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

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

Simple harmonic motion (SHM) is a special type of periodic motion where an object oscillates back and forth about an equilibrium position. Understanding SHM is essential for mastering waves, oscillations, and many mechanical systems in the IGCSE Edexcel Physics syllabus. This article summarises all the key points you need to learn, from definitions and graphs to energy changes and pendulum theory.

简谐运动（SHM）是一种特殊的周期性运动，物体围绕平衡位置来回振荡。掌握简谐运动对于在 IGCSE Edexcel 物理大纲中理解波动、振动以及许多力学系统至关重要。本文总结了你需要学习的所有重点，从定义、图像到能量变化和单摆理论。

1. What Is Simple Harmonic Motion? | 什么是简谐运动？

Simple harmonic motion is defined as an oscillation in which the acceleration of the object is directly proportional to its displacement from a fixed point (the equilibrium position) and is always directed towards that fixed point.

简谐运动定义为一种振荡，其中物体的加速度与其离开固定点（平衡位置）的位移成正比，并且始终指向该固定点。

The mathematical condition for SHM is a ∝ −x, which can be written as a = −(2πf)²x or a = −ω²x, where a is acceleration, x is displacement, f is frequency, and ω is angular frequency. The negative sign shows that acceleration and displacement are in opposite directions.

简谐运动的数学条件为 a ∝ −x，可写成 a = −(2πf)²x 或 a = −ω²x，其中 a 是加速度，x 是位移，f 是频率，ω 是角频率。负号表示加速度与位移方向相反。

Any system that obeys this rule is undergoing SHM. Common examples include a simple pendulum (for small angles) and a mass attached to a spring.

任何遵守此规则的系统都在进行简谐运动。常见例子包括单摆（小角度下）和连接在弹簧上的质量块。

2. Key Features of SHM | 简谐运动的关键特征

SHM has several defining features that appear frequently in exam questions. You must be able to identify and describe these features clearly.

简谐运动有几个定义特征，在考试题目中经常出现。你必须能够清楚地识别和描述这些特征。

The motion is periodic: the object repeats the same path in equal time intervals.
运动是周期性的：物体在相等的时间间隔内重复相同的路径。
The acceleration is zero at the equilibrium position and maximum at the extremes.
加速度在平衡位置为零，在端点最大。
The velocity is maximum at the equilibrium position and zero at the extremes.
速度在平衡位置最大，在端点为零。
The restoring force always acts towards the equilibrium position.
回复力始终指向平衡位置。
The amplitude, period, and frequency are constant in an ideal system (no energy loss).
在理想系统中（无能量损失），振幅、周期和频率是恒定的。

These features allow you to distinguish SHM from other types of oscillation, such as damped motion or circular motion.

这些特征使你能够将简谐运动与其他类型的振荡区分开来，例如阻尼运动或圆周运动。

3. Amplitude, Period, and Frequency | 振幅、周期和频率

Three fundamental quantities describe any SHM: amplitude (A), period (T), and frequency (f).

三个基本量描述任何简谐运动：振幅（A）、周期（T）和频率（f）。

Amplitude is the maximum displacement from the equilibrium position. It is measured in metres (m). The total distance travelled in one complete oscillation is four times the amplitude (4A).

振幅是平衡位置的最大位移，单位为米（m）。一次完整振荡所经过的总距离是振幅的四倍（4A）。

Period is the time taken for one complete oscillation. It is measured in seconds (s). Frequency is the number of complete oscillations per second, measured in hertz (Hz). The relationship is T = 1/f or f = 1/T.

周期是完成一次完整振荡所需的时间，单位为秒（s）。频率是每秒完成的完整振荡次数，单位为赫兹（Hz）。关系为 T = 1/f 或 f = 1/T。

In experiments, you can measure the period by timing a number of oscillations (say, 20) and then dividing by the count to reduce the uncertainty.

在实验中，你可以通过测量多次振荡（例如 20 次）的时间，然后除以次数来计算周期，以减少不确定度。

4. Displacement–Time Graphs | 位移–时间图像

For an object starting from the equilibrium position and moving in the positive direction, the displacement–time graph is a sine curve. If the motion starts from the maximum positive displacement, the graph is a cosine curve.

对于从平衡位置开始向正方向运动的物体，位移–时间图像是一条正弦曲线。如果运动从最大正位移开始，图像是余弦曲线。

The equation for displacement can be written as x = A sin(2πft) or x = A cos(2πft), depending on the starting point. The graph shows how displacement varies smoothly between +A and −A.

位移方程可以写为 x = A sin(2πft) 或 x = A cos(2πft)，取决于起始点。图像显示位移如何在 +A 和 −A 之间平滑变化。

From this graph you can read the amplitude A and the period T directly. The gradient at any point gives the velocity of the object.

从该图像中你可以直接读出振幅 A 和周期 T。任意一点的斜率给出物体的速度。

Common exam questions ask you to sketch or interpret these graphs and label key points such as the equilibrium position and the extremes.

常见的考试题目要求你画出或解释这些图像，并标注平衡位置和端点等关键点。

5. Velocity in SHM | 简谐运动中的速度

The velocity of an oscillator changes continuously. It is greatest when the object passes through the equilibrium position, and it is zero at the turning points (x = ±A).

振荡器的速度不断变化。当物体通过平衡位置时速度最大，在转向点（x = ±A）速度为零。

The velocity is proportional to the gradient of the displacement–time graph. For a sine-wave displacement, the velocity is a cosine wave (if we ignore phase shifts).

速度与位移–时间图像的斜率成正比。对于正弦波位移，速度是余弦波（忽略相位差）。

Mathematically, the maximum speed vₘₐₓ occurs at x = 0 and is given by vₘₐₓ = 2πfA = ωA. The speed at any displacement x can be found using v = ±2πf√(A² − x²).

数学上，最大速度 vₘₐₓ 出现在 x = 0 处，由 vₘₐₓ = 2πfA = ωA 给出。在任意位移 x 处的速度可用 v = ±2πf√(A² − x²) 计算。

The direction of velocity changes when the object reverses at the extremes, so the graph of velocity against time also crosses zero at the extremes.

当物体在端点反向时速度方向改变，因此速度–时间图像也在端点处穿过零。

6. Acceleration in SHM | 简谐运动中的加速度

Acceleration in SHM is directly proportional to the displacement from the equilibrium position and is always directed towards the equilibrium point. This is the defining equation a = −(2πf)²x.

简谐运动中的加速度与离开平衡位置的位移成正比，并且始终指向平衡点。这是定义方程 a = −(2πf)²x。

The acceleration is zero at the equilibrium position (x = 0) and reaches its maximum magnitude aₘₐₓ at the extremes (x = ±A), where aₘₐₓ = (2πf)²A.

加速度在平衡位置（x = 0）为零，在端点（x = ±A）达到最大值 aₘₐₓ = (2πf)²A。

The acceleration–time graph is also sinusoidal. If displacement is a sine function, acceleration is a sine curve but inverted (180° out of phase). This shows that when displacement is at a maximum positive value, acceleration is at a maximum negative value.

加速度–时间图像也是正弦形的。如果位移是正弦函数，加速度则是反转的正弦曲线（相位差 180°）。这表明当位移为最大正值时，加速度为最大负值。

Understanding this relationship helps explain why a mass–spring system and a simple pendulum both obey SHM for small amplitudes.

理解这种关系有助于解释为什么质量–弹簧系统和小角度的单摆都遵循简谐运动。

7. Restoring Force and Hooke’s Law | 回复力与胡克定律

The condition for SHM is that the resultant force (often called the restoring force) must be proportional to the displacement and act towards equilibrium. For a spring, Hooke’s law states F = −kx, where k is the spring constant.

简谐运动的条件是合力（常称为回复力）必须与位移成正比并指向平衡位置。对于弹簧，胡克定律指出 F = −kx，其中 k 是弹簧常数。

Combining F = ma and F = −kx gives ma = −kx, so a = −(k/m)x. Comparing with a = −(2πf)²x shows that (2πf)² = k/m, hence T = 2π√(m/k). This is the period of a mass–spring oscillator.

结合 F = ma 和 F = −kx 得到 ma = −kx，因此 a = −(k/m)x。与 a = −(2πf)²x 对比，表明 (2πf)² = k/m，因此 T = 2π√(m/k)。这就是质量–弹簧振荡器的周期。

The minus sign indicates that the force is opposite to the direction of displacement. This is the hallmark of SHM.

负号表示力与位移方向相反。这是简谐运动的标志。

For a simple pendulum, the restoring force is a component of the weight (mg sinθ). For small angles, sinθ ≈ θ in radians, giving a similar linear relationship and the period T = 2π√(l/g).

对于单摆，回复力是重力的一个分量（mg sinθ）。对于小角度，sinθ ≈ θ（以弧度计），给出类似的线性关系，周期为 T = 2π√(l/g)。

8. Energy in SHM | 简谐运动中的能量

In SHM, the total mechanical energy remains constant (if no external damping force acts). Energy continuously exchanges between kinetic energy (KE) and potential energy (PE).

在简谐运动中，总机械能保持不变（如果没有外部阻尼力作用）。能量在动能（KE）和势能（PE）之间不断转换。

At the equilibrium position, the object moves at maximum speed, so KE is maximum and PE is minimum (usually taken as zero). At the extreme positions, the object momentarily stops, so KE is zero and PE is maximum.

在平衡位置，物体以最大速度运动，因此动能最大，势能最小（通常取为零）。在端点位置，物体瞬间静止，因此动能为零，势能最大。

For a mass–spring system, the potential energy is stored as elastic potential energy, PE = ½kx². For a simple pendulum, the potential energy is gravitational potential energy, PE = mgh.

对于质量–弹簧系统，势能以弹性势能的形式储存，PE = ½kx²。对于单摆，势能是重力势能，PE = mgh。

At any point, the sum KE + PE = total energy E = ½kA² (for the spring) or mg × (maximum height) (for the pendulum). This constant total energy can be used to find speed at various displacements.

在任意一点，动能与势能之和等于总能量 E = ½kA²（对弹簧）或 mg ×（最大高度）（对单摆）。这个恒定的总能量可以用来求不同位移处的速度。

9. The Simple Pendulum | 单摆

A simple pendulum consists of a small mass (bob) suspended by a light, inextensible string. For small-amplitude oscillations (typically less than about 10°), the motion is simple harmonic.

单摆由一个悬挂在轻质、不可伸长的细绳上的小质量块（摆球）组成。对于小振幅振荡（通常小于约 10°），运动是简谐运动。

The period of a simple pendulum is given by T = 2π√(l/g), where l is the length of the string from the pivot to the centre of the bob, and g is the acceleration due to gravity (9.8 m/s² on Earth).

单摆的周期由 T = 2π√(l/g) 给出，其中 l 是从悬点到摆球中心的绳长，g 是重力加速度（地球上为 9.8 m/s²）。

Notice that the period does not depend on the mass of the bob or the amplitude (as long as it is small). This is called isochronism. That is why pendulums are used in clocks.

注意，周期不依赖于摆球的质量或振幅（只要振幅小）。这被称为等时性。这也是为什么单摆被用于时钟的原因。

In experiments, you can vary the length l and measure the period T. Plotting T² against l gives a straight line through the origin, with gradient = 4π²/g, allowing you to determine g.

在实验中，你可以改变绳长 l 并测量周期 T。绘制 T² 对 l 的图像可得一条过原点的直线，斜率为 4π²/g，从而可以测定 g。

10. The Mass–Spring System | 质量–弹簧系统

A mass attached to a spring, moving horizontally on a frictionless surface or vertically under gravity, performs SHM if the spring obeys Hooke’s law. The equilibrium position shifts in the vertical case, but the SHM condition still holds.

如果一个质量块连接在弹簧上，在无摩擦表面上水平运动或在重力下垂直运动，只要弹簧服从胡克定律，该物体就进行简谐运动。在垂直情况下平衡位置会发生偏移，但简谐运动的条件仍然成立。

The period for a mass–spring system is T = 2π√(m/k), where m is the mass (in kg) and k is the spring constant (in N/m). A stiffer spring (larger k) gives a smaller period; a larger mass gives a longer period.

质量–弹簧系统的周期为 T = 2π√(m/k)，其中 m 是质量（kg），k 是弹簧常数（N/m）。较硬的弹簧（较大的 k）使周期变短；较大的质量使周期变长。

The period does not depend on the amplitude of oscillation, again demonstrating the isochronous nature of SHM.

周期不依赖于振荡的振幅，这再次证明了简谐运动的等时性。

A classic investigation involves adding slotted masses to a spring, measuring the extension to find k, then timing oscillations to verify the period formula.

经典的探究活动包括在弹簧上增加槽码，测量伸长量以求出 k，然后测量振荡时间来验证周期公式。

11. Damping and Realistic Oscillations | 阻尼与实际振荡

In real systems, oscillations slowly die away due to friction, air resistance, or internal forces. This phenomenon is called damping. With light damping, the amplitude gradually decreases, but the period remains nearly constant.

在实际系统中，由于摩擦、空气阻力或内部力，振荡会慢慢衰减。这种现象称为阻尼。在弱阻尼下，振幅逐渐减小，但周期几乎保持不变。

The IGCSE syllabus may ask you to describe the effect of damping on amplitude, but you are not required to calculate the damping coefficient. In critical damping, the system returns to equilibrium without oscillating; in over-damping, it returns slowly.

IGCSE 大纲可能要求你描述阻尼对振幅的影响，但不需要计算阻尼系数。在临界阻尼下，系统不回地振荡便返回平衡位置；在过阻尼下，返回得很慢。

Damping is important in shock absorbers, door closers, and earthquake-resistant buildings where controlled damping prevents excessive swaying.

阻尼在减震器、闭门器和抗震建筑中很重要，在这些场合可控阻尼能防止过度摇摆。

12. Common Exam Mistakes and Tips | 常见考试错误与提示

Many students confuse amplitude with the total distance travelled in one oscillation. Remember: total distance = 4A, not 2A.

许多学生将振幅与一次振荡中走过的总距离混淆。记住：总距离 = 4A，而不是 2A。

Always check the direction of acceleration: it is towards the equilibrium position, not towards the extreme. The acceleration vector always points opposite to the displacement vector.

始终检查加速度的方向：它指向平衡位置，而不是端点。加速度矢量始终与位移矢量方向相反。

When using the formula T = 2π√(l/g), make sure the pendulum length is measured to the centre of the bob and that the angle of swing is small enough for the formula to apply.

在使用公式 T = 2π√(l/g) 时，确保摆长测量到摆球中心，并且摆动角度足够小，公式才适用。

For energy calculations, do not forget that at the equilibrium position, potential energy is zero (in mass–spring horizontal cases) or can be set to a reference level (pendulum). Know how to equate ½mv² to the change in PE.

进行能量计算时，不要忘记在平衡位置，势能为零（在质量–弹簧水平情况下）或可以设置为参考水平（单摆）。要明白如何将 ½mv² 等同于势能的变化。

Draw graphs accurately: displacement, velocity, and acceleration against time are all sinusoidal but with phase differences. Velocity leads displacement by 90° (or π/2 radians), and acceleration leads displacement by 180° (or π radians).

准确绘制图像：位移、速度和加速度对时间的图像都是正弦形的，但存在相位差。速度超前位移 90°（或 π/2 弧度），加速度超前位移 180°（或 π 弧度）。

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Simple Harmonic Motion in IGCSE Edexcel Physics | IGCSE Edexcel 物理：简谐运动 考点精讲