📚 PH01-INS Experiment Analysis: Investigating Newton’s Second Law | PH01-INS 实验探究：验证牛顿第二定律
This article explores the practical investigation from the PH01-INS International AS Physics Insert, dated 9 January 2023. It provides a step-by-step analysis of an experiment designed to verify Newton’s second law of motion, using a trolley, pulley, and light gates. Understanding this investigation will strengthen your data-handling, graphing, and evaluation skills for the Unit 1 exam.
本文深入剖析 2023 年 1 月 9 日 PH01-INS 国际 AS 物理插页中的实验探究题。该实验旨在利用小车、滑轮和光电门验证牛顿第二定律。通过逐步解析装置、步骤、数据处理与误差分析，帮助考生扎实掌握单元一所需的实验技能与评估方法。
1. Introduction | 实验背景简介
The Insert presents a familiar dynamics experiment. A trolley of constant mass is pulled along a track by a falling mass, with the driving force varied by changing the hanging mass. Two light gates record the time taken for a card of known length to interrupt the beams, allowing the acceleration to be calculated. The core relationship under investigation is F = ma.
插页中展示了一个经典的动力学实验。保持小车总质量不变，通过改变跨过滑轮的悬挂砝码质量来调整拉力。轨道上安装两个光电门，利用已知宽度的遮光卡片记录通过两门的时间，从而计算出加速度。其本质是探究 F = ma 这一核心关系。
2. Apparatus and Setup | 实验器材与装置
The apparatus includes an air track or low-friction dynamics track, a trolley fitted with a double-interrupter card (e.g., width 10.0 cm), two light gates connected to a data logger or timer, a pulley clamped to the end of the track, a length of inextensible string, and a set of slotted masses. The trolley is attached by the string to a mass hanger passing over the pulley.
实验装置包括气垫导轨或低摩擦动力学轨道、装有双遮光卡片（例如宽度 10.0 cm）的小车、两个连接数据采集器或计时器的光电门、固定在轨道末端的滑轮、一根不可伸长的细绳以及一套槽码。小车通过细绳绕过滑轮与砝码吊盘相连。
3. Independent, Dependent and Control Variables | 自变量、因变量与控制变量
The independent variable is the accelerating force F, provided by the weight of the hanging mass (F = mg). The dependent variable is the acceleration a of the trolley. To keep the total mass of the accelerating system constant, any masses removed from the hanger should be transferred to the trolley. This ensures that the overall mass being accelerated (trolley + hanger + all added masses) does not change.
自变量是加速力 F，由悬挂砝码的重力提供 (F = mg)。因变量为小车的加速度 a。为保持加速系统总质量不变，从吊盘上移走的每一个砝码都必须转移到小车顶部。这样，整个被加速物体的总质量（小车 + 吊盘 + 所有附加砝码）保持恒定。
4. Experimental Procedure | 实验步骤
Set up the track horizontally, compensating for friction if necessary by slightly tilting the track until the trolley moves at constant speed when given a gentle push. Place one light gate at a fixed starting point and the second at a set distance s apart (e.g., 0.800 m). Start with the largest hanging mass (e.g., 100 g). Release the trolley from rest and record the times t₁ and t₂ for the card to pass through each light gate.
将轨道调至水平，必要时可微倾轨道补偿摩擦，直至轻推小车时它能匀速运动。将第一个光电门置于固定起点，第二个光电门与之相距一定距离 s（例如 0.800 m）。使用最大悬挂质量（如 100 g）开始实验。从静止释放小车，记录遮光卡片通过两个光电门的时间 t₁ 和 t₂。
Repeat each measurement twice to check reliability and calculate an average. Then move one slotted mass from the hanger to the trolley, reducing F but keeping total mass unchanged, and repeat the procedure. Collect data for at least six different forces.
每项测量重复两次以检验可靠性并取平均值。然后将一个槽码从吊盘移到小车上，减小 F 但同时保持总质量不变，重复上述过程。至少收集六组不同力下的数据。
5. Data Collection and Recording | 数据收集与记录
Design a table with columns for hanging mass m (kg), force F (N), times t₁ and t₂ (s), initial velocity u (m s⁻¹), final velocity v (m s⁻¹), and acceleration a (m s⁻²). The Insert might show a sample table that candidates need to complete. Use the card width d to calculate u = d / t₁ and v = d / t₂, then find acceleration using v² = u² + 2as.
设计一个表格，列明悬挂质量 m (kg)、力 F (N)、时间 t₁ 和 t₂ (s)、初速度 u (m s⁻¹)、末速度 v (m s⁻¹) 以及加速度 a (m s⁻²)。插页可能给出了需要考生填充的示范表格。用遮光卡片宽度 d 计算 u = d / t₁ 和 v = d / t₂，再通过 v² = u² + 2as 求出加速度。
A typical processed data set from the Insert could be:
插页中一个典型的数据集可能如下：
| m / kg | F = mg / N | t₁ / s | t₂ / s | u / m s⁻¹ | v / m s⁻¹ | a / m s⁻² |
|---|---|---|---|---|---|---|
| 0.100 | 0.981 | 0.362 | 0.241 | 0.276 | 0.415 | 0.900 |
| 0.080 | 0.785 | 0.405 | 0.270 | 0.247 | 0.370 | 0.712 |
| 0.060 | 0.589 | 0.458 | 0.315 | 0.218 | 0.317 | 0.497 |
6. Graph Plotting and Interpretation | 作图与图线解读
Plot a scatter graph of acceleration a (on the y-axis) against force F (on the x-axis). Use large, well-chosen scales to occupy more than half the grid. Plot points with small crosses and draw the line of best fit. According to Newton’s second law, a = F / M, so the graph should be a straight line through the origin with gradient equal to 1 / M, where M is the total system mass.
绘制加速度 a（在 y 轴）随力 F（在 x 轴）变化的散点图。选用占据一半以上网格的大尺度坐标。用小十字标出数据点，并画出最佳拟合线。根据牛顿第二定律 a = F / M，图像应是一条过原点的直线，其斜率等于 1 / M，M 为系统总质量。
If the line does not pass through the origin, a small positive intercept on the F-axis suggests friction or other systematic errors. The gradient can be calculated from a large triangle and should be compared with the theoretical gradient using the known total mass.
若直线不过原点而在 F 轴上有微小正截距，则表明存在摩擦或其他系统误差。斜率应通过大三角形计算，并与根据已知总质量得出的理论斜率进行比较。
7. Calculating Uncertainty | 不确定度计算
The Insert may ask for percentage uncertainty in the acceleration. Since a = (v² – u²) / (2s), the uncertainty in a depends on the uncertainties in the times t₁ and t₂, the card width d, and the distance s. The absolute uncertainty in a single time measurement is typically the resolution of the timer, e.g. ±0.001 s, but it is more realistic to use the spread of repeat readings to calculate a ±absolute uncertainty.
插页可能要求计算加速度的百分比不确定度。由于 a = (v² – u²) / (2s)，a 的不确定度取决于时间 t₁ 与 t₂、卡片宽度 d 以及距离 s 的不确定度。单次时间测量的绝对不确定度通常是计时器的分辨率，例如 ±0.001 s，但更实际的做法是利用重复读数的偏差范围来计算 ±绝对不确定度。
For example, if two measurements of t₁ are 0.362 s and 0.366 s, the average is 0.364 s and the absolute uncertainty is (0.366 – 0.362)/2 = ±0.002 s. The percentage uncertainty in a can be found by combining percentage uncertainties in u and v. Large uncertainties suggest a particular data point could be an outlier.
例如，若 t₁ 的两次测量值为 0.362 s 和 0.366 s，平均值为 0.364 s，绝对不确定度为 (0.366 – 0.362)/2 = ±0.002 s。a 的百分比不确定度可通过合成 u 和 v 的百分比不确定度求得。较大的不确定度可能意味着某个数据点为异常值。
8. Sources of Error and Improvements | 误差来源与改进
Common systematic errors include friction in the pulley or between the trolley and track, the string not being parallel to the track, and the mass of the string itself. These can cause the graph not to pass through the origin and lead to an overestimate of the total mass. To reduce friction, use an air track or ensure the track is properly lubricated and compensated.
常见的系统误差包括滑轮摩擦或小车与轨道间摩擦、细绳未与轨道平行以及细绳本身的质量。这些误差会导致图线不过原点并可能高估系统总质量。为减小摩擦，可使用气垫导轨或确保轨道得到充分润滑及补偿。
Random errors arise from reaction times if manual release is used, parallax errors when reading the position of light gates, and timing uncertainties. Improvements include using an electromagnet for clean release, ensuring the card is rigid and vertical, and increasing the separation s of the gates to make time intervals larger, reducing the fractional uncertainty.
随机误差来源于手动释放的反应时间、读取光电门位置时的视差以及计时不确定度。改进方法包括使用电磁铁实现无扰动释放，确保遮光卡片坚固且垂直，并增大两个光电门的间距 s 以使时间间隔增大，从而降低分数不确定度。
9. Safety Precautions | 安全注意事项
Secure the track on a stable bench. Place a soft mat or padded landing area beneath the falling masses to prevent impact damage. Keep fingers clear of the pulley and string during release. Do not use excessively large hanging masses that could cause the trolley to accelerate violently off the track.
将轨道固定在稳定的实验台上。在悬挂的砝码下方放置软垫或缓冲垫，防止撞击损坏。释放时手指远离滑轮与细绳。切勿使用过大的悬挂质量，以免小车猛烈加速冲出轨道。
10. Conclusion | 实验结论
The investigation verifies Newton’s second law within the limits of experimental uncertainty. The a-F graph yields a straight line through the origin, confirming proportionality between acceleration and resultant force when total mass remains constant. Comparing the experimental gradient with 1 / M allows evaluation of accuracy and highlights the influence of systematic errors in a real laboratory environment.
在实验不确定度范围内，该探究验证了牛顿第二定律。a-F 图线呈现过原点的直线，证实了总质量不变时加速度与合力成正比。将实验斜率与 1 / M 相比较，可以评估实验的准确度，并突显真实实验环境中系统误差的影响。
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