Simple Harmonic Motion in IB & WJEC Physics | IB WJEC 物理：简谐运动考点精讲

📚 Simple Harmonic Motion in IB & WJEC Physics | IB WJEC 物理：简谐运动考点精讲

Simple harmonic motion (SHM) is a fundamental type of periodic oscillation that appears everywhere in physics — from pendulums and vibrating strings to alternating currents and quantum wave functions. For IB and WJEC Physics candidates, mastering the conditions, equations, energy transfers, and graphical interpretations of SHM is essential for solving exam problems with confidence. This article unpacks all key learning objectives, reinforcing each concept with clear English explanations and their Chinese equivalents to support bilingual learners.

简谐运动（SHM）是一种基础的周期性振动，在物理中随处可见——从单摆和弦振动到交流电以及量子波函数。对于 IB 和 WJEC 物理考生而言，掌握简谐运动的条件、方程、能量转换和图像分析是自信解题的关键。本文将逐一剖析所有重点学习目标，每个概念均配有清晰的英文解释和中文对照，帮助双语学习者巩固理解。

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

A body performs simple harmonic motion when its acceleration is directly proportional to its displacement from a fixed equilibrium position, and is always directed towards that equilibrium point. Mathematically, this means a ∝ −x, where x is the displacement measured from equilibrium. The constant of proportionality is the square of the angular frequency ω, so we write a = −ω²x. If these conditions hold, the motion is sinusoidal and periodic, with amplitude A and period T.

当物体的加速度与其相对于固定平衡位置的位移成正比，并且加速度方向始终指向该平衡点时，物体就在做简谐运动。数学上意味着 a ∝ −x，其中 x 是相对平衡位置的位移。比例常数为角频率 ω 的平方，因此我们写作 a = −ω²x。如果满足这些条件，运动将是正弦式的、周期性的，具有振幅 A 和周期 T。

In IB and WJEC exams, you must be able to state the two defining features: acceleration proportional to displacement and acceleration in the opposite direction to displacement. Remember, the presence of a linear restoring force (F = −kx) ensures SHM if the system is undamped.

在 IB 和 WJEC 考试中，你必须能够陈述两个定义特征：加速度与位移成正比，且加速度方向与位移方向相反。记住，若系统无阻尼，则线性回复力（F = −kx）的存在保证了简谐运动。

2. The Acceleration–Displacement Relationship | 加速度与位移的关系

The hallmark equation a = −ω²x gives the acceleration at any instant. Since ω = 2π/T = 2πf, you can substitute to see how the acceleration changes with the oscillation frequency. A graph of a against x yields a straight line through the origin with negative gradient −ω². This linear graph is often used to verify that an oscillator is truly executing SHM.

标志性方程 a = −ω²x 给出了任意时刻的加速度。由于 ω = 2π/T = 2πf，你可以代入理解加速度如何随振荡频率变化。a 关于 x 的图像是一条通过原点、斜率为负 −ω² 的直线。这个线性图像常被用来验证振动体是否确实在做简谐运动。

A common exam task is to find ω or k from experimental data. If a mass–spring system yields a linear a–x graph, the slope’s absolute value gives ω². For a simple pendulum, the restoring acceleration is approximately −(g/L)x for small angles, showing why its motion is SHM only when the amplitude is small.

常见的考题是从实验数据中求 ω 或 k。如果弹簧振子系统得到的 a–x 图是线性的，斜率绝对值就给出 ω²。对于单摆，小角度时回复加速度约为 −(g/L)x，这解释了为什么单摆运动仅在振幅较小时才是简谐运动。

3. Equations of Motion for SHM | 简谐运动的运动方程

The displacement of an SHM oscillator can be expressed as x = A sin(ωt + φ) or x = A cos(ωt + φ), depending on the starting phase φ. The choice of sine or cosine depends on where the oscillator begins at t = 0. If the motion starts from equilibrium moving in the positive direction, x = A sin(ωt) is convenient. If it starts from maximum positive displacement, x = A cos(ωt) is used.

简谐振子的位移可以表示为 x = A sin(ωt + φ) 或 x = A cos(ωt + φ)，取决于初相位 φ。选用正弦还是余弦取决于振子 t = 0 时的初始位置。若从平衡位置向正方向开始运动，通常用 x = A sin(ωt)；若从正向最大位移处开始，则用 x = A cos(ωt)。

x = A sin(ωt + φ) or x = A cos(ωt + φ)

The phase constant φ shifts the waveform horizontally. In IB and WJEC problems, you often use initial conditions to determine φ. For instance, if at t = 0, x = 0 and the velocity is positive, φ = 0 for a sine function. If at t = 0, x = A, then φ = π/2 for a sine function, which is equivalent to a cosine with φ = 0.

初相 φ 使波形水平移动。在 IB 和 WJEC 的题目中，经常利用初始条件来确定 φ。例如，若 t = 0 时 x = 0 且速度为正，则对于正弦函数 φ = 0。若 t = 0 时 x = A，则对于正弦函数 φ = π/2，这等同于初相为 0 的余弦函数。

4. Velocity and Acceleration in SHM | 简谐运动中的速度和加速度

Differentiating the displacement equation gives the velocity: v = dx/dt = ωA cos(ωt + φ). The maximum speed v_max occurs as the oscillator passes through equilibrium, where cos(ωt + φ) = ±1, so v_max = ωA. Acceleration comes from a second derivative: a = −ω²A sin(ωt + φ) = −ω²x, with maximum magnitude a_max = ω²A occurring at the extreme displacements x = ±A.

对位移方程求导可得速度：v = dx/dt = ωA cos(ωt + φ)。最大速率 v_max 出现在振子经过平衡位置时，此时 cos(ωt + φ) = ±1，故 v_max = ωA。加速度由二次求导得到：a = −ω²A sin(ωt + φ) = −ω²x，最大加速度幅值 a_max = ω²A 出现在极端位移 x = ±A 处。

v = ± ω √(A² − x²)

An alternative expression for speed in terms of displacement is v = ± ω √(A² − x²). This is especially useful for energy-based problems. Make sure you can derive it by combining the displacement and velocity equations or by energy conservation.

用位移表示速率的另一个表达式是 v = ± ω √(A² − x²)。这在基于能量的题目中特别有用。务必确保你能够通过联立位移和速度方程或能量守恒推导出该式。

5. Energy Changes in SHM | 简谐运动中的能量变化

In undamped SHM, the total mechanical energy remains constant, being continuously exchanged between kinetic energy (K.E.) and potential energy (P.E.). For a horizontal mass–spring system, K.E. = ½ m v² and P.E. = ½ k x². At equilibrium (x = 0), K.E. is maximum and P.E. is zero; at maximum displacement (x = ±A), K.E. is zero and P.E. is maximum. The total energy is E_tot = ½ k A² = ½ m ω² A².

在无阻尼的简谐运动中，总机械能保持不变，动能（K.E.）与势能（P.E.）不断相互转化。对于水平弹簧振子系统，K.E. = ½ m v²，P.E. = ½ k x²。在平衡位置（x = 0）处，动能最大而势能为零；在最大位移处（x = ±A），动能为零而势能最大。总能量为 E_tot = ½ k A² = ½ m ω² A²。

Energy graphs for SHM show how K.E. and P.E. vary with displacement, both being parabolic. It is important to note that the total energy is proportional to A². Doubling the amplitude quadruples the total energy. Exam questions may ask you to find the speed at a certain displacement using energy conservation.

简谐运动的能量图像展示了动能和势能随位移的变化，二者均为抛物线形状。重要的是总能量与 A² 成正比。振幅加倍会使总能量变为四倍。考试题可能要求你利用能量守恒求某位移处的速率。

For a simple pendulum, the potential energy is gravitational (mgh) and is approximately proportional to x² for small angles, which also yields SHM with equivalent spring constant k = mg/L.

对于单摆，势能为重力势能（mgh），在小角度下近似与 x² 成正比，等效劲度系数为 k = mg/L，因此同样符合简谐运动。

6. The Simple Pendulum | 单摆

A simple pendulum consists of a point mass suspended by a light, inextensible string. For small angular amplitudes (θ < about 10°), the restoring force is mg sin θ ≈ mg θ, and the motion approximates SHM. The period T is independent of amplitude and mass, given by T = 2π √(L / g), where L is the length of the string. This relationship is used to determine gravitational acceleration g experimentally.

单摆由一个通过轻质且不可伸长的细线悬挂的质点构成。在角振幅较小时（θ 约小于 10°），回复力为 mg sin θ ≈ mg θ，运动近似为简谐运动。周期 T 与振幅和质量无关，由 T = 2π √(L / g) 给出，其中 L 为摆长。这一关系常被用来通过实验测定重力加速度 g。

T = 2π √(L / g)

IB and WJEC specifications require understanding the derivation of the period from a = −(g/L)x. They also emphasize that the period does depend slightly on amplitude for larger angles; this is an example of anharmonicity. In practical work, you may plot T² against L to obtain a straight line through the origin, whose slope is 4π²/g.

IB 和 WJEC 考纲要求理解由 a = −(g/L)x 推导周期的方法。它们还强调，当角度较大时周期会轻微依赖于振幅；这是非简谐性的一例。在实验操作中，你可能会绘制 T² 对 L 的图像，得到一条通过原点的直线，其斜率为 4π²/g。

7. The Mass–Spring System | 弹簧振子

A mass attached to an ideal spring undergoing SHM has angular frequency ω = √(k / m), where k is the spring constant. The period T = 2π √(m / k) is independent of amplitude. Two common arrangements are horizontal and vertical mass–spring oscillators. In the vertical case, gravity shifts the equilibrium position but does not affect the period, provided the spring obeys Hooke’s law.

连接在理想弹簧上的振子做简谐运动时，角频率为 ω = √(k / m)，其中 k 为弹簧劲度系数。周期 T = 2π √(m / k) 与振幅无关。常见的两种布置是水平弹簧振子和竖直弹簧振子。在竖直情况下，重力会改变平衡位置，但只要弹簧遵从胡克定律，就不会影响周期。

In an exam, you may be asked to determine k from a graph of T² against m, or to calculate v_max and a_max for a given amplitude. Remember that for a vertical oscillator, the equilibrium extension x₀ satisfies kx₀ = mg.

考试中可能会要求你从 T² 对 m 的图像中求出 k，或者计算给定振幅下的 v_max 和 a_max。记住对于竖直振子，平衡时的伸长量 x₀ 满足 kx₀ = mg。

8. Period, Frequency, and Angular Frequency | 周期、频率和角频率

The period T is the time for one complete oscillation. Frequency f is the number of oscillations per unit time: f = 1/T, measured in hertz (Hz). Angular frequency ω has units of rad/s and is related to f and T by ω = 2πf = 2π/T. These fundamental quantities connect the time domain to circular motion analogies, where one complete oscillation corresponds to a 2π rotation in the phasor model.

周期 T 是完成一次完整振动所需的时间。频率 f 是单位时间内的振动次数：f = 1/T，单位为赫兹（Hz）。角频率 ω 的单位是 rad/s，通过 ω = 2πf = 2π/T 与 f 和 T 相联系。这些基本量将时域与圆周运动的类比联系起来，在相量模型中一次完整振动对应 2π 的旋转。

Many SHM problems begin by identifying the given quantities and selecting the appropriate formula. For a pendulum, T = 2π √(L/g); for a mass–spring, T = 2π √(m/k). Watch out for conversions between angular frequency and frequency, and ensure you know that the acceleration equation a = −ω²x uses ω, not f.

许多简谐运动问题都始于识别已知量并选择合适的公式。对于单摆，T = 2π √(L/g)；对于弹簧振子，T = 2π √(m/k)。注意角频率与频率之间的换算，并确保知道加速度方程 a = −ω²x 使用的是 ω，而非 f。

9. Resonance and Damping | 共振与阻尼

Damping removes energy from an oscillator, gradually reducing the amplitude. Light damping (underdamping) results in oscillations with an exponentially decaying envelope. Critical damping brings the system to equilibrium in the shortest possible time without oscillating, which is important in car suspensions and galvanometers. Overdamping produces a very slow return to equilibrium.

阻尼会从振动系统中去除能量，使振幅逐渐减小。弱阻尼（欠阻尼）产生具有指数衰减包络的振动。临界阻尼使系统以最短时间回到平衡位置而不发生振荡，这在汽车悬挂和检流计中很重要。过阻尼则导致非常缓慢地回复到平衡。

When a periodic driving force is applied at a frequency close to the natural frequency f₀ of an oscillator, the amplitude becomes very large. This phenomenon is called resonance. The amplitude–frequency graph shows a sharp peak at the resonant frequency, with the peak becoming broader and lower as damping increases. Resonance explains many real-world effects, from shattering wine glasses to bridge oscillations.

当周期性驱动力以接近振子固有频率 f₀ 的频率施加时，振幅将变得非常大，这种现象称为共振。振幅–频率曲线在共振频率处显示尖锐的峰值，随着阻尼增加，峰值变宽且降低。共振解释了许多现实世界中的效应，从震碎酒杯到桥梁振荡。

In IB and WJEC, you should be able to sketch resonance curves for different damping levels and discuss practical examples of forced oscillations and damping, such as in buildings during earthquakes or tuned circuits in radios.

在 IB 和 WJEC 中，你应该能够画出不同阻尼水平下的共振曲线，并讨论受迫振动和阻尼的实际例子，例如地震中的建筑物或无线电中的调谐电路。

10. Graphical Analysis of SHM | 简谐运动的图像分析

Graphs of x, v and a against time t are sinusoidal and out of phase with each other. The displacement–time graph starts at a value determined by the initial phase. The velocity–time graph leads the displacement by π/2 rad, and the acceleration–time graph is anti-phase (π rad out of phase) with the displacement. This phase relationship is a direct consequence of differentiation.

x、v 和 a 随时间 t 变化的图像均为正弦形，且彼此之间有相位差。位移–时间图像的起始值由初相决定。速度–时间图像领先位移 π/2 弧度，而加速度–时间图像与位移反相（相差 π 弧度）。这种相位关系是求导运算的直接结果。

Energy–displacement graphs show the constant total energy line and the parabolic forms of K.E. and P.E. Being able to sketch these graphs for an arbitrary amplitude and to identify key points (x = 0, x = ±A) is a common exam requirement. Similarly, the v–x graph is an ellipse, reflecting the relationship v²/ω² + x² = A².

能量–位移图像显示恒定的总能量线以及动能和势能的抛物线形状。能够针对任意振幅画出这些草图并标出关键点（x = 0，x = ±A）是常见的考试要求。类似地，v–x 图像为椭圆形，反映了关系式 v²/ω² + x² = A²。

When tackling multiple-choice or structured questions, quickly translating the situation described into a phasor (reference circle) model can often clarify phase differences and facilitate calculations of time or displacement.

在解答选择题或结构题时，将描述的情景快速转化为相量（参考圆）模型常常能理清相位差，并简化时间或位移的计算。

11. Damping Investigations and Logarithmic Decrement | 阻尼研究与对数衰减

While not always required at the core level, IB Higher Level and some WJEC extended assessments may explore the logarithmic decrement δ, which measures the rate of amplitude decay per cycle: δ = ln(xₙ / xₙ₊₁). This helps determine the damping constant. The amplitude envelope is given by A(t) = A₀ e^{−γt}, where γ is the damping coefficient.

尽管在核心层不总是要求，IB 高等级和部分 WJEC 拓展评估中可能会探讨对数衰减 δ，它衡量每周期振幅衰减的速率：δ = ln(xₙ / xₙ₊₁)。这有助于确定阻尼常数。振幅包络由 A(t) = A₀ e^{−γt} 给出，其中 γ 为阻尼系数。

Understanding exponential decay of amplitude in lightly damped systems reinforces the distinction between energy and amplitude decay. Energy decays as E(t) = E₀ e^{−2γt}, meaning energy falls twice as fast as amplitude in a logarithmic sense.

理解弱阻尼系统中振幅的指数衰减能够加深对能量衰减与振幅衰减区别的认识。能量按 E(t) = E₀ e^{−2γt} 衰减，意味着在对数意义上能量衰减速度是振幅的两倍。

12. Common Misconceptions and Exam Tips | 常见误区与备考建议

One frequent error is assuming all periodic motion is SHM. Circular motion is periodic but not SHM unless projected onto a diameter. Another is confusing the direction of acceleration: it always points toward equilibrium, regardless of whether the mass is moving away or toward it. In energy questions, students sometimes forget that total energy remains constant only in undamped systems.

一个常见误区是认为所有的周期性运动都是简谐运动。圆周运动是周期性的，但除非投影到直径上，否则不是简谐运动。另一个误区是混淆加速度的方向：无论物体是远离还是靠近平衡位置，加速度始终指向平衡点。在能量问题中，学生有时会忘记总能量仅在无阻尼系统中才是常数。

Always check that your calculator is in radian mode when evaluating sin and cos in SHM equations. Use significant figures appropriately, and show clearly how you derive ω from k/m or g/L. For pendulum timing experiments, measure multiple periods to reduce reaction-time uncertainty and plot a straight-line graph to find g.

在使用简谐运动方程计算正弦和余弦时，务必确保计算器处于弧度模式。适当地使用有效数字，并清楚展示如何从 k/m 或 g/L 推导出 ω。在进行单摆计时实验时，测量多个周期以降低反应时间的不确定度，并绘制直线图像来求 g。

Finally, practise converting between sinusoidal forms, interpreting slope and intercept values, and linking graphical features back to the underlying physics. This integrated skill is frequently assessed in both IB and WJEC examinations.

最后，练习在不同正弦表达式之间转换，解读斜率和截距值，并将图像特征与背后的物理原理联系起来。这种综合性技能在 IB 和 WJEC 考试中频繁受到考查。

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Simple Harmonic Motion in IB & WJEC Physics | IB WJEC 物理：简谐运动考点精讲