📚 AS-level Physics Unit 3 Question Paper Jun19 Formula Derivation | AS 物理：2019年6月单元3试卷公式推导
In the AS Physics Unit 3 examination (especially the June 2019 paper), candidates often need to derive formulas from experimental data, linearise equations, and calculate physical quantities along with their uncertainties. This article revisits the essential formula derivations that appear regularly, helping you build the analytical skills required for practical assessments.
在 AS 物理单元 3 考试（特别是 2019 年 6 月试卷）中，考生经常需要从实验数据推导公式、将方程线性化，并计算物理量及其不确定度。本文重温经常出现的重点公式推导，帮助您建立实验评估所需的解析技能。
1. Deriving g from a Simple Pendulum | 单摆求重力加速度
The period T of a simple pendulum of length L is given by T = 2π√(L/g). To find g experimentally, we square both sides and obtain a linear relationship between T² and L.
单摆周期 T 与摆长 L 的关系为 T = 2π√(L/g)。为了实验测定 g，我们将该式两边平方，得到 T² 与 L 之间的线性关系。
Squaring yields T² = (4π²/g)L. If we plot T² on the y‑axis against L on the x‑axis, the data points should lie on a straight line passing through the origin with slope = 4π²/g.
平方后得到 T² = (4π²/g)L。以 T² 为纵轴、L 为横轴作图，数据点应落在一条通过原点的直线上，斜率 = 4π²/g。
From the graph, the experimental value of g is g = 4π² / slope. If the measured slope is m, then g = 4π²/m. The percentage uncertainty in g can be estimated by combining the uncertainty in the slope with any small uncertainty in π (usually negligible) using standard uncertainty propagation rules.
从图中可得 g 的实验值 g = 4π² / 斜率。若测得的斜率为 m，则 g = 4π²/m。g 的百分不确定度可通过合成斜率的不确定度与 π 的不确定度（通常可忽略）得出。
T = 2π√(L/g) → T² = (4π²/g)L → g = 4π² / slope
2. Resistivity of a Wire | 导线电阻率
The resistance R of a uniform metal wire depends on its resistivity ρ, length L, and cross‑sectional area A: R = ρL/A. For a wire of diameter d, A = πd²/4, so R = 4ρL/(πd²).
均匀金属导线的电阻 R 取决于其电阻率 ρ、长度 L 和横截面积 A：R = ρL/A。对于直径为 d 的导线，A = πd²/4，因此 R = 4ρL/(πd²)。
Rearranging gives the working formula for resistivity: ρ = Rπd²/(4L). In the experiment, R is obtained from V/I, L is measured with a metre rule, and d is measured with a micrometer screw gauge. If several wires of different L but identical d and material are tested, plotting R against L gives a straight line of slope ρ/A, from which ρ can be extracted. For a single wire, the formula is used directly.
整理后得到电阻率的工作公式：ρ = Rπd²/(4L)。实验中，R 由 V/I 求得，L 用米尺测量，d 用螺旋测微器测量。如果测试多根不同 L 但相同 d 和材料的导线，绘制 R 对 L 的图线是一条斜率为 ρ/A 的直线，由此可求出 ρ。对于单根导线，直接使用该公式。
The relative uncertainty in ρ is found by adding relative uncertainties: Δρ/ρ = ΔR/R + 2Δd/d + ΔL/L. This follows from the multiplication/division rule of error propagation.
ρ 的相对不确定度通过合成相对不确定度求得：Δρ/ρ = ΔR/R + 2Δd/d + ΔL/L。这源自误差传递的乘除规则。
ρ = RA/L = Rπd²/(4L) ; Δρ/ρ = ΔR/R + 2Δd/d + ΔL/L
3. Young’s Modulus from a Wire Extension | 金属丝杨氏模量
Young’s modulus E is defined as stress/strain = (F/A) / (ΔL/L) = FL/(A ΔL). For a metal wire of diameter d, A = πd²/4, and the stretching force F = mg, so E = 4mgL/(πd² ΔL).
杨氏模量 E 定义为应力/应变 = (F/A) / (ΔL/L) = FL/(A ΔL)。对于直径为 d 的金属丝，A = πd²/4，拉伸力 F = mg，因此 E = 4mgL/(πd² ΔL)。
In the experiment, a series of masses m is added and the corresponding extension ΔL is recorded. Plotting F (mg) on the y‑axis against ΔL on the x‑axis yields a straight line whose slope = EA/L. Hence E = (slope × L)/A = (slope × L) / (πd²/4). The percentage uncertainty in E combines uncertainties in slope, L, and d.
实验中，依次增加质量 m 并记录相应的伸长量 ΔL。以 F (mg) 为纵轴、ΔL 为横轴作图，得到一条斜率为 EA/L 的直线。因此 E = (斜率 × L)/A = (斜率 × L) / (πd²/4)。E 的百分不确定度需合成斜率、L 和 d 的不确定度。
E = FL/(A ΔL) = 4mgL/(πd² ΔL) ; slope = EA/L → E = (slope × L)/A
4. Acceleration from a Ticker Tape | 打点纸带求加速度
A ticker‑timer operating at 50 Hz produces dots at intervals of T = 0.02 s. One common method to find acceleration is to measure the length of two successive tape segments, s₁ and s₂, each spanning the same number of time intervals n.
频率为 50 Hz 的打点计时器产生时间间隔 T = 0.02 s 的点。求加速度的一种常用方法是测量两段连续的纸带长度 s₁ 和 s₂，每段涵盖相同的时间间隔数 n。
The initial velocity is u at the start of s₁. Using s = ut + ½at², for the first segment of duration nT we have s₁ = u(nT) + ½a(nT)². For the next segment, the initial velocity is u + a(nT), so s₂ = (u + a(nT))(nT) + ½a(nT)². Subtracting the two equations gives s₂ – s₁ = a (nT)². Therefore the acceleration a = (s₂ – s₁) / (nT)². When n = 1, the formula simplifies to a = (s₂ – s₁) / T².
设 s₁ 起点初速度为 u。根据 s = ut + ½at²，对于持续时间为 nT 的第一段纸带有 s₁ = u(nT) + ½a(nT)²。对于下一段，初速度为 u + a(nT)，因此 s₂ = (u + a(nT))(nT) + ½a(nT)²。两式相减得到 s₂ – s₁ = a (nT)²。因此加速度 a = (s₂ – s₁) / (nT)²。当 n = 1 时，公式简化为 a = (s₂ – s₁) / T²。
Alternatively, average velocities v₁ = s₁/(nT) and v₂ = s₂/(nT) can be used with a = (v₂ – v₁) / (nT). Both approaches are acceptable in AS practical exams.
或者，可以用平均速度 v₁ = s₁/(nT) 和 v₂ = s₂/(nT)，再通过 a = (v₂ – v₁) / (nT) 计算。这两种方法在 AS 实验考试中均可接受。
s₂ – s₁ = a T² (for n=1) or a = (s₂ – s₁)/(nT)²
5. Propagation of Uncertainties | 不确定度的传递
When a quantity Q is derived from measured quantities x, y, … , the uncertainty ΔQ must be calculated from the individual uncertainties. The rules depend on the mathematical operation.
当一个量 Q 由测量值 x、y … 导出时，必须根据各个不确定度计算 ΔQ。规则取决于数学运算类型。
For addition or subtraction, Q = x ± y, absolute uncertainties add: ΔQ = Δx + Δy. For multiplication or division, Q = xy or Q = x/y, relative uncertainties add: ΔQ/Q = Δx/x + Δy/y. For a power law, Q = xⁿ, the relative uncertainty multiplies: ΔQ/Q = n Δx/x. These results follow from calculus approximations and are standard in the AS syllabus.
对于加减运算，Q = x ± y，绝对不确定度相加：ΔQ = Δx + Δy。对于乘除运算，Q = xy 或 Q = x/y，相对不确定度相加：ΔQ/Q = Δx/x + Δy/y。对于幂运算，Q = xⁿ，相对不确定度乘以指数：ΔQ/Q = n Δx/x。这些结果源自微积分近似，是 AS 大纲的标准内容。
The table below summarises the most frequently used rules.
下表总结了最常用的规则。
| Operation | Formula for Q | Uncertainty rulePublished by TutorHao | AS Physics Revision Series | aleveler.com
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