📚 Simple Harmonic Motion for IGCSE WJEC Physics: Key Concepts | IGCSE WJEC物理：简谐运动考点精讲
Simple Harmonic Motion (SHM) is a fundamental topic in the IGCSE WJEC Physics course, connecting the ideas of forces, energy and waves. This revision guide will walk you through every essential concept, from the defining condition a ∝ −x to practical experiments like measuring g with a pendulum. You will find clear explanations, worked equations and exam-focused tips to help you master SHM and avoid common pitfalls.
简谐运动是 IGCSE WJEC 物理课程中的一个核心主题，它把力、能量和波动的概念联系起来。此复习指南将带你梳理所有重要考点，从定义条件 a ∝ −x 到用单摆测量 g 的实验，配备清晰的讲解、重要公式和应试技巧，助你掌握 SHM 并避开常见失分点。
1. What is Simple Harmonic Motion? | 什么是简谐运动？
Simple Harmonic Motion (SHM) is a special type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and always acts towards that equilibrium.
简谐运动是一种特殊的周期性运动，其回复力与离开平衡位置的位移成正比，且方向始终指向平衡位置。
Mathematically, this condition is expressed as a ∝ re a is the acceleration, x is the displacement, and ω is the angular frequency of the motion.
数学上，该条件表示为 a ∝ −x，更精确的形式为 a = −ω²x，其中 a 是加速度，x 是位移，ω 是运动的角频率。
For IGCSE WJEC, you must be able to identify SHM by examining an acceleration–displacement graph: if the graph is a straight line passing through the origin with a negative gradient, the motion is SHM.
在 IGCSE WJEC 考试中，你需要通过加速度-位移图像来识别 SHM：若图像是通过原点且斜率为负的直线，则该运动为简谐运动。
2. Key Terms in SHM | 简谐运动的关键术语
Before working with equations, make sure you are confident with these definitions:
在使用公式之前，请先确保你熟悉以下定义：
- Amplitude (A): The maximum displacement from the equilibrium position, measured in metres (m). 振幅（A）：离开平衡位置的最大位移，单位为米（m）。
- Period (T): The time taken for one complete oscillation, measured in seconds (s). 周期（T）：完成一次全振动所需的时间，单位为秒（s）。
- Frequency (f): The number of complete oscillations per second, measured in hertz (Hz). f = 1/T. 频率（f）：每秒钟完成的全振动次数，单位为赫兹（Hz）。f = 1/T。
- Angular frequency (ω) The rate of change of angular position, given by ω = 2πf or ω = 2π/T, with units rad s⁻¹. 角频率（ω）：角位置的变化率，由 ω = 2πf 或 ω = 2π/T 给出，单位为 rad s⁻¹。
The maximum speed in SHM occurs as the object passes through equilibrium and is given by vmax = ωA.
物体经过平衡位置时速率最大，最大速率为 vmax = ωA。
3. The Defining Equation and Acceleration–Displacement Relationship | 定义方程与加速度-位移关系
a = −ω²x
This equation shows that the acceleration is always proportional to the displacement but in the opposite direction. The constant ω² determines how quickly the system oscillates.
该方程表明加速度始终与位移成正比但方向相反，常数 ω² 决定了系统振动的快慢。
For a horizontal mass–spring system, we can combine Hooke's law (F = −kx) with Newton's second law to obtain a = −(k/m)x, which means ω = √k/m.
对于水平弹簧振子，结合胡克定律 (F = −kx) 和牛顿第二定律可得 a = −(k/m)x，因此 ω = √k/m。
In the exam, you may be asked to explain why a certain motion is or is not SHM - always check whether the restoring force (or acceleration) is directly proportional to displacement and directed towards equilibrium.
在考试中，你可能需要解释某个运动是否为 SHM——请始终检查回复力（或加速度）是否与位移成正比且指向平衡位置。
4. Graphical Representations of SHM | 简谐运动的图像表示
against time, assuming the object starts at maximum positive displacement (t = 0, x = +A):
假设物体从最大正位移开始 (t = 0, x = +A)，可以通过以下三个重要图像来理解 SHM：
- Displacement–time is a cosine curve starting at +A. 位移-时间图是一条起始于 +A 的余弦曲线。
- Velocity–time is a negative sine curve, starting at zero, reaching maximum speed at zero displacement. 速度-时间图是一条负正弦曲线，初始速度为零，在位移为零时达到最大速率。
- Acceleration–time is a negative cosine curve, starting at −ω²A (maximum acceleration). 加速度-时间图是一条负余弦曲线，初始值为 −ω²A（最大加速度）。
The graphs are phase-shifted by T/4: velocity leads displacement by a quarter period, and acceleration is always opposite to displacement.
各图像之间相位差 T/4：速度超前位移四分之一周期，加速度始终与位移反相。
You can determine the period T and amplitude A directly from the displacement–time graph. The frequency is found from f = 1/T.
你直接从位移-时间图像上读出周期 T 和振幅 A，再由 f = 1/T 得到频率。
5. Energy Changes in SHM | 简谐运动中的能量变化
In an undamped SHM system, the total mechanical energy remains constant and is proportional to the square of the amplitude: Etotal = ½ k A² for a mass–spring system.
在无阻尼的简谐运动中，总机械能守恒，且与振幅的平方成正比；对于弹簧振子，Etotal = ½ k A²。
At the equilibrium position, the kinetic energy is maximum and the potential energy is minimum (zero for a horizontal spring). At the extreme displacements, all energy is stored as potential energy and kinetic energy is zero.
在平衡位置，动能最大而势能最小（水平弹簧的势能为零）；在最大位移处，所有能量以势能形式储存，动能为零。
tween kinetic and potential energy happens twice per cycle: KE → PE → KE → PE each period.
每个周期内，动能和势能的转换发生两次：KE → PE → KE → PE。
Understanding energy changes helps you explain why the amplitude of a freely oscillating system stays constant in the absence of friction and why damping reduces it over time.
理解能量变化有助于解释为什么无摩擦时自由振动的振幅保持不变，以及阻尼为何会随时间减小振幅。
6. The Simple Pendulum | 单摆
A simple pendulum consists of a point mass suspended by a light, inextensible string. For small angular displacements (usually less than about 10°), the motion approximates SHM.
单摆由一个质点悬挂在轻质、不可伸长的细线上构成。在小角度（通常小于约10°）下，运动可近似为简谐运动。
T = 2π√l/g
The period depends only on the length l and the gravitational field strength g; it is independent of the mass of the bob and of the amplitude (as long as the swing is small).
周期仅取决于摆长 l 和重力加速度 g，与摆球的质量及振幅（小角度下）无关。
In an exam, you may be asked to explain how changing the length affects the period, or to calculate g from measured T and l values.
考试中可能会要求你解释改变摆长如何影响周期，或根据测得的 T 和 l 值计算 g。
7. The Mass–Spring Oscillator | 弹簧振子
When a mass m is attached to a spring of force constant k, the system undergoes SHM described by:
当质量为 m 的物体连接在劲度系数为 k 的弹簧上时，系统作简谐运动，其周期为：
T = 2π√m/k
Here, the period depends on mass and spring stiffness. A heavier mass gives a longer period; a stiffer spring gives a shorter period.
这里，周期取决于质量和弹簧劲度：质量越大周期越长，弹簧越硬周期越短。
For a vertical mass–spring system, the equilibrium position is shifted by the weight mg, but the oscillations about that new equilibrium are still SHM with the same period as the horizontal case.
在竖直弹簧振子中，平衡位置因重力 mg 而下移，但围绕新平衡位置的振动仍然是简谐运动，且周期与水平情况相同。
8. Damping and Resonance (Brief Overview) | 阻尼与共振（概述）
Damping occurs when energy is gradually removed from an oscillating system by forces such as friction or air resistance. This causes the amplitude to decrease over time, though the period remains nearly unchanged for light damping.
当摩擦或空气阻力等力从振动系统中逐渐耗散能量时，便会产生阻尼。振幅随时间减小，但在轻度阻尼下周期基本不变。
Resonance happens when a periodic driving force matches the natural frequency of the system, causing a dramatic increase in amplitude. Examples include pushing a swing or the tuning of a radio circuit.
当周期性驱动力与系统的固有频率
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