IGCSE CIE Physics: Simple Harmonic Motion | IGCSE CIE 物理：简谐运动考点精讲

📚 IGCSE CIE Physics: Simple Harmonic Motion | IGCSE CIE 物理：简谐运动考点精讲

Simple harmonic motion (SHM) is a fundamental type of oscillatory motion that appears in many physical systems, from pendulums to vibrating molecules. In the IGCSE CIE Physics syllabus, you are expected to understand what defines SHM, interpret displacement–time and velocity–time graphs, calculate key quantities such as amplitude, period, and frequency, and apply the relevant equations. This article covers all the core concepts, worked examples, and exam tips you need to master SHM.

简谐运动（SHM）是许多物理系统（从钟摆到振动分子）中都会出现的一种基本振动形式。在 IGCSE CIE 物理考纲中，你需要理解什么是简谐运动，会解释位移–时间和速度–时间图像，能计算振幅、周期和频率等关键量，并能运用相关公式。本文将涵盖所有核心概念、典型例题和备考技巧，帮助你彻底掌握 SHM。

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

Simple harmonic motion is defined as oscillatory motion in which the acceleration is directly proportional to the displacement from the equilibrium position and is always directed towards that equilibrium point. Mathematically, we write:

简谐运动被定义为加速度与偏离平衡位置的位移成正比，且加速度方向总是指向平衡位置的振动。数学上表示为：

a ∝ −x or a = −ω²x

The minus sign shows that acceleration and displacement act in opposite directions. The constant ω is the angular frequency of the motion. This relationship gives rise to sinusoidal displacement and velocity patterns.

负号表示加速度和位移方向相反。常数 ω 是运动的角频率。这一关系产生了正弦式的位移和速度变化模式。

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2. Key Quantities: Amplitude, Period, and Frequency | 基本量：振幅、周期和频率

Amplitude (x₀ or A) is the maximum displacement from the equilibrium position. It is always a positive value and measured in metres (m). The total distance travelled in one complete oscillation is 4A.

振幅（x₀ 或 A）是物体偏离平衡位置的最大位移，始终取正值，单位为米（m）。一次完整振动经过的总路程为 4A。

The period (T) is the time taken for one complete oscillation. It is measured in seconds (s). Frequency (f) is the number of oscillations per second, measured in hertz (Hz). The relationship is:

周期（T）是完成一次完整振动所需的时间，单位秒（s）。频率（f）是每秒振动的次数，单位赫兹（Hz）。两者关系为：

T = 1 / f

In SHM, the period is independent of the amplitude (isochronous), provided the amplitude is small for a pendulum and within the elastic limit for a spring.

在简谐运动中，只要摆角很小（对单摆）或弹簧处于弹性限度内，周期与振幅无关（等时性）。

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3. Displacement–Time Graph for SHM | 位移–时间图像

For an object starting at the equilibrium position and moving in the positive direction, the displacement–time graph is a sine curve. If it starts at maximum positive displacement, the graph is a cosine curve. The graph oscillates between +A and –A.

若物体从平衡位置向正方向开始运动，位移–时间图像为正弦曲线；若从正最大位移处开始，则为余弦曲线。图像在 +A 和 –A 之间振荡。

On such a graph, the amplitude is the maximum height of the curve, and the period is the time between two successive peaks or two successive troughs. The gradient of the displacement–time graph at any point gives the velocity at that instant.

在这样的图像中，振幅是曲线的最大高度，周期是相邻两个波峰或波谷之间的时间间隔。位移–时间图线上任意一点的斜率代表该时刻的瞬时速度。

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4. Velocity–Time Graph and Phase Difference | 速度–时间图像与相位差

The velocity–time graph for SHM is also sinusoidal but leads the displacement graph by a quarter of a cycle (π/2 radians or 90°). When the displacement is zero, the speed is maximum; when the displacement is maximum, the speed is zero.

简谐运动的速度–时间图像也是正弦曲线，但比位移图像超前四分之一周期（π/2 弧度或 90°）。当位移为零时，速率最大；当位移最大时，速率为零。

The maximum speed v₀ occurs as the object passes through equilibrium and is given by:

最大速率 v₀ 出现在物体通过平衡位置时，由下式给出：

v₀ = ω x₀

You can find the speed at any displacement x using the equation:

任意位移 x 处的速率可由下式求得：

v = ± ω √(x₀² − x²)

The sign indicates direction. This equation is derived from energy conservation.

正负号表示方向。这个方程由能量守恒导出。

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5. Acceleration in SHM | 简谐运动中的加速度

Acceleration in SHM is always directed towards the equilibrium position. The maximum acceleration a₀ occurs at the extremes of the motion, when x = ± x₀:

简谐运动中的加速度总是指向平衡位置。最大加速度 a₀ 出现在运动端点，即 x = ± x₀ 时：

a₀ = ω² x₀

The acceleration–displacement graph is a straight line through the origin with a negative gradient of −ω². This linear relationship is a defining feature of SHM.

加速度–位移图像是一条通过原点、斜率为 −ω² 的直线。这种线性关系是简谐运动的定义特征之一。

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6. The Mass–Spring System | 弹簧振子系统

A mass attached to a horizontal spring on a frictionless surface performs SHM when displaced from equilibrium. The restoring force is given by Hooke’s Law: F = −k x, where k is the spring constant. The angular frequency is:

一个连在轻弹簧上的物体，在光滑水平面上偏离平衡位置后将做简谐运动。回复力由胡克定律给出：F = −k x，其中 k 为劲度系数。角频率为：

ω = √(k / m)

Hence the period of a mass–spring system is:

因此弹簧振子的周期为：

T = 2π √(m / k)

This equation shows that a larger mass leads to a longer period, while a stiffer spring (larger k) reduces the period. The amplitude does not affect T.

这个公式表明，质量越大周期越长，而弹簧越硬（k 越大）周期越短。振幅不影响周期。

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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 (less than about 10°), the motion approximates SHM. The restoring force is a component of the bob’s weight.

单摆由一个质点（摆球）通过轻质且不可伸长的细线悬挂而成。当角位移很小（约小于 10°）时，运动可近似为简谐运动。回复力是摆球重力沿弧切线方向的分量。

The period of a simple pendulum is given by:

单摆的周期为：

T = 2π √(l / g)

where l is the length of the string and g is the acceleration due to gravity. Note that the period is independent of the bob’s mass and of the amplitude (for small angles).

其中 l 为摆长，g 为重力加速度。注意周期与摆球质量和小角度下的振幅无关。

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8. Energy Changes in SHM | 简谐运动的能量变化

In an ideal undamped SHM system, the total mechanical energy is conserved and continuously transforms between kinetic energy and potential energy. At the equilibrium position, kinetic energy is maximum and potential energy is minimum (usually taken as zero for horizontal spring). At maximum displacement, kinetic energy is zero and potential energy is maximum.

在理想无阻尼的简谐运动中，总机械能守恒，并在动能和势能之间连续转化。在平衡位置，动能最大、势能最小（水平弹簧通常取为零）；在最大位移处，动能为零、势能最大。

For a horizontal mass–spring system, the total energy is:

对水平弹簧振子，总能量为：

E_total = ½ k x₀² = ½ m ω² x₀²

Both kinetic energy and potential energy vary sinusoidally with time, but their sum remains constant.

动能和势能都随时间按正弦平方规律变化，但二者之和保持不变。

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9. Damping and Its Effects | 阻尼及其影响

Damping occurs when an external force (such as air resistance or internal friction) removes energy from the oscillating system. The amplitude of oscillation decreases gradually over time. Light damping slightly reduces the frequency; heavy damping can stop oscillation before it completes a cycle.

当外力（如空气阻力或内摩擦）从振动系统中带走能量时，就发生了阻尼。振幅随时间逐渐减小。弱阻尼会使频率略微减小；强阻尼可能导致系统在完成一个周期之前就停止振动。

Critical damping is the minimum amount of damping that brings the system back to equilibrium without oscillating. It is used in car suspension systems and door closers.

临界阻尼是使系统在不发生振动的情况下最快回到平衡位置的最小阻尼量，应用于汽车悬架和关门器等。

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10. Forced Oscillations and Resonance | 受迫振动与共振

When a periodic external force drives an oscillating system, the system vibrates at the driving frequency. When the driving frequency equals the natural frequency of the system, resonance occurs. At resonance, the amplitude of oscillation becomes dramatically large because energy is transferred most efficiently.

当一个周期性的外力驱动振动系统时，系统会以外力的频率振动。当驱动频率等于系统的固有频率时，发生共振。共振时，振幅急剧增大，因为能量传递效率最高。

Resonance can be useful (e.g., in tuning a radio) or destructive (e.g., bridges collapsing). Damping reduces the sharpness of resonance.

共振可能有益（如收音机调谐），也可能有害（如桥梁坍塌）。阻尼会降低共振的尖锐程度。

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11. Graphical Analysis and Exam Tips | 图像分析与应试技巧

Always label axes on displacement–time, velocity–time, and acceleration–displacement graphs. Mark amplitude, period, and equilibrium clearly. For velocity–time graphs, remember that the speed is zero at maximum displacement and maximum at equilibrium. The acceleration–displacement graph must be a straight line with a negative gradient.

在位移–时间、速度–时间和加速度–位移图像上，一定要标注坐标轴。清楚标出振幅、周期和平衡位置。对于速度–时间图，记住在最大位移处速度为零，平衡位置处速度最大。加速度–位移图必须是一条斜率为负的直线。

Use the equations carefully, paying attention to whether a quantity is a vector (displacement, velocity, acceleration) or a scalar (speed, energy). Practice converting between periods, frequencies, and angular frequencies.

使用公式时要细心，注意量是矢量（位移、速度、加速度）还是标量（速率、能量）。练习周期、频率和角频率之间的换算。

For the pendulum, remember that g is constant in a particular location, so T depends only on l. For the spring, T depends on m and k. These relationships are frequently tested in multiple-choice questions.

对于单摆，记住在特定地点 g 为常量，因此周期只取决于摆长 l。对于弹簧振子，周期取决于质量 m 和劲度系数 k。这些关系在选择题中经常考察。

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12. Common Mistakes to Avoid | 常见错误提醒

A common error is to confuse displacement and distance. Displacement is the directed distance from equilibrium, while distance is the total path length. In one complete oscillation, the displacement is zero but the distance is 4A. Another frequent mistake is forgetting the negative sign in a = −ω²x; the negative sign is essential for oscillation.

一个常见错误是混淆位移和路程。位移是相对于平衡位置的有向距离，而路程是总路径长度。一次全振动中，位移为零，但路程是 4A。另一个常见错误是遗忘 a = −ω²x 中的负号；负号对于形成振动至关重要。

When calculating speed at a certain displacement, use the equation v = ± ω √(x₀² − x²). Do not assume that the speed is constant; in SHM it varies continuously. Also, ensure that you use consistent units: metres for displacement, seconds for time, kilograms for mass.

在计算某位移处的速率时，要用 v = ± ω √(x₀² − x²)。不要认为速率是恒定的；在简谐运动中速率不断变化。此外，确保单位一致：位移用米，时间用秒，质量用千克。

Finally, remember that the pendulum formula applies only for small angles—roughly under 10°—otherwise the motion is not simple harmonic and the period depends on amplitude.

最后，记住单摆公式仅适用于小角度——大致在 10° 以内——否则运动不是简谐运动，周期将与振幅有关。

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