📚 Newton's Second Law Case Study Drill | 牛顿第二定律案例分析实战演练
This WJEC AS Physics case study walks you through a classic practical investigation into Newton's second law using an air track, light gates and a pulley system. You will analyse real-world data, carry out graphical analysis, evaluate uncertainties and answer the types of questions that appear on examination papers. The focus is on building the skills needed to score full marks on practical-based case study questions.
本案例针对 WJEC AS 物理考试，带你亲历一个利用气垫导轨、光门和滑轮系统验证牛顿第二定律的经典实验。你将分析真实数据、绘制图形、评定不确定度，并回答考试中常见的题型。重点在于训练你在实验案例分析题中获取满分的实战能力。
1. Case Study Overview | 案例概览
This case study centres on verifying Newton' how the acceleration of a trolley depends on the net force when the total mass of the system is kept constant. A linear air track is used to minimise friction, and a light inextensible string passing over a pulley transfers the weight of a small hanging mass to the trolley. By moving masses between the trolley and the hanger, the total accelerating mass remains unchanged, which simplifies the relationship to a ∝ F.
本案例聚焦于验证牛顿第二定律 F = ma，方法是在保持系统总质量不变的前提下，探究滑块加速度与净力之间的关系。实验采用气垫导轨以减小摩擦，一根轻质且不可伸长的细线跨过滑轮，将悬挂小重物的重力传递给滑块。通过在滑块与挂钩之间转移槽码，保持加速系统的总质量不变，从而使 a 与 F 成正比关系得以简化验证。
2. Apparatus and Setup | 仪器与装置
The apparatus includes a linear air track, a glider fitted with an interrupt card of known width d, two light gates connected to a data logger, a low-friction pulley, an inextensible string, a set of slotted masses, a metre rule and an electronic balance. The track must be carefully levelled and then tilted very slightly to compensate for any residual friction, so the glider moves with constant velocity when given a gentle push without the hanging mass attached.
主要仪器包括气垫导轨、装有已知宽度 d 遮光板的滑块、两个与数据记录器相连的光门、一个低摩擦滑轮、一根不可伸长的细线、一组槽码、一把米尺和一台电子天平。导轨需要仔细调平，然后略微倾斜以补偿残余摩擦，使得滑块在不连接悬挂重物时受到轻推后能匀速运动。
The interrupt card width d is measured with vernier callipers, and the distance s between the light gates is measured with a metre rule. The total mass of the glider plus all available slotted masses is measured as Mtotal before the experiment begins. Keeping Mtotal constant throughout the experiment is a key design feature.
遮光板宽度 d 用游标卡尺测量，两光门间距 s 用米尺测量。实验开始前，测量滑块与全部可用槽码的总质量 Mtotal。在整个实验过程中保持 Mtotal 恒定是本设计的关键特征。
3. Experimental Procedure | 实验步骤
Place all slotted masses on the glider initially. Transfer a small mass mhanging to the hanger at the end of the string; this provides an accelerating force F = mhanging rvals t₁ and t₂ for the interrupt card to pass through the first and second light gates. The instantaneous speeds are v₁ = d / t₁ and v₂ = d / t₂, and the acceleration is calculated using a = (v₂² − v₁²) / (2s).
开始时将所有槽码置于滑块上。移取一小块质量 mhanging 挂到细线末端的挂钩上，该质量产生加速力 F = mhanging g，其中 g = 9.81 m s⁻²。每次释放滑块时，需从第一个光门前方的相同起始位置释放。数据记录器记录遮光板通过第一和第二光门的时间间隔 t₁ 和 t₂。瞬时速度由 v₁ = d / t₁ 和 v₂ = d / t₂ 算得，加速度则通过 a = (v₂² − v₁²) / (2s) 求出。
For the next trial, increase mhanging and simultaneously remove an equal mass from the glider, placing it aside so it is no longer part of the moving system. This keeps the total accelerating mass constant. Repeat the measurement for at least six different values of F. Record all data in a suitable table.
下一轮试验中，增大 mhanging，同时从滑块上取下等质量的槽码放置一旁，使其不再属于运动系统的一部分。这样即可保持总加速质量不变。至少改变六次 F 值并重复测量，将所有数据记录在合适的表格中。
4. Data Collection and Presentation | 数据收集与呈现
The fixed parameters are: d = 0.100 m, s = 0.800 m, total system mass Mtotal re shown in the table below. The acceleration has been calculated from the measured times.
固定参数为：d = 0.100 m，s = 0.800 m，系统总质量 Mtotal = 0.400 kg。下表中的加速度已由测得的时间算出。
| mhanging / kg | F / N | t₁ / s | t₂ / s | v₁ / m s⁻¹ | v₂ / m s⁻¹ | a / m s⁻² |
|---|---|---|---|---|---|---|
| 0.020 | 0.196 | 0.452 | 0.110 | 0.221 | 0.909 | 0.486 |
| 0.040 | 0.392 | 0.319 | 0.078 | 0.313 | 1.282 | 0.966 |
| 0.060 | 0.589 | 0.261 | 0.063 | 0.383 | 1.587 | 1.483 |
| 0.080 | 0.785 | 0.226 | 0.055 | 0.442 | 1.818 | 1.944 |
An initial inspection of the table shows that as F increases, a also increases, suggesting a proportional relationship. The next step is to plot a graph to confirm this quantitatively.
初步观察表格可以发现，随着 F 增大，a 也增大，暗示两者呈正比关系。下一步是绘制图形以定量验证。
5. Graphical Analysis and Line of Best Fit | 图形分析与最佳拟合线
Plot a graph of a (y-axis) against F (x-axis). The four data points should lie close to a straight line passing through the origin, consistent with a ∝ F for a constant mass. Draw a line of best fit that has an equal number of points on either side and passes through as many points as possible. Do not force the line through the origin unless the data strongly suggest it.
绘制 a（纵轴）随 F（横轴）变化的图形。四个数据点应分布在一条通过原点的直线附近，这与质量恒定时 a ∝ F 的预期一致。画出最佳拟合线，使直线两侧点数大致相等，并尽可能通过较多的点。除非数据强烈支持，否则不要强制直线通过原点。
e line. For the data above, points (0.200, 0.500) and (0.800, 2.000) might lie on the best-fit line, giving gradient = (2.000 − 0.500) / (0.800 − 0.200) = 1.500 / 0.600 = 2.50. The expected gradient is 1 / Mtotal = 1 / 0.400 = 2.50, so the agreement is excellent.
在最佳拟合线上选取两个间距较大的点计算斜率，若数据点不完全在直线上则不要直接使用原始数据点。上述数据中，点 (0.200, 0.500) 和 (0.800, 2.000) 可能落在最佳拟合线上，得出斜率 = (2.000 − 0.500) / (0.800 − 0.200) = 1.500 / 0.600 = 2.50。预期斜率是 1 / Mtotal = 1 / 0.400 = 2.50，两者吻合极佳。
intercept may indicate a systematic error such as incomplete friction compensation. In this case the intercept is approximately 0.00 m s⁻², confirming a good experimental design.
最佳拟合线的 y 截距理论上应为零，可由 F = 0 时的 a 值读取。微小的非零截距可能暗示系统误差，如摩擦补偿不完全。本例中截距约为 0.00 m s⁻²，证实实验设计较好。
6. Calculating the Gradient and Intercept | 梯度与截距的计算
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