📚 A-Level Edexcel Physics: Simple Harmonic Motion – Key Concepts and Exam Tips | A-Level Edexcel 物理：简谐运动考点精讲
Simple harmonic motion (SHM) is a core topic in the Edexcel A-Level Physics specification, building on your understanding of forces, energy and waves. It describes a very particular type of oscillation that appears in pendulums, mass-spring systems and even molecular vibrations. This article breaks down every essential concept, from the defining equation to resonance graphs, pairing clear English explanations with Chinese translations to support bilingual learners. Master these ideas and you will handle both calculation and explanation questions with confidence.
简谐运动（SHM）是爱德思 A-Level 物理课程中的核心主题，它建立在你对力、能量和波的理解之上。它描述了一种非常特殊的振动类型，出现在单摆、弹簧振子甚至分子振动中。本文逐一剖析从定义方程到共振图像的每一个基本概念，将清晰的英文解释与中文翻译配对，以支持双语学习者。掌握这些思想，你将能够自信地应对计算题和解释题。
1. What is Simple Harmonic Motion? | 什么是简谐运动？
An oscillation is any back-and-forth motion about an equilibrium position. Simple harmonic motion is a specific, mathematically neat oscillation that occurs when the restoring force (and therefore acceleration) is directly proportional to the displacement from equilibrium, but acts in the opposite direction.
振动是围绕平衡位置的任何来回运动。简谐运动是一种特定的、数学上简洁的振动，当恢复力（因此加速度）与离开平衡位置的位移成正比，但方向相反时发生。
In practice, this means the further you pull a system away from the middle, the stronger it tries to snap back. A pendulum swinging with small angles and a mass on a light spring are the classic examples you will meet in the exam.
在实践中，这意味着你把系统拉离中心越远，它试图弹回的力量就越强。小角度摆动的单摆和轻质弹簧上的物体是你在考试中会遇到的两个典型例子。
Not all periodic motions are SHM – bouncing a basketball is not SHM because the force is not proportional to displacement. Recognising the conditions for SHM is the first skill examiners test.
并非所有周期性运动都是 SHM——拍篮球就不是 SHM，因为作用力不与位移成正比。识别 SHM 的条件是考官测试的第一项技能。
2. The Defining Equation of SHM | 简谐运动的定义方程
The hallmark of SHM is the relationship between acceleration a and displacement x:
SHM 的标志是加速度 a 与位移 x 之间的关系：
a = − ω² x
Here ω is the angular frequency (in rad s⁻¹), a constant for a given oscillator. The minus sign tells you that acceleration always points towards the equilibrium position – the hallmark of a restoring force.
这里 ω 是角频率（单位 rad s⁻¹），对于给定的振子是一个常数。负号告诉你加速度始终指向平衡位置——这是恢复力的标志。
he spring constant or effective stiffness. This linear force law is the origin of the term ‘harmonic’.
根据牛顿第二定律，恢复力为 F = m a = − m ω² x，通常写成 F = − k x，其中 k = m ω² 是弹簧常数或有效刚度。这种线性力定律是“谐”这个词的来源。
You must be able to identify SHM from graphs or data: if a graph of a against x is a straight line through the origin with a negative slope, the motion is simple harmonic.
你必须能够从图像或数据中识别 SHM：如果 a 对 x 的图是一条通过原点且斜率为负的直线，那么该运动就是简谐运动。
3. Displacement, Velocity and Acceleration Equations | 位移、速度和加速度方程
The displacement of an SHM oscillator can be described by a sine or cosine function. The most common form, assuming the motion starts from the maximum displacement (amplitude A), is:
SHM 振子的位移可以用正弦或余弦函数描述。最常见的形态，假设运动从最大位移（振幅 A）开始，是：
x = A cos(ω t)
A general expression with a phase constant φ is x = A sin(ω t + φ) or x = A cos(ω t + φ).
如果从平衡位置向正方向运动开始计时，则用 x = A sin(ω t)。带有相位常数 φ 的一般表达式为 x = A sin(ω t + φ) 或 x = A cos(ω t + φ)。
The velocity is the gradient of the displacement–time graph. Differentiating gives:
速度是位移-时间图像的梯度。微分可得：
v = − ω A sin(ω t) or v = ± ω √A² − x²
The second form is particularly useful because it relates speed directly to displacement without knowing time. The maximum speed vmax = ω A occurs at x = 0.
第二种形式特别有用，因为它直接将速率与位移联系起来而不需要知道时间。最大速率 vmax = ω A 出现在 x = 0 处。
ks back to the defining equation: a = − ω² A cos(ω t) = − ω² x. The maximum acceleration amax = ω² A occurs at the extremes of the motion.
加速度回归到定义方程：a = − ω² A cos(ω t) = − ω² x。最大加速度 amax = ω² A 出现在运动的端点。
- Displacement x: zero at equilibrium, ±A at extremes
- Speed v: zero at extremes, maximum at equilibrium
- Acceleration a: zero at equilibrium, maximum magnitude at extremes
- 位移 x：在平衡位置为零，在端点处为 ±A
- 速率 v：在端点处为零，在平衡位置最大
- 加速度 a：在平衡位置为零，在端点处幅度最大
4. The Reference Circle and Phase Angles | 参考圆与相位角
SHM can be visualised as the projection of uniform circular motion onto a diameter. The radius of the circle equals the amplitude A, and the angular speed of the rotating point is ω. This geometrical link is why ω is called the angular frequency.
SHM 可以想象为匀速圆周运动在直径上的投影。圆的半径等于振幅 A，旋转点的角速度是 ω。这种几何联系就是 ω 被称为角频率的原因。
The phase of the oscillation describes where the oscillator is in its cycle. A phase difference of π/2 radians between displacement and velocity means velocity leads displacement by a quarter cycle. Acceleration leads velocity by π/2, and is in antiphase (π radians) with displacement.
振动的相位描述了振子在其周期中的位置。位移与速度之间 π/2 弧度的相位差意味着速度领先位移四分之一周期。加速度领先速度 π/2，并与位移反相（相差 π 弧度）。
In exam questions, you may need to compare two oscillators with a phase difference or deduce the equation from initial conditions. Always draw a reference circle if you are stuck.
在考试题目中，你可能需要比较有两个相位差的振子，或者根据初始条件推导方程。万一卡住，可以画一个参考圆。
5. Energy Changes in SHM | 简谐运动中的能量变化
During SHM, energy is continuously exchanged between kinetic and potential forms, while the total mechanical energy stays constant (if undamped). For a mass-spring system:
在 SHM 过程中，能量在动能和势能之间不断转化，而总机械能保持不变（若无阻尼）。对于弹簧振子系统：
Etotal = ½ k A² = ½ m ω² A²
The kinetic energy at displacement x is:
在位移 x 处的动能为：
K = ½ m v² = ½ m ω² (A² − x²)
And the potential energy stored in the spring is:
而储存在弹簧中的势能为：
U = ½ k x² = ½ m ω² x²
l energy is kinetic. A graph of K and U against displacement forms two parabola-like curves that sum to a constant horizontal line for total energy.
在端点（x = ±A）处，所有能量为势能；在平衡位置（x = 0），所有能量为动能。动能和势能对位移的图形成两条抛物线形状的曲线，它们相加得到表示总能量的水平直线。
For a simple pendulum, the potential energy is gravitational, but the mathematics is analogous: Etotal = ½ m ω² A² still holds, and the energy transformations follow the same pattern.
对于单摆，势能是重力势能，但数学上类似：Etotal = ½ m ω² A² 仍然成立，能量转化遵循相同模式。
6. Mass–Spring System | 弹簧振子系统
A mass m attached to a light spring of spring constant k executes SHM when displaced from equilibrium. The angular frequency and period are:
一个质量 m 连接在劲度系数为 k 的轻弹簧上，当偏离平衡位置时做 SHM。其角频率和周期为：
ω = √k/m and T = 2π √m/k
Note that the period does not depend on the amplitude, a key feature of SHM called isochronism. This makes mass-spring systems excellent for timing devices.
注意周期不依赖于振幅，这是 SHM 的一个关键特征，叫做等时性。这使得弹簧振子系统非常适合计时装置。
spring to a new equilibrium position where mg = k e, but the period formula remains T = 2π √m/k. The oscillation occurs about that new equilibrium.
如果弹簧竖直悬挂，质量会将弹簧拉伸到一个新的平衡位置，满足 mg = k e，但周期公式仍为 T = 2π √m/k。振动围绕该新平衡位置发生。
For combinations of springs, you must find the effective spring constant. For springs in parallel, keff = k₁ + k₂; for springs in series, 1/keff = 1/k₁ + 1/k₂. This directly affects the period.
对于弹簧的组合，你必须求出有效劲度系数。并联时 keff = k₁ + k₂；串联时 1/keff = 1/k₁ + 1/k₂。这直接影响周期。
7. The Simple Pendulum | 单摆
A simple pendulum consists of a point mass (bob) suspended by a light, inextensible string. For small angular displacements (θ < about 10°), the motion approximates SHM. The period is given by:
单摆由一个点质量（摆球）通过轻质且不可伸长的细线悬挂而成。对于小角位移（θ < 约 10°），运动近似为 SHM。周期由下式给出：
T = 2π √l/g
Notice that the mass of the bob does not appear – the period depends only on the length l of the string and the gravitational field strength g. This is one of the most tested facts in multiple-choice questions.
注意，摆球的质量并不出现——周期仅取决于绳长 l 和重力场强度 g。这是选择题中最常考查的事实之一。
To achieve accurate timing, the amplitude must be kept small. At larger angles the restoring force is no longer proportional to displacement, and the motion ceases to be simple harmonic; the period begins to increase slightly.
为了获得准确的计时，振幅必须保持较小。在较大角度下，恢复力不再与位移成正比，运动不再是简谐的；周期开始略有增加。
You may be asked to determine g from a pendulum experiment: measure T for different lengths l, plot T² against l, and use the gradient (4π²/g) to calculate g.
你可能会遇到通过单摆实验测定 g 的题目：测量不同长度 l 的周期 T，画出 T² 对 l 的图像，利用梯度 (4π²/g) 计算 g。
8. Damping and Its Effects | 阻尼及其影响
In real oscillators, resistive forces (such as air resistance or internal friction) remove energy from the system, causing the amplitude to decay gradually. This is called damping.
在真实振子中，阻力（如空气阻力或内摩擦）会从系统中带走能量，导致振幅逐渐衰减。这称为阻尼。
Light damping: the system still oscillates with a slightly reduced frequency, and the amplitude decreases exponentially. The motion remains approximately SHM over a few cycles.
轻阻尼：系统仍以略微降低的频率振动，振幅按指数衰减。在几个周期内，运动仍近似为 SHM。
Critical damping: the system returns to equilibrium in the shortest possible time without overshooting. This is desirable in car shock absorbers and door-closing mechanisms.
临界阻尼：系统在尽可能短的时间内回到平衡位置且没有过冲。这在汽车减震器和门闭合装置中是理想的。
Heavy damping (overdamping): the return to equilibrium is very slow, with no oscillation. The displacement decays monotonically.
过阻尼：返回平衡位置非常缓慢，没有振动。位移单调衰减。
or light damping, the period T ≈ 2π/ω remains nearly unchanged for many exam calculations.
阻尼不仅减小振幅，还会略微降低振子的固有频率。不过，对于轻阻尼，在许多考试计算中周期 T ≈ 2π/ω 几乎保持不变。
9. Forced Oscillations and Resonance | 受迫振动与共振
When a periodic external force is applied to an oscillator, the system vibrates at the driving frequency, not at its own natural frequency. The amplitude of the forced oscillation depends on how close the driving frequency is to the natural frequency f₀.
当一个周期性的外力施加于振子时，系统以驱动频率振动，而不是其自身的固有频率。受迫振动的振幅取决于驱动频率与固有频率 f₀ 的接近程度。
Resonance occurs when the driving frequency equals the natural frequency. At resonance, the energy transfer is most efficient, and the amplitude of oscillation becomes very large – potentially causing damage (think of the Tacoma Narrows Bridge or shattering a wine glass).
当驱动频率等于固有频率时发生共振。在共振时，能量传递效率最高，振幅变得非常大——可能造成破坏（想想塔科马海峡大桥或震碎酒杯）。
The sharpness of the resonance peak is described by the Q-factor. Lightly damped systems have a tall, narrow resonance curve, while heavily damped systems show a broad, low peak. Damping thus reduces the destructive potential of resonance.
共振峰的尖锐程度由品质因数 Q 描述。轻阻尼系统具有高而窄的共振曲线，而过阻尼系统则显示出宽而低的峰。因此，阻尼降低了共振的破坏潜力。
Exam questions often show amplitude‑frequency graphs. Read the axes carefully: the peak occurs at the natural frequency, and increasing damping lowers and broadens the peak while shifting it slightly to lower frequencies.
考试题经常展示振幅-频率图像。仔细读取坐标轴：峰出现在固有频率处，增大阻尼会降低并加宽峰，同时使其略微向低频移动。
10. Graphical Representations of SHM | 简谐运动的图像表示
You are expected to sketch and interpret displacement–time, velocity–time, acceleration–time and energy–displacement graphs. The three kinematic graphs share the same period T = 2π/ω.
你需要能够绘制并解释位移-时间、速度-时间、加速度-时间以及能量-位移图像。三个运动学图像具有相同的周期 T = 2π/ω。
Displacement–time: a sine or cosine curve with amplitude A. The gradient at any point gives the velocity.
位移-时间：振幅为 A 的正弦或余弦曲线。任一点的梯度给出速度。
Velocity–time: also sinusoidal but shifted by a quarter period (π/2 rad) ahead of displacement. Its amplitude is ω A.
速度-时间：同样是正弦形状，但比位移超前四分之一周期（π/2 rad）。其振幅为 ω A。
Acceleration–time: sinusoidal, shifted by π rad relative to displacement (i.e. inverted) with amplitude ω² A.
加速度-时间：正弦形状，相对于位移移动了 π rad（即翻转），振幅为 ω² A。
Energy–displacement: K and U curves are both parabolic, K concave down, U concave up, crossing at |x| = A/√2 where K = U. Total energy line is horizontal at ½ k A².
能量-位移：K 和 U 曲线均为抛物线，K 开口向下，U 开口向上，在 |x| = A/√2 处相交，此时 K = U。总能量线是高度为 ½ k A² 的水平线。
| Quantity | Maximum value | Phase relative to x |
| Displacement x | A | 0 |
| Velocity v | ω A | +π/2 |
| Acceleration a | ω² A | π (antiphase) |
Remember: the area under a velocity–time graph equals displacement, and the area under an acceleration–time graph equals change in velocity. These rarely appear directly but can be handy for understanding.
记住：速度-时间图像下的面积等于位移，加速度-时间图像下的面积等于速度的变化量。这些虽然很少直接考查，但有助于理解。
11. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧
Mistake 1: Confusing angular frequency ω with velocity v. ω is a constant (rad s⁻¹) that tells you how rapidly the phase changes, while v changes continuously and is measured in m s⁻¹.
错误 1：混淆角频率 ω 与速度 v。ω 是一个常数（rad s⁻¹）表示相位变化的快慢，而 v 不断变化且以 m s⁻¹ 为单位。
Mistake 2: g the small-angle approximation for a pendulum when the angle is large. The formula T = 2π √l/g is only valid for small oscillations.
错误 2：在大角度时对单摆使用小角近似。公式 T = 2π √l/g 仅对小振动有效。
Mistake 3: = 0 where acceleration is zero; maximum acceleration occurs at x = ±A where speed is zero.
错误 3：认为最大加速度出现在最大速率处。在 SHM 中，最大速率在 x = 0 处而加速度为零；最大加速度在 x = ±A 处而速率为零。
Tip: t everything else. Identify whether you are given amplitude, period, frequency or damping information, and choose the energy or kinematic formula accordingly.
技巧：始终先写下定义方程 a = − ω² x。结合 ω = 2π f = 2π/T，你几乎可以推导出其他所有内容。识别题目给出的是振幅、周期、频率还是阻尼信息，然后选择相应的能量或运动学公式。
Tip: For multi‑step problems, list knowns and unknowns in a table. Convert to SI units early (e.g., mass in kg, length in m, time in s). Check whether the spring is vertical or horizontal – the period is the same, but the equilibrium position shifts.
技巧：对于多步问题，用表格列出已知量和未知量。尽早转换为国际单位（如质量用 kg，长度用 m，时间用 s）。检查弹簧是竖直还是水平——周期相同，但平衡位置有偏移。
12. Quick Reference: Key SHM Formulas | 速查：简谐运动关键公式
| Quantity | Formula | Notes |
|---|---|---|
| Defining relation | a = − ω² x | Negative sign essential |
| Angular frequency (spring) | ω = √k/m | k = spring constant |
| Period (spring) | T = 2π √m/k | Independent of amplitude |
| Period (pendulum) | T = 2π √l/g |
