材料 5 中的数据使学生能够绘制 h 对 t² 的图像，从而根据斜率求出 g。该实验假设空气阻力可忽略，并且小球在计时器启动的瞬间从静止精确释放。

2. Experimental Apparatus | 实验器材

The key components shown in Insert 5 are an electromagnet powered by a d.c. supply, a steel ball bearing, a trapdoor mechanism that opens on impact, an electronic timer or stopwatch, a metre rule, and clamps and stands.

插入材料 5 中展示的关键部件包括由直流电源供电的电磁铁、一个钢球、碰撞时打开的触发门装置、电子计时器或秒表、一把米尺及铁架台和夹具。

The electromagnet holds the ball at a measured height h above the trapdoor. When the switch is opened, the magnetic field collapses and the ball begins to fall. The timer starts simultaneously and stops when the ball hits the trapdoor, breaking a contact.

电磁铁将小球保持在触发门上方已测定的高度 h 处。断开开关时，磁场消失，小球开始下落。计时器同步启动，当小球击中触发门并断开触点时计时停止。

3. Procedure and Measurements | 实验步骤与测量

First, set the height h to a small value, using the metre rule to measure from the bottom of the ball to the trapdoor. Ensure the electromagnet is directly above the trapdoor to minimise sideways motion. Record the fall time t three times for each height and calculate an average t to reduce random errors.

首先，将高度 h 设为一个较小值，用米尺从球的底部量到触发门。确保电磁铁位于触发门正上方以减少侧向运动。每个高度记录下落时间 t 三次，计算平均 t 以减小随机误差。

Repeat for at least five different heights, up to about 1.0 m. Insert 5 shows h values ranging from 0.200 m to 1.000 m. For each h, the timer reading t is recorded to 0.001 s. All raw data should be entered into a suitable table.

对至少五个不同的高度重复实验，最高约 1.0 m。插入材料 5 显示 h 值范围从 0.200 m 到 1.000 m。对每个 h，计时器读数 t 记录到 0.001 s。所有原始数据应填入合适的表格。

4. Data Table and Key Calculations | 数据表与关键计算

Insert 5 gives the following typical data. You are expected to calculate t² and then plot h against t². A sample table is shown below:

插入材料 5 给出如下典型数据。你需要计算 t²，然后绘制 h 对 t² 的图像。以下是一个示例表格：

h / m	t₁ / s	t₂ / s	t₃ / s	mean t / s	t² / s²
0.200	0.206	0.199	0.201	0.202	0.0408
0.400	0.288	0.284	0.286	0.286	0.0818
0.600	0.352	0.348	0.350	0.350	0.1225
0.800	0.405	0.403	0.404	0.404	0.1632
1.000	0.454	0.450	0.452	0.452	0.2043

The scatter in repeated readings is small, indicating good precision. The table is used to construct the graph required by Insert 5.

重复读数的离散程度很小，表明精密度良好。这个表格用于构建插入材料 5 所要求的图像。

5. Graph Plotting | 绘图

According to the insert, you must plot a graph of h on the y-axis against t² on the x-axis. Use sensible scales that utilise at least half the graph paper. Label axes as ‘h / m’ and ‘t² / s²’. Plot the points carefully using small crosses and draw a line of best fit.

根据插入材料，你必须绘制 h 在 y 轴、t² 在 x 轴的图像。使用合理的刻度，至少占据坐标纸的一半。坐标轴标注为 ‘h / m’ 和 ‘t² / s²’。用小叉号仔细标出数据点，并画出最佳拟合线。

The expected relationship is a straight line through the origin, because the equation of motion for free fall from rest is:

预期关系是一条过原点的直线，因为从静止自由下落的运动方程为：

h = ½ g t²

Comparing with y = m x, the gradient of the line is ½ g. If the line does not pass through the origin, a small systematic error is present.

与 y = m x 比较，直线的斜率（梯度）为 ½ g。如果直线不通过原点，则存在小的系统误差。

6. Determining g from the Graph | 从图像确定 g

To find g, select two well-separated points on the best-fit line, not the data points. For example, using (t² = 0.040 s², h = 0.196 m) and (t² = 0.200 s², h = 0.980 m):

为了求 g，在最佳拟合线上选取两个间隔较远的点，而非数据点。例如，使用 (t² = 0.040 s², h = 0.196 m) 和 (t² = 0.200 s², h = 0.980 m)：

gradient = Δh / Δ(t²) = (0.980 – 0.196) / (0.200 – 0.040) = 0.784 / 0.160 = 4.90 m/s²

Then g = 2 × gradient = 2 × 4.90 = 9.80 m/s². This agrees well with the standard value of 9.81 m/s².

则 g = 2 × 斜率 = 2 × 4.90 = 9.80 m/s²。这与标准值 9.81 m/s² 吻合得很好。

The percentage difference from the accepted value is |9.81 – 9.80| / 9.81 × 100% ≈ 0.1%, which shows excellent accuracy for this method.

与公认值的百分差为 |9.81 – 9.80| / 9.81 × 100% ≈ 0.1%，表明该方法具有极高的准确度。

7. Uncertainty and Error Analysis | 不确定度与误差分析

Insert 5 may ask for an estimate of uncertainty. The main instrumental uncertainty comes from the metre rule (±1 mm) and the timer (±0.001 s). For the smallest h = 0.200 m, the percentage uncertainty in h is (0.001/0.200)×100% = 0.5%.

插入材料 5 可能要求估计不确定度。主要的仪器不确定度来自米尺（±1 mm）和计时器（±0.001 s）。对于最小的 h = 0.200 m，h 的相对不确定度为 (0.001/0.200)×100% = 0.5%。

For time t ≈ 0.202 s, the percentage uncertainty in t is (0.001/0.202)×100% ≈ 0.5%, but t² has double the percentage uncertainty, about 1.0%. The combined uncertainty in the gradient can be found using error bars or by plotting the steepest and shallowest acceptable lines.

对于时间 t ≈ 0.202 s，t 的相对不确定度为 (0.001/0.202)×100% ≈ 0.5%，但 t² 的相对不确定度加倍，约 1.0%。可以借助误差棒或绘制最陡与最浅可接受直线来计算斜率的不确定度。

If the steepest gradient is 5.00 and the shallowest is 4.80, then g = 2 × gradient would range from 9.6 to 10.0 m/s², giving g = 9.80 ± 0.20 m/s². This range includes the true value, indicating the experiment is valid.

如果最陡斜率为 5.00，最浅斜率为 4.80，那么 g = 2 × 斜率范围为 9.6 至 10.0 m/s²，得出 g = 9.80 ± 0.20 m/s²。该区间包含真值，说明实验是有效的。

8. Significant Sources of Error | 误差的主要来源

The most significant systematic error is the delay in releasing the ball when the circuit is broken. Residual magnetism in the electromagnet can hold the ball for a few milliseconds after the timer starts, effectively increasing the measured fall time and reducing the value of g.

最显著的系统误差是电路断开时小球释放的延迟。电磁铁的剩磁可能在计时器启动后仍吸住小球几毫秒，有效增加了测量的下落时间，使 g 值偏小。

Another source is the reaction time of the trapdoor switch. If the trapdoor does not open instantly, the timer runs slightly longer. Parallax error when measuring height h with a ruler can also affect accuracy, as can any sideways movement causing the ball to miss the trapdoor centre.

另一个来源是触发门开关的反应时间。如果触发门不能瞬间打开，计时器会多运行极短时间。用米尺测量高度 h 时的视差也可能影响准确度，而侧向移动导致小球未击中触发门中心也会引入误差。

Air resistance is usually negligible for steel balls over these heights, but for a very light ball it would reduce the net acceleration, giving a lower apparent g.

对于钢球，在这些高度下空气阻力通常可忽略，但如果球很轻，阻力会减小合加速度，使表观 g 偏小。

9. Improvements to the Experiment | 实验改进

Insert 5 may ask for improvements. Place a thin piece of earthing or use a demagnetised core to reduce residual magnetism. Use a dual-beam oscilloscope or a light gate placed just below the release point to confirm that the ball starts falling exactly when the timer triggers.

插入材料 5 可能要求提出改进措施。可放置一个薄接地片或使用消磁铁芯以减小剩磁。使用双迹示波器或在释放点正下方放置一个光电门，以确认小球在计时器触发的同时开始下落。

Measure the height h with a vernier calliper or digital height gauge for better precision. To minimise the trapdoor reaction time, use an optical sensor that breaks a light beam without any mechanical delay. Repeat measurements more times and use a computer-based data logger to capture the fall time automatically.

使用游标卡尺或数字高度规测量 h，以提高精密度。为减小触发门反应时间，可使用无机械延迟的光束传感器。增加测量重复次数，并使用计算机数据采集器自动记录下落时间。

Finally, always check that the steel ball is perfectly clean and the electromagnet surface is flat to ensure instantaneous release. Drawing a graph with h vs t² naturally averages out random timing errors.

最后，始终检查钢球是否清洁、电磁铁表面是否平整，以确保瞬间释放。绘制 h-t² 图像能够自然平均掉随机的计时误差。

10. Conclusion | 结论

The experimental investigation in June 2018 Insert 5 successfully demonstrates a free-fall method to determine g using simple apparatus. By plotting h against t², the gradient equals ½ g, and the value obtained is in strong agreement with the accepted 9.81 m/s².

2018 年 6 月插入材料 5 中的实验探究成功地演示了用简单器材通过自由落体测定 g 的方法。通过绘制 h-t² 图像，斜率等于 ½ g，得到的数值与公认值 9.81 m/s² 高度吻合。

Mastering the skills of identifying errors, calculating uncertainties, and suggesting valid improvements is essential for A-Level practical-based questions. Referring to Insert 5 data and applying standard techniques such as using a best-fit line, changing variables, and controlling conditions will help you achieve full marks in such investigations.

掌握识别误差、计算不确定度并提出有效改进措施的技能，对 A-Level 实验类题目至关重要。参照插入材料 5 的数据，运用诸如最佳拟合线、变量变换和条件控制等标准技巧，将帮助你在这类探究中获得满分。

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