A-Level Physics Unit 3 Past Paper Jan19 Formula Derivation: Young’s Modulus | 2019年1月A-Level物理单元3真题公式推导：杨氏模量

📚 A-Level Physics Unit 3 Past Paper Jan19 Formula Derivation: Young’s Modulus | 2019年1月A-Level物理单元3真题公式推导：杨氏模量

In the January 2019 A-Level Physics Unit 3 practical examination, candidates were required to derive a key formula from first principles and apply it to experimental data. One of the most common tasks of this type involves the determination of the Young modulus of a metal wire. This article revisits that specific question, breaking down the step-by-step derivation of E = 4FL / (π d² ΔL), clarifying the underlying mechanics concepts, and linking each algebraic stage to the actual measurements made in the laboratory. Whether you are preparing for your own Unit 3 assessment or reviewing past paper techniques, mastering this derivation will sharpen both your mathematical and practical analysis skills.

在2019年1月的A-Level物理单元3实验考试中，考生需要从基本原理出发推导一个关键公式，并将其应用于实验数据。这类任务中最常见的一种就是测定金属丝的杨氏模量。本文回顾那道试题，逐步拆解公式 E = 4FL / (π d² ΔL) 的推导过程，阐明背后的力学概念，并将每个代数步骤与实际实验测量联系起来。无论你是在备考单元3，还是在复习真题技巧，掌握这个推导都能提升你的数学分析和实验数据处理能力。

1. Understanding the Unit 3 Practical Context | 理解单元3实验背景

Unit 3 papers in International A-Level Physics (such as Edexcel WPH03) are designed to assess practical skills, including planning, implementation, analysis, and evaluation. A typical question provides a scenario where a student hangs masses from a vertical wire, measures the resulting extension, and must derive the expression for Young modulus E. The January 2019 paper followed this format, asking for a full derivation before numerical substitution.

国际A-Level物理单元3试卷（例如爱德思WPH03）旨在考查实验技能，包括实验规划、实施、分析和评估。一道典型试题设定学生将砝码悬挂在竖直金属丝上，测量相应的伸长量，然后要求推导出杨氏模量E的表达式。2019年1月的试卷就采用了这一形式，要求先完成完整推导，再进行数值代入。

2. The Young’s Modulus Experiment Setup | 杨氏模量实验装置

The standard school laboratory setup consists of a long thin metal wire clamped at one end, running over a frictionless pulley, with a hanger for slotted masses at the free end. A fiducial marker or a travelling microscope is used to read the wire’s elongation as loads are added or removed. Two key lengths are recorded: the original length L of the wire under a small reference load, and the diameter d measured at several points with a micrometer screw gauge.

标准的学校实验室装置包括一根一端被夹紧的长细金属丝，绕过无摩擦的滑轮，自由端挂有槽码吊架。用一个指针标记或移测显微镜来读取加载增减时金属丝的伸长量。记录两个关键长度：在微小基准负载下金属丝的原长L，以及用千分尺在多个位置测量的直径d。

3. Key Measurements and Variables | 关键测量与变量

The physical quantities directly measured are force F (calculated from mass and gravitational field strength g), original length L, extension ΔL, and wire diameter d. The derived quantity is the Young modulus E. In the January 2019 exam, a data table typically listed loads in newtons and corresponding extensions in millimetres, requiring students to identify the independent variable (force) and dependent variable (extension).

直接测量的物理量是力F（由质量和重力场强g计算）、原长L、伸长量ΔL以及金属丝直径d。而推导出的量是杨氏模量E。在2019年1月的考试中，通常会有一张数据表列出以牛顿为单位的负载和以毫米为单位的对应伸长量，要求考生识别自变量（力）和因变量（伸长量）。

4. Derivation of Stress Formula | 应力公式推导

Stress σ is defined as the force applied perpendicularly per unit cross-sectional area. For a wire of cross‑sectional area A, we write σ = F / A. Here F is the tensile force, which in the experiment equals the weight of the hanging masses only if the reference load is subtracted appropriately. The derivation starts with this fundamental relationship: σ = F / A.

应力σ的定义是垂直施加在单位横截面积上的力。对于横截面积为A的金属丝，我们写为σ = F / A。这里的F是拉伸力，在实验中只要适当减去基准负载，它就等于悬挂砝码的重量。推导就从这一基本关系式开始：σ = F / A。

5. Derivation of Strain Formula | 应变公式推导

Strain ε is a dimensionless ratio describing relative deformation. For a tensile deformation, ε = extension / original length, i.e., ε = ΔL / L. The extension ΔL is the increase in length measured from the wire under zero additional load to the wire under the applied load F. The original length L is the unstretched length between the clamped and pulley contact points.

应变ε是一个无量纲的比值，用于描述相对形变。对于拉伸形变，ε = 伸长量 / 原长，即ε = ΔL / L。伸长量ΔL是从零附加负载到施加负载F时测得的金属丝长度增加量。原长L是夹紧点到滑轮接触点之间未拉伸的长度。

6. Combining Stress and Strain: Young’s Modulus | 合并应力与应变：杨氏模量

The Young modulus E is a material property defined by the ratio of tensile stress to tensile strain within the elastic limit. Hence:

E = σ / ε

Substituting the expressions for stress and strain gives:

E = (F / A) ÷ (ΔL / L) = (F / A) × (L / ΔL) = F L / (A ΔL)

This is the core formula, but it still contains the cross‑sectional area A. The next step is to replace A with the measured diameter.

杨氏模量E是一个材料属性，被定义为在弹性限度内拉伸应力与拉伸应变之比。因此：E = σ / ε。代入应力和应变的表达式便可得到：E = (F / A) ÷ (ΔL / L) = F L / (A ΔL)。这是核心公式，但仍包含横截面积A。下一步就是把A用测量的直径表示出来。

7. Expressing Cross-Sectional Area in Terms of Diameter | 用直径表示横截面积

For a wire with a circular cross‑section, the area A is given by the familiar geometry formula:

A = π (d/2)² = π d² / 4

where d is the average diameter obtained from multiple micrometer readings. Substituting this into the Young modulus equation yields:

E = F L / ((π d² / 4) × ΔL) = (F L × 4) / (π d² ΔL)

Thus the final working formula emerges.

对于圆形横截面的金属丝，面积A由熟悉的几何公式给出：A = π (d/2)² = π d² / 4，其中d是由多次千分尺读数得到的平均直径。将此代入杨氏模量方程便得：E = F L / ((π d² / 4) × ΔL) = 4 F L / (π d² ΔL)。这样最终的计算公式就出现了。

8. Final Derived Formula: E = 4FL/(π d² ΔL) | 最终推导公式

The fully derived relationship is:

E = 4 F L / (π d² ΔL)

In the January 2019 paper, students were expected to present this step‑by‑step logic neatly, often using symbols and showing cancellation explicitly. Marks are allocated for clear substitution of area and for retaining the factor of 4. Many candidates lose marks by omitting the factor 4 or misplacing π.

完整推导出的关系式为：E = 4 F L / (π d² ΔL)。在2019年1月的试卷中，学生应能清晰地展示上述分步逻辑，通常要使用符号并明确显示约分过程。正确代入面积表达式并保留因子4是得分点。许多考生因遗漏因子4或π的位置出错而失分。

9. Practical Example Using Sample Data | 使用样本数据的案例

Imagine the following data obtained in a Unit 3 practical: original length L = 2.550 m, wire diameter d = 0.276 mm = 2.76 × 10⁻⁴ m, load F = 8.40 N produced an extension ΔL = 1.82 mm = 1.82 × 10⁻³ m. Substituting into the derived formula:

E = (4 × 8.40 N × 2.550 m) / (π × (2.76 × 10⁻⁴ m)² × 1.82 × 10⁻³ m)

Calculating stepwise gives E ≈ 1.96 × 10¹¹ Pa, a typical value for steel. This aligns with the January 2019 style, where calculators must be used carefully with standard form conversions.

假设在一次单元3实验中获得了以下数据：原长L = 2.550 m，金属丝直径d = 0.276 mm = 2.76 × 10⁻⁴ m，负载F = 8.40 N 产生的伸长量ΔL = 1.82 mm = 1.82 × 10⁻³ m。代入推导出的公式：E = (4 × 8.40 N × 2.550 m) / (π × (2.76 × 10⁻⁴ m)² × 1.82 × 10⁻³ m)。逐步计算得到E ≈ 1.96 × 10¹¹ Pa，这是钢的典型值。这与2019年1月的命题风格一致，要求考生谨慎使用计算器并进行标准形式转换。

10. Uncertainty Analysis in the Derived Formula | 推导公式中的不确定性分析

Unit 3 marking schemes often require candidates to propagate percentage uncertainties. For the formula E = 4FL/(π d² ΔL), the fractional uncertainty is:

ΔE/E = √[(ΔF/F)² + (ΔL/L)² + (2 Δd/d)² + (Δ(ΔL)/ΔL)²]

Note that the uncertainty in d² is double the relative uncertainty in d, hence the factor 2. Examiners expect students to identify that the dominant contribution usually comes from the diameter measurement because it is squared and the precision of the micrometer may be smaller in relative terms.

单元3的评分方案常要求考生传递百分比不确定度。对于公式E = 4FL/(π d² ΔL)，合成不确定度为：ΔE/E = √[(ΔF/F)² + (ΔL/L)² + (2 Δd/d)² + (Δ(ΔL)/ΔL)²]。注意d²的不确定度是d相对不确定度的两倍，因此出现因子2。考官期望学生识别出主要贡献通常来自直径测量，因为它被平方了，而且千分尺的精度在相对意义上可能较短。

11. Common Mistakes and Tips | 常见错误与提示

Many students in the January 2019 sitting forgot to convert millimetres to metres, leading to unrealistic E values. Others used diameter instead of radius for area, ending up with a formula lacking the factor 4. A frequent error is to treat extension ΔL as the final length rather than the change. To avoid these, always draw a clear diagram of the wire, mark L and (L+ΔL), and double‑check unit conversions before substituting into the final expression.

在2019年1月的考试中，许多考生忘记将毫米换算为米，导致计算出不切实际的E值。另一些考生用直径代替半径来计算面积，最终得到缺少因子4的公式。一个常见错误是将伸长量ΔL当作最终长度，而不是长度的变化量。为避免这些错误，务必画出清晰的金属丝示意图，标出L和(L+ΔL)，并在代入最终表达式之前仔细检查单位换算。

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12. Conclusion and Exam Relevance | 结论与考试相关性

Mastering the derivation of E = 4FL/(π d² ΔL) is not just a box‑ticking exercise for Unit 3; it reinforces the entire structure of stress, strain, and material properties. The January 2019 past paper rewards candidates who can logically deconstruct the experiment, link measurements to physical quantities, and manipulate algebraic equations with confidence. Use this walkthrough as a template when tackling similar derivations, such as those for the resistivity of a wire (ρ = RA/L) or the acceleration due to gravity from a simple pendulum (g = 4π²L/T²).

掌握E = 4FL/(π d² ΔL)的推导不仅是单元3的一道必做题，它还巩固了应力、应变和材料性质的整体知识框架。2019年1月的真题奖励那些能够合乎逻辑地解构实验、将测量量与物理量联系起来，并自信地进行代数运算的考生。在应对类似的推导题时（比如金属丝电阻率ρ = RA/L或由单摆测量重力加速度g = 4π²L/T²），可以把这篇推演当作模板。

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