📚 PH04 IAL Physics Jan 2023 Derivations Explained | PH04 国际A Level物理 2023年1月 公式推导解析
Understanding how to derive key equations is essential for success in the IAL Physics Unit 4 paper. The January 2023 PH04 exam included several questions that tested students’ ability to construct derivations, particularly in gravitational fields and simple harmonic motion. This article breaks down these derivations step by step, providing the logical chain and mathematical techniques required.
理解如何推导关键方程对于在IAL物理第四单元考试中取得优异成绩至关重要。2023年1月的PH04试卷中包含了若干考查学生推导能力的题目，主要集中在引力场和简谐运动领域。本文将逐步拆解这些推导过程，展示所需的逻辑链和数学方法。
1. Exam Context and Derivation Requirements | 考试背景与推导要求
The PH04 paper, covering further mechanics, fields, and particles, routinely awards marks for showing the derivation of standard results. In the Jan 2023 sitting, Question 5 targeted the derivation of satellite motion equations, while Question 9 assessed the derivation of displacement in simple harmonic motion from Newton’s second law.
PH04试卷涵盖进阶力学、场和粒子物理，常对标准结论的推导过程设分。在2023年1月考卷中，第5题针对卫星运动方程的推导，第9题则考查了从牛顿第二定律出发推导简谐运动位移方程的能力。
These derivations not only test mathematical manipulation but also a deep understanding of physical principles such as circular motion, gravitational force, and restoring forces.
这些推导不仅考查数学变换，还考查对圆周运动、万有引力和回复力等物理原理的深刻理解。
2. Core Derivation Topics | 核心推导主题
The two main derivation blocks covered in the paper were: (1) orbital mechanics – deriving orbital speed, Kepler’s third law, and energy; (2) simple harmonic motion – deriving x = A cos(ωt + φ) from the second-order differential equation.
试卷涵盖的两个主要推导模块为：（1）轨道力学——推导轨道速度、开普勒第三定律及能量；（2）简谐运动——从二阶微分方程推导 x = A cos(ωt + φ) 。
3. Fundamental Law: Newton’s Law of Gravitation | 基本定律：牛顿万有引力定律
The starting point for gravitational field derivations is Newton’s law of universal gravitation: Fg = G M m / r², where G is the gravitational constant, M and m are the masses, and r is the distance between their centres.
引力场推导的起点是牛顿万有引力定律：Fg = G M m / r² ，其中 G 为引力常数，M 和 m 为质量，r 为两物体中心距离。
This inverse-square law provides the force that holds planets and satellites in their orbits. It is always attractive and acts along the line joining the two masses.
这一定律遵循平方反比关系，提供了维系行星和卫星在其轨道运行的力。引力始终为吸引力，并且作用在两物体连线上。
4. Deriving Centripetal Force for Orbital Motion | 推导轨道运动所需的向心力
Any object moving in a circular path of radius r with constant speed v experiences a centripetal acceleration ac = v² / r directed towards the centre. Hence the required centripetal force is Fc = m v² / r.
任何物体以恒定速率 v 在半径为 r 的圆周上运动时，都会受到指向圆心的向心加速度 ac = v² / r 。因此所需的向心力为 Fc = m v² / r 。
For a satellite of mass m orbiting a central mass M, the centripetal force is provided entirely by the gravitational attraction.
对于质量为 m 且环绕中心质量 M 的卫星，向心力完全由万有引力提供。
5. Equating Gravitational Force and Centripetal Force | 引力与向心力平衡
Setting the gravitational force equal to the centripetal force gives: G M m / r² = m v² / r.
令万有引力等于向心力，得到：G M m / r² = m v² / r 。
Cancelling one factor of m from both sides and one r from the denominator yields a simplified expression.
从两边约去因子 m ，再约去分母中的一个 r ，可得到简化式。
G M / r = v²
6. Obtaining Orbital Speed v = √(GM/r) | 得出轨道速度公式
Taking the square root of both sides provides the well-known expression for the orbital speed:
两边开平方得到著名的轨道速度公式：
v = √(G M / r)
This result shows that the orbital speed of a satellite does not depend on its own mass m but only on the mass of the central body and the orbital radius. The closer the satellite, the greater its speed.
这一结果表明卫星的轨道速度与其自身质量 m 无关，只取决于中心天体的质量和轨道半径。卫星离中心越近，速度越大。
7. Deriving Kepler’s Third Law T² ∝ r³ | 推导开普勒第三定律
The orbital period T is the time taken for one complete revolution. For circular motion, T = 2πr / v. Substituting v = √(GM / r) gives:
轨道周期 T 是卫星完成一整周所需的时间。对于圆周运动，T = 2πr / v 。代入 v = √(GM / r) 得到：
T = 2πr / √(G M / r) = 2π √(r³ / G M)
Squaring both sides leads to T² = (4π² / G M) r³, demonstrating that T² is proportional to r³, exactly as Kepler stated for planetary motion.
两边平方可得 T² = (4π² / G M) r³ ，这表明 T² 与 r³ 成正比，正与开普勒对行星运动的陈述一致。
8. Deriving Total Energy of an Orbiting Satellite | 推导轨道卫星的总能量
The total mechanical energy E of a satellite is the sum of its kinetic energy K = ½ m v² and its gravitational potential energy U = – G M m / r. Substituting v² = G M / r into K gives K = ½ m (G M / r) = G M m / (2r).
卫星的总机械能 E 为动能 K = ½ m v² 与引力势能 U = – G M m / r 之和。将 v² = G M / r 代入动能表达式得 K = ½ m (G M / r) = G M m / (2r) 。
Therefore, the total energy is E = K + U = G M m / (2r) – G M m / r = – G M m / (2r).
因此总能量为 E = K + U = G M m / (2r) – G M m / r = – G M m / (2r) 。
The negative sign indicates the satellite is in a bound orbit. To escape, a total energy of at least zero is required, leading directly to the escape velocity formula.
负号表明卫星处于束缚轨道。要逃逸，总能量至少需为零，这直接导出逃逸速度公式。
9. Simple Harmonic Motion: Restoring Force Equation | 简谐运动：回复力方程
Simple harmonic motion (SHM) occurs when the net force on an object is directly proportional to its displacement from equilibrium and acts in the opposite direction: F = – k x, where k is a constant.
当物体所受合力与其离开平衡位置的位移成正比且方向相反时，即发生简谐运动（SHM）：F = – k x ，
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