Simple Harmonic Motion for CCEA A-Level Physics | CCEA A-Level 物理：简谐运动 考点精讲

📚 Simple Harmonic Motion for CCEA A-Level Physics | CCEA A-Level 物理：简谐运动 考点精讲

Simple harmonic motion (SHM) lies at the heart of CCEA A-Level Physics, providing the theoretical framework for oscillatory systems ranging from clock pendulums to vibrations in molecules. A confident grasp of SHM requires combining kinematic descriptions with dynamic principles, energy analysis and an understanding of how resonance arises.

简谐运动(SHM)是CCEA A-Level物理的核心内容，为从钟摆到分子振动的各种振荡系统提供了理论框架。要扎实掌握简谐运动，需要将运动学描述与动力学原理、能量分析以及对共振如何产生的理解结合起来。

1. Defining Simple Harmonic Motion | 简谐运动的定义

An object exhibits simple harmonic motion when its acceleration is directly proportional to its displacement from a fixed equilibrium position and is always directed towards that equilibrium. In scalar form the condition is a ∝ −x, and the defining equation is a = −ω² x, where ω is the angular frequency.

当物体的加速度与它偏离固定平衡位置的位移成正比，并且始终指向该平衡位置时，物体做简谐运动。标量形式可表示为 a ∝ −x，其定义方程为 a = −ω² x，其中 ω 是角频率。

The necessary and sufficient condition for SHM is the presence of a linear restoring force that obeys Hooke’s law for small displacements, such as F = −kx. Any system that satisfies a = − (constant) × x will oscillate sinusoidally with a characteristic natural frequency.

简谐运动的充分必要条件是存在一个线性回复力，在小位移下满足胡克定律，例如 F = −kx。任何满足 a = − (常数) × x 的系统都将以固有的自然频率做正弦振荡。

a = − ω² x

2. The Equations of SHM | 简谐运动的基本方程

For an oscillator of amplitude A, angular frequency ω and initial phase φ, the displacement from equilibrium at time t is given by either a sine or cosine function. The most common representation is x = A sin(ωt + φ). Velocity is the first derivative: v = dx/dt = ωA cos(ωt + φ). The acceleration follows as a = d²x/dt² = −ω² A sin(ωt + φ) = −ω² x.

对于一个振幅为 A、角频率为 ω、初相为 φ 的振子，t 时刻的位移可用正弦或余弦函数表示，最常见的形式为 x = A sin(ωt + φ)。速度是一阶导数：v = dx/dt = ωA cos(ωt + φ)。加速度则为 a = d²x/dt² = −ω² A sin(ωt + φ) = −ω² x。

In many exam questions it is useful to express speed in terms of displacement: v = ± ω √(A² − x²). The maximum speed occurs as the particle passes through equilibrium (x = 0): v_max = ωA. The maximum acceleration occurs at the extremes of the motion (x = ±A): a_max = ω²A. The period T and frequency f are linked to ω by ω = 2πf = 2π/T.

在许多考题中，用位移表示速率更为方便：v = ± ω √(A² − x²)。最大速率出现在振子经过平衡位置时 (x = 0)：v_max = ωA。最大加速度出现在运动的两端 (x = ±A)：a_max = ω²A。周期 T 和频率 f 与 ω 的关系为 ω = 2πf = 2π/T。

x = A sin(ωt + φ)   v = ± ω √(A² − x²)   a = −ω² x

3. Displacement–Time and Velocity–Time Graphs | 位移–时间与速度–时间图像

The displacement–time graph for an oscillator starting from equilibrium with zero initial phase is a sine wave. When the motion starts at maximum positive displacement (x = +A) the graph is a cosine wave. CCEA questions often ask you to sketch or interpret these curves and to identify key points.

对于从平衡位置开始且初相为零的振子，位移–时间图像是一条正弦曲线。若运动从最大正位移处 (x = +A) 开始，则图像为余弦曲线。CCEA考题经常要求你画出或解读这些曲线并识别关键点。

The velocity–time graph leads the displacement graph by a quarter of a period (phase difference of π/2). When displacement is at a maximum, the velocity is momentarily zero; when the oscillator sweeps through the equilibrium position, the speed is greatest. The acceleration–time graph is always opposite in sign to the displacement, illustrating a = −ω² x directly.

速度–时间图像超前位移图像四分之一周期（相位差为 π/2）。当位移最大时，速度瞬时为零；当振子经过平衡位置时，速率最大。加速度–时间图像总是与位移符号相反，直接体现出 a = −ω² x 的关系。

4. Acceleration in SHM | 简谐运动中的加速度

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