📚 Year 12 Edexcel Physics: Case Study Workout | Year 12 Edexcel 物理案例分析实战演练
In this case study, we step through a practical investigation of Newton's second law using a light-gate system, a trolley and a pulley. You will follow the real process of collecting data, calculating acceleration, analysing uncertainties and interpreting a force–acceleration graph. This kind of hands‑on problem is typical of the Edexcel Year 12 Physics core practicals and builds essential skills for both written papers and internal assessment.
在本案例分析中，我们将一步步走过一个利用光门、小车与滑轮系统研究牛顿第二定律的实验。你将跟随真实的数据采集、加速度计算、不确定度分析以及力–加速度图像的解读过程。这类动手问题常见于 Edexcel Year 12 物理核心实验，能够为纸笔考试和校内评估打下扎实的技能基础。
1. Introduction to the Case Study | 案例研究介绍
total mass. The experiment uses a linear air track to reduce friction and a hanging mass that provides the accelerating force. By carefully transferring small masses from the trolley to the hanger, the net force changes while the total mass of the system (trolley + all added masses) stays the same. This design allows a direct test of the proportionality between force and acceleration.
一名学生被要求验证恒总质量系统中 F = ma 的关系。实验使用线性气垫导轨以减小摩擦，并通过悬挂质量提供加速力。通过将小车上的小质量逐次转移到挂钩上，净力发生变化而系统总质量（小车加所有附加质量）保持不变。这一设计使得力与加速度的正比关系能够直接被检验。
2. Experimental Setup and Equipment | 实验装置与设备
The main apparatus is a linear air track with a blower that creates a thin cushion of air, effectively removing contact friction. A trolley fitted with a rectangular card of precisely known width sits on the track. Two light gates are placed a measured distance apart along the track, each connected to a digital timer that records the interval during which the card interrupts the light beam.
主要装置是一条带吹气机的线性气垫导轨，它产生薄薄的空气垫层，有效消除了接触摩擦。一辆小车固定在导轨上，车上装有宽度精确已知的矩形挡光片。沿导轨相隔已知距离放置两个光门，每个光门连接至数字计时器，计时器记录挡光片阻断光束的时间间隔。
A light, inextensible string is attached to the trolley, passes over a low‑ ating force while keeping the total moving mass constant. A metre ruler and a set of calibrated masses (±0.1 g) are also needed.
一根轻质、不可伸长的细绳系在小车上，绕过轨道末端的低摩擦滑轮，并悬挂一个小质量挂钩。可以将质量块在小车与挂钩之间转移，以改变加速力并保持总运动质量不变。还需要一把米尺和一套已校准的质量块（±0.1 g）。
3. Data Collection Method | 数据收集方法
Before taking measurements, the track is carefully levelled. With the hanger empty, the air blower is switched on and the trolley is given a gentle push. If the track is exactly horizontal, the trolley would eventually slow down due to residual drag, so a small tilt is introduced until the trolley moves with constant velocity when given a light tap. This compensates for any residual friction.
在测量之前，需要仔细调平导轨。在挂钩空载的情况下，开启吹气机并轻推小车。如果导轨完全水平，小车会因残余阻力最终减速，因此需要引入微小倾角，直到轻推小车后它能匀速运动为止。这补偿了剩余的摩擦力。
The card width d is measured five times with a digital calliper and the average is recorded. The distance s between the mid‑points of the two light gates is measured with a metre ruler. Starting with 0.010 kg on the hanger and the remaining additional masses on top of the trolley, the system is released from rest. The timers record the intervals t₁ and t₂ for the card to pass through the first and second light gates. The measurement is repeated three times for each hanger mass, and the mean times are calculated.
挡光片宽度 d 用数字游标卡尺测量五次并取平均值记录。用米尺测量两个光门中点之间的距离 s。在挂钩上放置 0.010 kg 质量，其余附加质量则放在小车顶上，将系统从静止释放。计时器记录挡光片通过第一和第二个光门的时间间隔 t₁ 和 t₂。每种挂钩质量下重复测量三次，计算平均时间。
4. Raw Data and Initial Observations | 原始数据与初步观察
The total system mass is kept constant at Mtotal nt hanger masses m, together with the corresponding accelerating force F = mg (taking g = 9.81 m s⁻²).
系统总质量保持恒定，Mtotal = 1.000 kg。挡光片宽度 d = 0.0200 m。光门间距 s = 0.800 m。下表给出了五种不同挂钩质量 m 所测得的平均时间，以及相应的加速力 F = mg（取 g = 9.81 m s⁻²）。
| Trial | m / kg | F / N | t₁ (s) | t₂ (s) |
|---|---|---|---|---|
| 1 | 0.010 | 0.0981 | 0.0667 | 0.0407 |
| 2 | 0.020 | 0.1962 | 0.0543 | 0.0355 |
| 3 | 0.030 | 0.2943 | 0.0480 | 0.0330 |
| 4 | 0.040 | 0.3924 | 0.0438 | 0.0312 |
| 5 | 0.050 | 0.4905 | 0.0408 | 0.0298 |
ement with an uncertainty of ±0.002 m, and the calliper gave an uncertainty of ±0.0001 m for the card width.
多次读数显示 t₁ 和 t₂ 的变动可达 ±0.0005 s，这表明存在随机误差。米尺的距离测量不确定度为 ±0.002 m，游标卡尺对挡光片宽度的不确定度为 ±0.0001 m。
5. Calculating Acceleration from Light Gates | 利用光门计算加速度
time. Applying this to both light gates gives the initial speed v₁ = d / t₁ and the final speed v₂ = d / t₂. The acceleration a is then found from the kinematic relation for uniformly accelerated motion:
小车通过光门时的瞬时速度用 v = d / t 计算，其中 d 为挡光片宽度，t 为阻断时间。将此应用于两个光门，便得到初速度 v₁ = d / t₁ 和末速度 v₂ = d / t₂。然后利用匀加速运动的运动学关系求得加速度 a：
a = (v₂² − v₁²) / (2s)
he results for all trials are summarised in a table together with the derived accelerations.
以试验 1 为例，v₁ = 0.0200 / 0.0667 = 0.300 m s⁻¹，v₂ = 0.0200 / 0.0407 = 0.492 m s⁻¹，得到 a = (0.492² − 0.300²) / (2 × 0.800) = 0.095 m s⁻²。所有试验的结果连同导出的加速度汇总于表格中。
6. Uncertainty Analysis in Measurements | 测量中的不确定度分析
s = ±0.002 m, and Δt = ±0.0005 s for each time reading. The relative uncertainty in each speed v is approximately Δv/v = Δd/d + Δt/t. For a typical trial, this gives Δv/v around 1.5–2.0 %.
每个被测量都带有会传递至加速度计算的不确定度。绝对不确定度为：Δd = ±0.0001 m，Δs = ±0.002 m，以及每个时间读数的 Δt = ±0.0005 s。每个速度 v 的相对不确定度近似为 Δv/v = Δd/d + Δt/t。对于一次典型试验，Δv/v 约在 1.5–2.0 % 左右。
in a, the expression a = (v₂² − v₁²)/(2s) is differentiated. A simple approach uses worst‑ ed with the uncertainty in s, the resulting absolute uncertainty in a is typically of the order of ±0.005 m s⁻². Error bars on the graph will reflect these values.
为估算 a 的不确定度，需对 a = (v₂² − v₁²)/(2s) 进行微分。一种简单的方法是使用最不利情况界限：v² 的绝对误差约为 2vΔv，而差值 v₂² − v₁² 的误差为各绝对误差之和。再与 s 的不确定度结合，得到 a 的绝对不确定度通常在 ±0.005 m s⁻² 量级。图中的误差棒将反映这些数值。
7. Plotting Force vs Acceleration Graph | 绘制力与加速度关系图
or bars for both a (horizontal) and F (vertical, arising from the small uncertainty in the masses, ±0.0001 kg). A best‑fit straight line is drawn through the origin, as predicted by Newton's second law when the total mass is constant.
将加速力 F = mg 绘于
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