Simple Harmonic Motion: Key Exam Points | 简谐运动考点精讲

📚 Simple Harmonic Motion: Key Exam Points | 简谐运动考点精讲

Simple harmonic motion (SHM) is a cornerstone of oscillations and waves in both IB and Edexcel Physics specifications. It describes systems where the restoring force is proportional to displacement, leading to sinusoidal motion. Mastering SHM means understanding its kinematic equations, energy transformations, and the behaviour of mass‑spring and pendulum systems. This article unpacks every essential point you need for the exam.

简谐运动是IB和Edexcel物理大纲中振动与波的核心内容。它描述回复力与位移成正比的系统，从而产生正弦式运动。掌握简谐运动意味着理解其运动学方程、能量转化以及弹簧振子与单摆的行为。本文梳理了考试中必备的所有要点。

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

SHM is defined by a simple condition: the acceleration of an object is directly proportional to its displacement from a fixed equilibrium point and is always directed toward that point. Mathematically this is expressed as a ∝ –x. Introducing the angular frequency ω gives the hallmark equation of SHM:

简谐运动由一条简洁的条件定义：物体的加速度与其偏离固定平衡位置的位移成正比，且总指向该平衡点。数学上表示为 a ∝ –x。引入角频率 ω 后得到简谐运动的标志性方程：

a = –ω²x

Here x is the displacement measured from equilibrium, and ω is the angular frequency (unit: rad s⁻¹). The negative sign indicates that acceleration always opposes the displacement. In terms of force, a linear restoring force F = –kx acts on the mass, where k is the spring constant or an effective stiffness constant.

式中 x 是从平衡位置测量的位移，ω 是角频率（单位：rad s⁻¹）。负号表示加速度始终与位移反向。从力的角度看，物体受到线性回复力 F = –kx 的作用，其中 k 是劲度系数或等效的刚度常数。

2. Kinematic Equations of SHM | 简谐运动的运动学方程

The most general displacement function for SHM is a sinusoid. Depending on the starting point, we write:

简谐运动最一般的位移函数是正弦或余弦函数。根据计时起点的不同，我们可写成：

x = A cos(ωt + φ)

where A is the amplitude (maximum displacement), ω is the angular frequency, t is time and φ is the initial phase (phase constant) measured in radians. If oscillation starts from the equilibrium position with positive velocity, a sine function x = A sin(ωt) is often used. The period T and frequency f are linked by ω = 2πf = 2π/T.

其中 A 是振幅（最大位移），ω 是角频率，t 是时间，φ 是初相位，以弧度为单位。若振动从平衡位置以正速度开始，常使用正弦函数 x = A sin(ωt)。周期 T 与频率 f 的关系为 ω = 2πf = 2π/T。

IB and Edexcel papers frequently ask students to identify amplitude, period and phase from a graph or to write the corresponding equation. Always check whether the time axis is in seconds or cycles and remember to use radian mode in your calculator.

IB 和 Edexcel 试题常要求学生从图像中识别振幅、周期和相位，或者写出相应的振动方程。务必检查时间轴单位是秒还是周期数，并记得计算器需使用弧度制。

3. Velocity and Acceleration in SHM | 简谐运动的速度与加速度

Differentiating displacement once gives velocity, twice gives acceleration. For x = A cos(ωt + φ):

位移对时间求一次导数得速度，二次导数得加速度。对于 x = A cos(ωt + φ)：

v = –Aω sin(ωt + φ)

a = –Aω² cos(ωt + φ) = –ω²x

Two vital results for exam calculations are the maximum speed v_max = ωA (occurring as the object passes through the equilibrium) and maximum acceleration a_max = ω²A (occurring at the extreme points x = ±A). The speed at any displacement can also be expressed without time:

考试计算中最关键的两个结果是最大速度 v_max = ωA（当物体通过平衡位置时）和最大加速度 a_max = ω²A（在端点 x = ±A 处）。任意位置的速度还可以消去时间表达为：

v = ± ω √(A² – x²)

Both forms are required for solving problems, especially when linking energy and dynamics.

这两种形式在解题时都需要掌握，尤其是涉及能量和动力学的联系时。

4. Energy Transformations in SHM | 简谐运动中的能量转化

In the absence of damping, the total mechanical energy of an oscillator is constant. Kinetic energy Eₖ and potential energy Eₚ continually exchange, but their sum remains the same. For a mass‑spring system, the potential energy is elastic: Eₚ = ½ kx². Using k = mω², we can rewrite all energies in terms of ω and A:

在无阻尼条件下，振子的总机械能守恒。动能 Eₖ 和势能 Eₚ 不断相互转化，但总和恒定。对于弹簧振子，势能为弹性势能：Eₚ = ½ kx²。利用 k = mω²，可将所有能量用 ω 和 A 表达：

Eₖ = ½ m ω² (A² – x²)

Eₚ = ½ m ω² x²

E_total = ½ m ω² A² = ½ k A²

At the equilibrium position, Eₖ is maximum and Eₚ is zero; at the amplitude extremes, Eₚ is maximum and Eₖ is zero. These energy–displacement relationships frequently appear in multiple‑choice and structured questions. Be prepared to sketch or interpret energy‑displacement graphs: a parabolic well for total energy, an upward parabola for potential energy, and an inverted parabola for kinetic energy.

在平衡位置，Eₖ 最大、Eₚ 为零；在振幅端点，Eₚ 最大、Eₖ 为零。这些能量–位移关系常出现在选择题和结构题中，需要会画或解读能量–位移图像：总能量是水平线，势能为开口向上的抛物线，动能为开口向下的抛物线。

5. The Mass–Spring System | 弹簧振子

A horizontal mass–spring oscillator is the simplest realisation of SHM. The restoring force is F = –kx, so applying Newton’s second law yields a = –(k/m) x. Comparison with a = –ω²x gives ω = √(k/m). Therefore, the period is:

水平弹簧振子是实现简谐运动的最简单系统。回复力 F = –kx，应用牛顿第二定律得 a = –(k/m) x。与 a = –ω²x 对比，得到 ω = √(k/m)。因此周期为：

T = 2π √(m/k)

Key points for exams: the period depends on mass and spring constant, but not on amplitude (isochronism). In a vertical spring‑mass system, gravity merely shifts the equilibrium position, and the period formula remains unchanged as long as the spring obeys Hooke’s law.

考试的要点：周期只取决于质量和劲度系数，与振幅无关（等时性）。在竖直弹簧振子中，重力只改变平衡位置，只要弹簧遵循胡克定律，周期公式保持不变。

6. The Simple Pendulum | 单摆

For small angular displacements (θ ≤ about 10°), the pendulum’s motion approximates SHM. The restoring force is the tangential component of weight: F = –mg sinθ ≈ –mg (x/L), where L is the string length and x is the arc displacement. This gives ω = √(g/L) and the famous period:

对于小角度摆动（θ ≤ 约10°），单摆的运动近似为简谐运动。回复力是重力的切向分量：F = –mg sinθ ≈ –mg (x/L)，其中 L 为摆长，x 为弧位移。由此得到 ω = √(g/L) 和著名的周期公式：

T = 2π √(L/g)

Crucially, the period is independent of the bob’s mass and amplitude (for small angles). Common exam tasks include deriving this period, calculating g from experimental data, or explaining why large amplitudes lead to a period increase.

关键点：周期与摆锤质量和振幅（小角度时）无关。考试常见任务包括推导该周期公式、由实验数据计算 g 以及解释大振幅为什么会使周期增大。

7. Graphical Representations | 图像表示

Interpreting and sketching SHM graphs is a core skill. The most important curves are displacement–time (x–t), velocity–time (v–t) and acceleration–time (a–t). Their phase relationships are summarised below.

解读与绘制简谐运动图像是一项核心技能。最重要的曲线是位移–时间 (x–t)、速度–时间 (v–t) 和加速度–时间 (a–t)。它们的相位关系总结如下：

Quantity	Expression (φ=0)	Phase relative to x
Displacement x	x = A cos(ωt)	0 (reference)
Velocity v	v = –Aω sin(ωt)	Leads x by π/2
Acceleration a	a = –Aω² cos(ωt)	Leads x by π (out of phase)

Energy–displacement graphs are equally important: potential energy is a parabola opening upwards, kinetic energy is an upside‑down parabola, and total energy is a horizontal line.

能量–位移图像同样重要：势能是开口向上的抛物线，动能是开口向下的抛物线，总能量是水平线。

8. Phase and Phase Difference | 相位与相位差

The term (ωt + φ) is the phase of the oscillation, measured in radians. Phase difference between two oscillators, or between two quantities in the same oscillator, determines whether they are in phase, out of phase, or somewhere in between. In SHM, velocity always leads displacement by π/2 rad, and acceleration leads displacement by π rad (or is completely out of phase). These relationships underpin many wave superposition and interference ideas.

(ωt + φ) 称为振动相位，单位为弧度。两个振子之间或同一振子中两个物理量之间的相位差决定了它们是同相、反相还是其他关系。在简谐运动中，速度始终比位移超前 π/2 rad，加速度比位移超前 π rad（即完全反相）。这些关系是许多波动叠加和干涉概念的基础。

9. Damped Harmonic Motion | 阻尼振动

In real systems, resistive forces remove energy, causing the amplitude to decay over time. The degree of damping is classified as:

在真实系统中，阻力消耗能量，使振幅随时间衰减。按阻尼程度可分为：

Underdamping: The system oscillates with gradually decreasing amplitude. The amplitude envelope follows A(t) = A₀ e^–γt.
Critically damped: The system returns to equilibrium in the shortest possible time without oscillating.
Overdamped: The system returns to equilibrium more slowly, with no oscillation.

欠阻尼：系统做振幅逐渐减小的振荡，振幅包络线为 A(t) = A₀ e^–γt。
临界阻尼：系统在最短时间内回到平衡位置而不发生振荡。
过阻尼：系统不振荡，但回到平衡位置更慢。

Exam questions often ask you to identify damping types from a graph or to explain practical applications, such as shock absorbers that are critically damped.

考题常要求从图像中识别阻尼类型，或解释实际应用，例如汽车减震器通常设计为临界阻尼。

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

When a periodic external force drives an oscillator, the system vibrates at the driving frequency. Resonance occurs when the driving frequency matches the natural frequency of the system, producing a dramatic increase in amplitude. The resonance curve (amplitude vs. driving frequency) shows a sharp peak; the sharpness depends on the damping: lighter damping gives a higher, narrower peak.

当周期性外力驱动振子时，系统按驱动力频率振动。当驱动频率等于系统的固有频率时，发生共振，振幅急剧增大。共振曲线（振幅–驱动频率图）显示一个尖峰；峰的尖锐程度取决于阻尼：阻尼越小，峰值越高、越尖锐。

Famous examples include soldiers breaking step on a bridge and the collapse of the Tacoma Narrows Bridge. In exams, you may be asked to sketch resonance curves for different damping values or to explain why resonance is useful (e.g., in musical instruments) or destructive (e.g., in buildings).

著名例子包括士兵过桥时便步走和塔科马海峡大桥的倒塌。考试可能要求你画出不同阻尼下的共振曲线，或解释共振在乐器中有用而在建筑中有害的原因。

11. Experimental Determination of g Using a Pendulum | 用单摆测定重力加速度实验

A classic experiment uses a simple pendulum to measure the acceleration of free fall, g. By varying the length L and measuring the corresponding period T, you can plot a straight‑line graph. From T = 2π √(L/g), squaring both sides yields:

这是一项经典实验，利用单摆测量自由落体加速度 g。通过改变摆长 L 并测量对应的周期 T，可绘制一条直线图像。由 T = 2π √(L/g) 两边平方得：

T² = (4π²/g) L

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Simple Harmonic Motion: Key Exam Points | 简谐运动考点精讲