📚 A-Level Physics: Simple Harmonic Motion Key Points Revision | A-Level 物理：简谐运动 考点精讲
Simple Harmonic Motion (SHM) is a fundamental concept in A-Level Physics that describes oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. Understanding SHM is crucial for analysing a wide range of physical systems, from pendulums to vibrating molecules, and it forms the basis for wave theory. This revision guide systematically covers all the essential points you need to master for your exams, including defining conditions, displacement equations, velocity, acceleration, energy changes, phase angles, and real-world examples.
简谐运动（SHM）是 A-Level 物理中的核心概念，描述的是当回复力与偏离平衡位置的位移成正比且方向相反时的振荡运动。理解简谐运动对于分析从单摆到振动分子等各种物理系统至关重要，它也是波动理论的基础。这份复习指南系统地涵盖了你需要掌握的所有关键考点，包括简谐运动的定义条件、位移方程、速度、加速度、能量变化、相位角以及实际案例。
1. Defining Simple Harmonic Motion | 简谐运动的定义
Simple harmonic motion is defined as motion where the acceleration a of an object is directly proportional to its displacement x from a fixed equilibrium position and is always directed towards that equilibrium point. Mathematically, this is expressed as a ∝ -x, leading to the key equation a = -ω²x, where ω is the angular frequency of the motion. The negative sign indicates that the acceleration and displacement are in opposite directions.
简谐运动定义为物体的加速度 a 与其相对固定平衡位置的位移 x 成正比，且加速度方向始终指向该平衡点。数学上这表示为 a ∝ -x，从而得到关键方程 a = -ω²x，其中 ω 是运动的角频率。负号表示加速度和位移方向相反。
For any motion to be simple harmonic, two conditions must be met: the net force or acceleration must be proportional to the negative of the displacement, and the motion must repeat in a regular cycle. This proportionality constant is ω², which links the oscillation’s period and frequency. The defining equation a = -ω²x is a differential equation whose solutions are sinusoidal functions of time – sine or cosine – which is why SHM graphs are smooth waves.
任何要成为简谐运动的运动必须满足两个条件：回复力或加速度必须与位移的负值成正比，并且运动必须周期性地重复。这个比例常数就是 ω²，它将振荡的周期和频率联系起来。定义方程 a = -ω²x 是一个微分方程，其解是时间的正弦函数（正弦或余弦），因此 SHM 图像是平滑的波形。
2. Displacement, Velocity and Acceleration Equations | 位移、速度和加速度方程
The displacement x of an oscillator undergoing SHM can be described by either x = A cos(ωt) or x = A sin(ωt), depending on the initial conditions. If the oscillation starts at maximum displacement, we use cosine; if it starts from equilibrium moving in the positive direction, we use sine. Here, A is the amplitude (maximum displacement) and ω is the angular frequency given by ω = 2πf = 2π/T, with frequency f and period T.
简谐振子的位移 x 可以用 x = A cos(ωt) 或 x = A sin(ωt) 来描述，具体取决于初始条件。如果振动从最大位移开始，我们用余弦函数；如果从平衡位置向正方向开始运动，则用正弦函数。其中 A 是振幅（最大位移），ω 是角频率，由 ω = 2πf = 2π/T 给出，f 是频率，T 是周期。
Velocity v is the rate of change of displacement, obtained by differentiation: v = -Aω sin(ωt) for the cosine displacement, or v = Aω cos(ωt) for the sine form. The maximum speed vₘₐₓ occurs at the equilibrium position (x=0) and equals Aω. Acceleration a is the rate of change of velocity, leading to a = -Aω² cos(ωt) = -ω²x. Maximum acceleration aₘₐₓ occurs at maximum displacement and is given by Aω².
速度 v 是位移随时间的变化率，通过微分得到：对于余弦位移，v = -Aω sin(ωt)；对于正弦形式，v = Aω cos(ωt)。最大速度 vₘₐₓ 出现在平衡位置（x=0），大小等于 Aω。加速度 a 是速度的变化率，可得 a = -Aω² cos(ωt) = -ω²x。最大加速度 aₘₐₓ 出现在最大位移处，大小为 Aω²。
These equations are often visualised in exam questions. You should be able to sketch displacement-time, velocity-time and acceleration-time graphs for SHM, noting that velocity leads displacement by π/2 and acceleration is in anti-phase with displacement (phase difference of π).
这些方程常在考题中以图像形式出现。你要能够绘制 SHM 的位移-时间、速度-时间和加速度-时间图像，并注意速度领先位移 π/2，而加速度与位移反相（相位差为 π）。
3. The Physics of ω and the Time Period | 角频率 ω 和周期的物理意义
The angular frequency ω is a measure of how rapidly the oscillation occurs in radians per second. It is linked to the physical constants of the system: for a mass-spring system ω = √(k/m), where k is the spring constant and m is the mass; for a simple pendulum undergoing small oscillations, ω = √(g/L), where L is the length of the pendulum and g is the gravitational field strength.
角频率 ω 衡量振荡以弧度每秒为单位的快慢。它与系统的物理常量相关联：对于弹簧振子，ω = √(k/m)，其中 k 是弹簧劲度系数，m 是质量；对于做小角度摆动的单摆，ω = √(g/L)，其中 L 是摆长，g 是重力场强。
The period T, the time for one complete oscillation, is given by T = 2π/ω. Substituting the specific ω expressions, for a mass-spring system T = 2π√(m/k) and for a simple pendulum T = 2π√(L/g). These formulas are valid only when the oscillation is simple harmonic – meaning the pendulum’s angular amplitude is small (typically less than about 10°) so that sin θ ≈ θ, and the spring obeys Hooke’s law and has negligible mass.
周期 T 是一次完整振荡所用的时间，由 T = 2π/ω 给出。代入具体的 ω 表达式，对于弹簧振子 T = 2π√(m/k)，对于单摆 T = 2π√(L/g)。这些公式仅在振动为简谐运动时成立——即单摆的角振幅较小（通常小于约 10°），此时 sin θ ≈ θ；弹簧必须遵循胡克定律且质量可忽略。
In many exam calculations, you will be asked to find k, m, L or g using these relationships. Always remember that the period of a simple pendulum is independent of its mass and amplitude (for small angles).
在许多考试计算中，你会被要求利用这些关系求 k、m、L 或 g。务必记住单摆的周期与它的质量和振幅（在小角度时）无关。
4. Mass-Spring Oscillator Details | 弹簧振子详解
A mass attached to a horizontal or vertical spring can exhibit SHM if the restoring force obeys Hooke’s law: F = -kx, where k is the spring constant. The equation of motion becomes m d²x/dt² = -kx, which directly gives a = -(k/m)x, so ω = √(k/m). The mass-spring system is a favourite experimental setup for investigating SHM because changing m or k allows direct measurement of T.
水平或垂直放置的弹簧振子，如果回复力遵循胡克定律 F = -kx（k 为弹簧劲度系数），就能产生简谐运动。运动方程变为 m d²x/dt² = -kx，直接给出 a = -(k/m)x，因此 ω = √(k/m)。弹簧振子是研究简谐运动的经典实验装置，因为改变 m 或 k 可以直接测量周期 T。
In a vertical mass-spring system, gravity causes a shift in the equilibrium position but does not affect the SHM nature or the period. The equilibrium extension x₀ satisfies kx₀ = mg. The oscillations occur about this new equilibrium with the same ω and T as the horizontal case. A common exam trick is to confuse students into thinking gravity changes the period.
在竖直弹簧振子中，重力会导致平衡位置下移，但不影响简谐运动的性质或周期。平衡时的伸长量 x₀ 满足 kx₀ = mg。振动围绕这一新的平衡位置发生，ω 和 T 与水平情形相同。一个常见的考试陷阱是让学生误认为重力会改变周期。
The energy in a mass-spring system continually transforms between kinetic energy (½mv²) and elastic potential energy (½kx²). At maximum displacement, all energy is potential; at equilibrium, all energy is kinetic. The total mechanical energy is constant and equals ½kA².
弹簧振子中的能量在动能（½mv²）和弹性势能（½kx²）之间不断转换。在最大位移处，能量全为势能；在平衡位置，能量全为动能。总机械能量保持不变，等于 ½kA²。
5. Simple Pendulum and Small-Angle Approximation | 单摆与小角度近似
The simple pendulum consists of a point mass (bob) suspended by a light, inextensible string. When displaced by a small angle, the restoring force is the tangential component of weight: F = -mg sin θ. For SHM we require F ∝ -θ, which is only true when sin θ ≈ θ (in radians). Thus the condition for SHM in a pendulum is that the angular amplitude is small, usually < 10°.
单摆由一个用轻质不可伸长细线悬挂的质点（摆球）构成。当偏离一个小角度时，回复力是重力的切向分量：F = -mg sin θ。要使运动为简谐运动，需要 F ∝ -θ，而这仅在 sin θ ≈ θ（弧度制）时成立。因此单摆产生简谐运动的条件是角振幅很小，通常小于 10°。
Under this approximation, the equation of motion becomes d²θ/dt² = -(g/L)θ, yielding ω² = g/L and T = 2π√(L/g). The period is independent of the bob mass – a fact replicated in the famous demonstration involving pendulums of different masses but equal length swinging in unison. It also explains why a pendulum clock’s rate depends on L and local g.
在此近似下，运动方程变为 d²θ/dt² = -(g/L)θ，从而 ω² = g/L，T = 2π√(L/g)。周期与摆球质量无关——这一点在著名的不同质量等长单摆同步摆动演示中得到验证。这也解释了为何摆钟的快慢取决于摆长 L 和当地重力加速度 g。
In a typical experiment, measuring T for various lengths and plotting T² against L yields a straight line through the origin with gradient 4π²/g, allowing determination of g. Make sure to measure the length from the point of suspension to the centre of the bob and to time multiple oscillations (e.g. 10 or 20) to reduce uncertainty.
在典型实验中，测量不同摆长的周期 T，并绘制 T²-L 图，得到一条过原点的直线，斜率为 4π²/g，由此可求出 g。务必从悬点测量到摆球中心，并计时多次振荡（如 10 次或 20 次）以减小不确定度。
6. Energy in Simple Harmonic Motion | 简谐运动中的能量
The total energy E of an undamped harmonic oscillator remains constant and is proportional to the square of the amplitude: E = ½mω²A². This energy is shared between kinetic energy K and potential energy U. For a mass-spring system, K = ½mω²(A² – x²) and U = ½mω²x². Notice that U is zero at equilibrium and maximum at x = ±A, while K is maximum at equilibrium and zero at the extremes.
无阻尼简谐振子的总能量 E 保持不变，且与振幅的平方成正比：E = ½mω²A²。这一能量在动能 K 和势能 U 之间分配。对于弹簧振子，K = ½mω²(A² – x²)，U = ½mω²x²。注意，势能在平衡位置为零，在 x = ±A 处最大；而动能在平衡位置最大，在端点处为零。
These energy expressions result in parabolic potential and kinetic energy graphs as functions of displacement. The sum K + U at any instant gives the constant total energy. An exam favourite is to ask for displacement when K = U, which occurs when x = A/√2. Similarly, you might be asked for the fraction of energy that is kinetic at a given displacement.
这些能量关系得出势能和动能随位移变化的抛物线图像。任一时刻 K + U 等于恒定的总能量。考试中常问当动能等于势能时的位移，这发生在 x = A/√2 时。类似地，可能会问在某一位移处动能占总能量的比例。
In a simple pendulum, the potential energy is gravitational: U = mgh where h is the vertical height above the lowest point. For small angles, h ≈ ½Lθ², so U = ½(mg/L)(Lθ)² = ½mω²x², consistent with the SHM energy form. This shows the generality of the SHM energy model.
在单摆中，势能是重力势能：U = mgh，其中 h 是相对最低点的竖直高度。对于小角度，h ≈ ½Lθ²，于是 U = ½(mg/L)(Lθ)² = ½mω²x²，与简谐运动的能量形式一致。这体现了简谐运动能量模型的普适性。
7. Phase and Phase Difference | 相位与相位差
The phase of an oscillation describes the stage in the cycle of motion. For displacement x = A sin(ωt + φ), the quantity (ωt + φ) is the phase at time t, and φ is the initial phase constant or phase offset. Two oscillators with the same frequency but different φ are said to have a phase difference. For example, the displacement and velocity in SHM differ in phase by π/2 radians; displacement and acceleration differ by π radians (anti-phase).
振动的相位描述了运动周期中所处的阶段。对于位移 x = A sin(ωt + φ)，量 (ωt + φ) 是 t 时刻的相位，而 φ 是初相或相位偏移。两个频率相同但 φ 不同的振子被称为有相位差。例如，简谐运动中位移与速度的相位差为 π/2 弧度；位移与加速度相差 π 弧度（反相）。
Understanding phase is essential when combining oscillatory motions or analysing wave superposition. In SHM graphs, you can read the phase difference from the time shift Δt between two waveforms: phase difference = ω Δt = 2π Δt / T. If one curve reaches a maximum earlier, it is said to lead the other.
理解相位对于合振动的分析或波的叠加至关重要。在简谐运动图像中，你可以从两个波形之间的时间偏移 Δt 得出相位差：相位差 = ω Δt = 2π Δt / T。如果一条曲线更早到达最大值，就说它领先另一条。
In practical experiments, phase differences can be observed using a double-beam oscilloscope with two voltage signals from sensors. You might be asked to state the phase relationship between two SHM quantities derived from the same motion.
在实际实验中，可以使用双踪示波器观察来自传感器的两个电压信号之间的相位差。考试可能会要求你表述同一运动中两个简谐运动量之间的相位关系。
8. Damping: Light, Critical and Heavy | 阻尼：弱阻尼、临界阻尼和过阻尼
Real oscillators lose energy over time due to resistive forces like friction or air resistance. This effect is called damping and causes the amplitude to decrease gradually. In light damping, the system still oscillates with a frequency slightly lower than the natural frequency, and the amplitude decays exponentially: A = A₀ e^(-γt), where γ is the damping constant.
实际的振子会因摩擦力或空气阻力等耗散力而逐渐损失能量。这种效应称为阻尼，它导致振幅逐渐减小。在弱阻尼情况下，系统仍会以略低于固有频率的频率振荡，振幅按指数衰减：A = A₀ e^(-γt)，其中 γ 是阻尼常数。
Critical damping occurs when the system returns to equilibrium in the shortest possible time without overshooting and without oscillating. This is the design goal for car suspension systems, galvanometers, and door-closing mechanisms. Heavy (overdamped) damping means the system returns to equilibrium very slowly without oscillating, taking longer than critical damping.
临界阻尼是指系统在尽可能短的时间内回到平衡位置，且没有超调和振荡。这是汽车悬挂系统、检流计和门闭合机构的设计目标。过阻尼（重阻尼）是指系统不振荡但极其缓慢地回到平衡位置，比临界阻尼耗时更长。
The damping condition is characterised by the relationship between the damping coefficient b and the natural angular frequency ω₀. In many A-Level syllabi, qualitative understanding and graph interpretation are expected rather than quantitative differential equation solutions. You should recognise the three damping regimes from amplitude-time and displacement-time graphs.
阻尼条件可由阻尼系数 b 与固有角频率 ω₀ 之间的关系来表征。在许多 A-Level 大纲中，要求的是定性理解和图像解读，而非微分方程的定量求解。你应该能从振幅-时间图和位移-时间图中识别出三种阻尼状态。
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 its natural frequency. This is called forced oscillation. As the driving frequency approaches the natural frequency of the system, the amplitude of oscillation increases dramatically – this phenomenon is resonance.
当对一个振子施加周期性的外力时，系统会以外力的驱动频率振动，而非其固有频率。这称为受迫振动。当驱动频率接近系统的固有频率时，振幅急剧增大——这种现象就是共振。
At resonance, the energy transfer from the driver to the oscillator is most efficient, and the amplitude is limited only by damping. A lightly damped system has a very sharp resonance peak, while heavy damping broadens the peak and reduces the maximum amplitude. Key examples include pushing a swing, the Tacoma Narrows Bridge collapse, and tuning a radio circuit.
共振时，从驱动源到振子的能量传递效率最高，振幅仅受阻尼限制。弱阻尼系统的共振峰非常尖锐，而强阻尼会使峰变宽并降低最大振幅。典型例子包括荡秋千、塔科马海峡大桥坍塌和调节无线电电路。
The phase relationship between the driver and the oscillator also changes across resonance. Well below resonance, the oscillator and driver are in phase; at resonance, the oscillator lags the driver by π/2; well above resonance, the lag approaches π. This phase behaviour is important in understanding mechanical and electrical resonant systems.
驱动源和振子之间的相位关系在共振点附近也会发生变化。远低于共振频率时，振子与驱动源同相；在共振点，振子落后驱动源 π/2；远高于共振频率时，相位滞后接近 π。这种相位行为对于理解机械和电路中的共振系统很重要。
10. Graphical Analysis and Experimental Skills | 图像分析与实验技能
A-Level exams often test graphical representations of SHM. You must be able to determine amplitude, period, frequency, and phase from a displacement-time graph. For velocity-time and acceleration-time graphs, remember the amplitude relationships: vₘₐₓ = ωA and aₘₐₓ = ω²A. The gradient of a displacement-time graph at any point gives the instantaneous velocity.
A-Level 考试经常考查简谐运动的图像表示。你必须能够从位移-时间图中读取振幅、周期、频率和相位。对于速度-时间图和加速度-时间图，要记住振幅关系：vₘₐₓ = ωA，aₘₐₓ = ω²A。位移-时间图中任意一点的斜率给出瞬时速度。
Experiments typically involve: using a pendulum or mass-spring to measure T and hence determine g or k; investigating factors affecting the period; studying damping by measuring amplitude decay; and observing resonance using a vibration generator. Common apparatus includes stopwatches, light gates, motion sensors, and data loggers.
典型实验包括：用单摆或弹簧振子测量 T，从而确定 g 或 k；研究影响周期的因素；通过测量振幅衰减研究阻尼；用振动发生器观察共振现象。常用器材包括秒表、光电门、运动传感器和数据采集器。
To reduce uncertainty, always measure time for many oscillations (e.g. 10 oscillations repeated three times), use fiducial markers to define the counting point, and ensure the amplitude is small. In pendulum experiments, measure length to the centre of the bob and correct for any non-point-mass effects. Graphical methods such as T² vs L offer a more accurate determination of g than a single measurement.
为了减小不确定度，务必测量多次振荡的时间（例如，测量 10 个周期并重复三次），使用参考标记来明确计数点，并确保振幅较小。在单摆实验中，测量摆长到摆球中心，并修正任何非质点效应的影响。与单次测量相比，T²-L 图像法等图解方法可以更精确地测定 g。
11. Common Misconceptions and Exam Tips | 常见误区与应试技巧
One common misconception is that the period of a pendulum depends on its mass or amplitude; it does not for small angles. Another is confusing displacement from equilibrium with the length of the arc travelled. In SHM, the displacement x is always measured from the equilibrium position and can be positive or negative.
一个常见的误区是认为单摆的周期取决于质量或振幅；在小角度下并非如此。另一个误区是混淆离开平衡位置的位移和摆球经过的弧长。在简谐运动中，位移 x 总是以平衡位置为参考点测量，可以是正值或负值。
Students sometimes think velocity is maximum at maximum displacement because the bob ‘stops and starts’ there. Actually, the instantaneous velocity is zero at the extremes and maximum when passing through equilibrium. Acceleration, however, is maximum at extremes because the net force is largest.
学生有时会认为速度在最大位移处最大，因为摆球在那里“停住又启动”。其实，在端点处瞬时速度为零，而通过平衡位置时速度最大。然而，加速度在端点处最大，因为此时回复力最大。
When using energy equations, don’t confuse the potential energy of a spring (½kx²) with gravitational potential energy of a pendulum. For a pendulum, you must convert height to displacement using geometry. Also, in a vertical spring system, the equilibrium position is already stretched; total elastic energy includes static extension plus dynamic displacement from that equilibrium.
使用能量方程时，不要混淆弹簧的势能（½kx²）和单摆的重力势能。对于单摆，你必须用几何关系将高度转换为位移。此外，在竖直弹簧系统中，平衡位置已有伸长；总弹性势能包括静态伸长加上相对于该平衡位置的动态位移。
A final tip: practice drawing free-body diagrams for oscillators at various points. They highlight the restoring force direction and help avoid sign errors. In the exam, show clear working, include units, and always check if your calculator is in radian mode when evaluating trigonometric functions for SHM problems.
最后一点建议：练习绘制振子在不同位置处的受力图。这些图能突显回复力方向，有助于避免符号错误。在考试中，要展示清晰的计算过程，注明单位，并在处理简谐运动的三角函数时始终检查计算器是否处于弧度模式。
12. Summary of Key Formulas | 关键公式汇总
The central SHM equations are collected here for quick reference. Each one is vital for quantitative problem-solving.
这里汇总了简谐运动的核心公式，方便快速查阅。每一个公式对定量解题都至关重要。
| Quantity 物理量 | Formula 公式 |
|---|---|
| Defining equation 定义方程 | a = -ω²x |
| Displacement 位移 | x = A cos(ωt) or x = A sin(ωt) |
| Velocity 速度 | v = ±ω√(A² – x²) |
| Maximum speed 最大速度 | vₘₐₓ = ωA |
| Maximum acceleration 最大加速度 | aₘₐₓ = ω²A |
| Angular frequency 角频率 | ω = 2πf = 2π/T |
| Period (mass-spring) 周期（弹簧振子） | T = 2π√(m/k) |
| Period (pendulum) 周期（单摆） | T = 2π√(L/g) |
| Total energy 总能量 | E = ½mω²A² |
| Kinetic energy 动能 | K = ½mω²(A² – x²) |
| Potential energy 势能 | U = ½mω²x² |
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