📚 Simple Harmonic Motion for IGCSE WJEC Maths | IGCSE WJEC 数学：简谐运动考点精讲
In the IGCSE WJEC Maths specification, simple harmonic motion (SHM) is a key topic that bridges algebra, trigonometry, and mechanics. SHM describes a special type of periodic motion where an object oscillates about an equilibrium position with a restoring force proportional to its displacement. Understanding SHM not only helps you solve exam questions but also deepens your appreciation of how mathematics models real-world phenomena like pendulums and vibrating springs. This article will guide you through the essential concepts, formulas, graphs, and problem-solving techniques needed to excel in this topic.
在IGCSE WJEC数学大纲中，简谐运动（SHM）是一个连接代数、三角学和力学的关键主题。简谐运动描述了一种特殊的周期性运动，物体围绕平衡位置振动，回复力与其位移成正比。理解简谐运动不仅能帮助你解答考试问题，还能加深你对数学如何模拟单摆和弹簧振动等现实世界现象的理解。本文将带你梳理这一主题所需的核心概念、公式、图像和解题技巧。
1. Defining Simple Harmonic Motion (SHM) | 定义简谐运动
Simple harmonic motion is defined as oscillatory motion in which the acceleration of the particle is directly proportional to its displacement from a fixed point and is always directed towards that point. Mathematically, this condition is expressed as a = -ω²x, where a is acceleration, x is displacement, and ω (omega) is a positive constant called the angular frequency. The negative sign indicates that acceleration and displacement are in opposite directions, which is the defining characteristic of SHM.
简谐运动被定义为一种振荡运动，其中粒子的加速度与其相对于固定点的位移成正比，并且始终指向该固定点。数学上，这一条件表示为 a = -ω²x，其中 a 是加速度，x 是位移，ω（欧米伽）是一个称为角频率的正常数。负号表示加速度与位移方向相反，这正是简谐运动的决定性特征。
If a system obeys a = -ω²x, it undergoes SHM regardless of whether it is a mass on a spring or a simple pendulum swinging through a small angle.
只要一个系统遵循 a = -ω²x，无论它是由弹簧连接的物块，还是小幅摆动的单摆，它所做的就是简谐运动。
2. Key Quantities in SHM | 简谐运动中的关键量
Several quantities are fundamental when describing SHM: amplitude (A) – the maximum displacement from equilibrium; period (T) – the time taken for one complete oscillation; frequency (f) – the number of oscillations per unit time; and angular frequency (ω) – related to frequency by ω = 2πf. The displacement x at any time t can be expressed as x = A cos(ωt) or x = A sin(ωt), depending on the starting conditions.
描述简谐运动时有几个基本量：振幅（A）—— 离开平衡位置的最大位移；周期（T）—— 完成一次完整振动所需的时间；频率（f）—— 单位时间内的振动次数；以及角频率（ω）—— 与频率的关系为 ω = 2πf。任意时刻 t 的位移 x 可以表示为 x = A cos(ωt) 或 x = A sin(ωt)，具体取决于初始条件。
In many WJEC exam questions, you will be given the equation x = A cos(ωt) with the motion starting at maximum displacement. Make sure you know that the relationship T = 1/f and ω = 2π/T also hold.
在许多 WJEC 考题中，会给出 x = A cos(ωt) 的方程，表示运动从最大位移处开始。确保你知道 T = 1/f 以及 ω = 2π/T 也成立。
3. The Equation a = -ω²x | 加速度与位移的关系式
The core equation a = -ω²x is the starting point for all SHM analysis. It tells us that the acceleration is always proportional to the negative of the displacement. If you can show that a system satisfies this relation (e.g., by deriving net force or acceleration in terms of x), you have proven the motion is simple harmonic. The constant ω² is positive, so ω is called the angular frequency.
核心方程 a = -ω²x 是所有简谐运动分析的出发点。它表明加速度始终与位移的负值成正比。如果你能证明一个系统满足这个关系（例如，通过推导出净力或加速度关于 x 的表达式），你就证明了运动是简谐运动。常数 ω² 为正，因此 ω 被称为角频率。
From a = -ω²x, using calculus or standard results, we can find expressions for velocity and displacement as functions of time. However, at IGCSE level, you may be given these derived formulas directly.
从 a = -ω²x 出发，利用微积分或标准结果，可以求出速度和位移随时间变化的表达式。不过，在 IGCSE 阶段，你可能会直接得到这些推导后的公式。
4. Displacement as a Function of Time | 位移随时间的变化
The displacement of a particle in SHM can be written as x = A cos(ωt) if it starts at the positive amplitude, or x = A sin(ωt) if it starts at the equilibrium position moving in the positive direction. In either case, the graph of x against t is a sinusoid with amplitude A and period T = 2π/ω. The phase constant may be added for different starting points, but WJEC questions typically stick to the simple forms.
做简谐运动的质点的位移可以写作 x = A cos(ωt)（如果从正振幅处开始），或 x = A sin(ωt)（如果从平衡位置开始向正方向运动）。无论是哪种形式，x 关于 t 的图像都是正弦曲线，振幅为 A，周期为 T = 2π/ω。对于不同的起始点可以添加相位常数，但 WJEC 考题通常使用这些简单的形式。
To find the displacement at a given time, simply substitute t into the equation. For example, if A = 0.5 m, ω = 2 rad/s, then at t = 1 s, x = 0.5 cos(2) using cosine form. Make sure your calculator is in radian mode.
要计算给定时刻的位移，只需将 t 代入方程。例如，若 A = 0.5 m，ω = 2 rad/s，那么在 t = 1 s 时，若使用余弦形式则 x = 0.5 cos(2)。确保你的计算器处于弧度模式。
5. Velocity in SHM | 简谐运动中的速度
The velocity v in SHM can be derived and is often given as v = ± ω√(A² – x²). This formula shows that speed is maximum when x = 0 (at equilibrium) and zero when x = ± A (at the extreme points). Alternatively, if you know the displacement function, you can use v = -Aω sin(ωt) for the cosine displacement function.
简谐运动中的速度 v 可以推导出来，通常给出为 v = ± ω√(A² – x²)。这个公式表明，当 x = 0（在平衡位置）时速率最大，而当 x = ± A（在端点）时速度为零。或者，如果你知道位移函数，可以使用 v = -Aω sin(ωt)（针对余弦位移函数）。
The sign indicates direction: positive when moving away from equilibrium in the positive direction, negative otherwise. In many problems, you need to find the speed at a particular displacement, so the formula v = ω√(A² – x²) (ignoring sign) is very useful.
正负号表示方向：当物体从平衡位置向正方向运动时为正，否则为负。在许多问题中，你需要求出特定位移处的速率，因此公式 v = ω√(A² – x²)（忽略符号）非常有用。
6. Maximum Speed and Acceleration | 最大速度和最大加速度
The maximum speed v_max occurs at the equilibrium position (x = 0) and is given by v_max = ωA. The maximum acceleration a_max occurs at the extreme positions (x = ± A) and is given by a_max = ω²A. These two results are frequently tested because they directly link the amplitude and angular frequency to the peak values.
最大速度 v_max 出现在平衡位置（x = 0），且 v_max = ωA。最大加速度 a_max 出现在端点位置（x = ± A），且 a_max = ω²A。这两个结果经常被考查，因为它们直接将振幅、角频率与峰值联系起来。
For instance, if a mass oscillates with amplitude 0.2 m and period 2 s, then ω = 2π/T = π rad/s, so v_max = π × 0.2 ≈ 0.628 m/s and a_max = π² × 0.2 ≈ 1.97 m/s².
例如，如果一个物体以振幅 0.2 m、周期 2 s 振动，则 ω = 2π/T = π rad/s，因此 v_max = π × 0.2 ≈ 0.628 m/s，a_max = π² × 0.2 ≈ 1.97 m/s²。
7. Period of a Mass-Spring System | 弹簧振子的周期
For a mass m attached to a spring with stiffness constant k that obeys Hooke’s law, the motion is simple harmonic with angular frequency ω = √(k/m). The period T is then T = 2π/ω = 2π √(m/k). This formula shows that the period depends only on mass and spring constant, not on amplitude, which is a key feature of SHM.
对于附着在劲度系数为 k 且遵循胡克定律的弹簧上的质量块 m，其运动是简谐运动，角频率为 ω = √(k/m)。周期 T 为 T = 2π/ω = 2π √(m/k)。该公式表明周期仅取决于质量和弹簧劲度系数，与振幅无关，这是简谐运动的一个关键特征。
You might be asked to calculate the period given m and k, or to determine k from the period and mass. Always ensure that units are consistent (e.g., m in kg, k in N/m).
你可能会被要求根据 m 和 k 计算周期，或根据周期和质量求 k。务必确保单位一致（例如，m 用 kg，k 用 N/m）。
8. Period of a Simple Pendulum | 单摆的周期
A simple pendulum consists of a point mass suspended by a light, inextensible string. For small angular displacements (typically less than about 10°), the motion approximates SHM with ω = √(g/L), where L is the length of the pendulum and g is the acceleration due to gravity. Hence, the period T = 2π √(L/g).
单摆由一根轻质、不可伸长的细线悬挂一个质点构成。在小角度位移下（通常小于约 10°），其运动可近似为简谐运动，ω = √(g/L)，其中 L 是摆长，g 是重力加速度。因此，周期 T = 2π √(L/g)。
Notice that, like the mass-spring system, the period is independent of the amplitude (for small angles) and also independent of the bob’s mass. This is why pendulum clocks can keep accurate time.
注意，与弹簧振子一样，周期（在小角度下）与振幅无关，也与摆球的质量无关。这就是摆钟能够精确计时的原因。
9. Energy in SHM | 简谐运动中的能量
In SHM, total mechanical energy E remains constant (assuming no damping) and is the sum of kinetic energy (KE) and potential energy (PE). At any displacement x, KE = ½ m v² and PE = ½ m ω² x², provided the PE is taken as zero at equilibrium. The total energy can be shown to be E = ½ m ω² A².
在简谐运动中，总机械能 E 保持恒定（假设没有阻尼），是动能（KE）和势能（PE）之和。在任意位移 x 处，KE = ½ m v²，而 PE = ½ m ω² x²，前提是选取平衡位置处势能为零。可以证明总能量为 E = ½ m ω² A²。
This means that at the amplitude extreme, all energy is potential (KE=0), and at equilibrium, all energy is kinetic (PE=0). Energy considerations
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