📚 Simple Harmonic Motion (SHM) – CCEA Physics | 简谐运动 – CCEA 物理考点精讲
Simple harmonic motion is a fundamental type of oscillation that appears in pendulums, vibrating springs, and molecular vibrations. In CCEA A-Level Physics, you are expected to define SHM precisely, analyse its kinematics and dynamics, apply energy considerations, and perform experiments to measure quantities such as acceleration due to gravity. This article revisits every major aspect of the topic with a clear, bilingual explanation.
简谐运动是出现在单摆、弹簧振动以及分子振动中的一种基本振动形式。在 CCEA A-Level 物理中，你需要精确定义简谐运动，分析其运动学和动力学，运用能量观点，并进行实验测量如重力加速度等物理量。本文以清晰的中英双语讲解，重新梳理该主题的每个重点。
1. Definition and Conditions for SHM | 简谐运动的定义与条件
SHM is defined as oscillatory motion in which the acceleration a is directly proportional to the displacement x from the equilibrium position, and is always directed towards that equilibrium point. Mathematically, a ∝ −x, or a = −ω²x, where ω is the angular frequency.
简谐运动定义为加速度 a 与离开平衡位置的位移 x 成正比，且加速度方向总是指向平衡位置。数学上表示为 a ∝ −x，或 a = −ω²x，其中 ω 是角频率。
The conditions for SHM require the restoring force F to obey Hooke’s law type relationship F = −kx, which leads to a = −(k/m)x. The system must be free from non-conservative forces such as friction in the ideal case, and the amplitude must be small enough that the restoring force remains linear.
产生简谐运动的条件是回复力 F 必须满足类似胡克定律的关系 F = −kx，从而导致 a = −(k/m)x。理想情况下系统不受摩擦等非保守力的影响，且振幅必须足够小，使回复力保持线性。
2. Displacement, Velocity and Acceleration | 位移、速度与加速度
In SHM, displacement x varies sinusoidally with time: x = A sin(ωt + φ) or x = A cos(ωt + φ). The velocity v is the time derivative: v = ωA cos(ωt + φ) or v = −ωA sin(ωt + φ). The acceleration a is the second derivative: a = −ω²A sin(ωt + φ) = −ω²x.
在简谐运动中，位移 x 随时间按正弦规律变化：x = A sin(ωt + φ) 或 x = A cos(ωt + φ)。速度 v 是位移对时间的导数：v = ωA cos(ωt + φ) 或 v = −ωA sin(ωt + φ)。加速度 a 是二阶导数：a = −ω²A sin(ωt + φ) = −ω²x。
The maximum speed occurs as the oscillator passes through equilibrium: vₘₐₓ = ωA. The maximum acceleration occurs at the extreme displacements: aₘₐₓ = ω²A. Note that velocity leads displacement by π/2 radians, while acceleration is π radians out of phase with displacement.
最大速度出现在振子经过平衡位置时：vₘₐₓ = ωA。最大加速度出现在最大位移处：aₘₐₓ = ω²A。注意速度的相位比位移超前 π/2 弧度，而加速度与位移相位相差 π 弧度（反向）。
3. Equations of SHM | 简谐运动的方程
Key equations for SHM include the defining equation a = −ω²x and the time equations x = A cos(ωt) (if starting from maximum displacement) or x = A sin(ωt) (if starting from equilibrium). The period T is the time for one complete oscillation, related to angular frequency by ω = 2πf = 2π/T.
简谐运动的关键方程包括定义方程 a = −ω²x，以及时间方程 x = A cos(ωt)（若从最大位移开始）或 x = A sin(ωt)（若从平衡位置开始）。周期 T 是完成一次全振动的时间，与角频率的关系为 ω = 2πf = 2π/T。
For a mass-spring system, ω = √(k/m) and T = 2π√(m/k). For a simple pendulum, ω = √(g/L) and T = 2π√(L/g). These formulas are derived from the restoring force expressions and are valid only for small angular amplitudes (<10°).
对于弹簧振子系统，ω = √(k/m)，T = 2π√(m/k)。对于单摆，ω = √(g/L)，T = 2π√(L/g)。这些公式是由回复力表达式推导而来的，仅在小角度振幅（<10°）下成立。
The velocity at any displacement can be found from energy conservation or by v = ± ω √(A² − x²). The acceleration can be written as a = −ω²x, giving a linear relationship between a and x with slope −ω².
任意位移处的速度可由能量守恒得到，或使用 v = ± ω √(A² − x²)。加速度可写为 a = −ω²x，表明 a 与 x 之间呈线性关系，斜率为 −ω²。
4. The Simple Pendulum | 单摆
A simple pendulum consists of a point mass suspended from a light, inextensible string. When displaced by a small angle θ, the restoring force is −mg sinθ ≈ −mgθ, producing an angular acceleration proportional to −θ. This leads to SHM about the lowest point.
单摆由悬挂于轻质、不可伸长的细线上的质点构成。当偏离小角度 θ 时，回复力为 −mg sinθ ≈ −mgθ，产生的角加速度与 −θ 成正比，从而使摆锤绕最低点作简谐运动。
The period of a simple pendulum is T = 2π√(L/g) and is independent of mass and amplitude (for small angles). This is often used to measure gravitational field strength g. A graph of T² against L yields a straight line through the origin with gradient 4π²/g.
单摆的周期为 T = 2π√(L/g)，与质量和振幅（小角度时）无关。这一性质常被用来测量重力加速度 g。绘制 T² 对 L 的图像，可得到一条过原点的直线，斜率为 4π²/g。
Remember that the formula assumes the small-angle approximation sinθ ≈ θ (in radians). For larger amplitudes, the period increases and motion is no longer simple harmonic. CCEA may ask you to suggest how to minimise uncertainties when measuring T.
请记住该公式基于小角度近似 sinθ ≈ θ（弧度制）。振幅较大时，周期会增大，运动不再是简谐运动。CCEA 可能会要求你提出如何减小测量 T 时的不确定度。
5. Mass-Spring System | 弹簧振子系统
A mass attached to a spring obeys Hooke’s law F = −kx when displaced. The resultant equation of motion m(d²x/dt²) = −kx gives an angular frequency ω = √(k/m) and period T = 2π√(m/k). This is true for both horizontal and vertical setups, provided the spring obeys Hooke’s law.
连接在弹簧上的物体偏离平衡位置时满足胡克定律 F = −kx。其运动方程 m(d²x/dt²) = −kx 给出角频率 ω = √(k/m)，周期 T = 2π√(m/k)。这适用于水平和竖直安装的弹簧，前提是弹簧遵守胡克定律。
In a vertical mass-spring system, gravity shifts the equilibrium position but does not affect the period. The spring constant k can be determined from static extension measurements: k = mg/e, where e is the extension at equilibrium.
在竖直弹簧振子中，重力会使平衡位置发生移动，但不影响周期。弹簧的劲度系数 k 可通过静态伸长量测量得到：k = mg/e，其中 e 为平衡时的伸长量。
Experiments often involve varying the mass and measuring T² to find k: T² = (4π²/k)m, which gives a linear graph. The energy in the system continuously interchanges between elastic potential energy and kinetic energy.
实验通常通过改变质量并测量 T² 来求出 k：T² = (4π²/k)m，由此可得线性图像。系统中的能量在弹性势能和动能之间连续转换。
6. Energy in SHM | 简谐运动中的能量
The total mechanical energy of an undamped SHM system is constant and proportional to A². For a mass-spring system, E_total = ½kA². At any position x, the kinetic energy is ½k(A² − x²) and the potential energy is ½kx².
无阻尼简谐运动系统的总机械能保持不变，且与 A² 成正比。对于弹簧振子，E_total = ½kA²。在任意位置 x 处，动能为 ½k(A² − x²)，势能为 ½kx²。
Energy graphs show that KE and PE both vary sinusoidally with time but with twice the frequency. When KE is maximum (at equilibrium), PE is zero; when KE is zero (at extremes), PE is maximum. The total energy line is horizontal in an undamped system.
能量图像显示动能和势能随时间均按正弦规律变化，但频率是位移频率的两倍。当动能最大时（平衡位置），势能为零；当动能为零时（最大位移），势能最大。在无阻尼系统中，总能量线为水平直线。
In a pendulum, the potential energy is mgh, where h = L(1 − cosθ). For small angles, PE ≈ ½mgLθ², analogous to the ½kx² form. Energy conservation arguments can be used to find speed at any point.
在单摆中，势能为 mgh，其中 h = L(1 − cosθ)。对于小角度，PE ≈ ½mgLθ²，类似于 ½kx² 的形式。利用能量守恒可以求出任意点的速度。
7. Phase Difference | 相位差
Phase difference between two oscillating quantities is expressed in radians or degrees. In SHM, displacement lags velocity by π/2, while acceleration leads displacement by π (or is anti-phase). When comparing two oscillators of the same frequency, phase difference Δφ = 2π(Δt/T).
两个振动量之间的相位差用弧度或度表示。在简谐运动中，位移比速度滞后 π/2，加速度比位移超前 π（或反相）。当比较两个相同频率的振子时，相位差 Δφ = 2π(Δt/T)。
Using rotating vector (phasor) diagrams can help visualise phase relationships. A phasor of length A rotates with angular speed ω; its horizontal component gives x = A cos(ωt). Velocity and acceleration phasors are rotated by 90° and 180° respectively.
使用旋转矢量（相量）图有助于可视化相位关系。长度为 A 的相量以角速度 ω 旋转，其水平分量给出 x = A cos(ωt)。速度和加速度相量分别旋转 90° 和 180°。
8. Damping and Resonance | 阻尼与共振
Damping causes the amplitude of oscillation to decay over time due to dissipative forces. Three types are identified: light damping (amplitude decreases gradually), critical damping (system returns to equilibrium in the shortest time without oscillating), and heavy damping (slow return without oscillating).
阻尼使振幅因耗散力而随时间衰减。阻尼可分为三类：轻阻尼（振幅逐渐减小）、临界阻尼（系统以最短时间回到平衡位置而无振荡）和过阻尼（缓慢回到平衡位置且无振荡）。
Forced oscillations occur when a periodic driving force is applied. Resonance happens when the driving frequency matches the natural frequency of the system, leading to maximum amplitude. The resonance curve shows amplitude vs. frequency, with the peak becoming sharper for lighter damping.
受迫振动发生在施加周期性驱动力时。当驱动力频率等于系统的固有频率时，发生共振，振幅达到最大。共振曲线显示振幅随频率的变化，阻尼越小，峰越尖锐。
Examples of resonance include a swing being pushed at its natural frequency, a wine glass shattered by sound, and the Tacoma Narrows Bridge collapse. Applications include tuning radios and microwave ovens.
共振的例子包括以固有频率推动秋千、声波震碎酒杯、以及塔科马海峡大桥的倒塌。应用包括调谐收音机和微波炉。
9. Graphical Representation | 图像表示
Typical CCEA questions ask you to sketch or interpret graphs of displacement, velocity, and acceleration against time. Displacement is a sine or cosine wave; velocity is also sinusoidal but shifted left by a quarter period; acceleration is a reflected sine wave (inverted relative to displacement).
典型的 CCEA 考题要求你绘制或解读位移、速度和加速度对时间的图像。位移是正弦或余弦波；速度同样是正弦波，但向左移动四分之一周期；加速度是位移的倒置正弦波（与位移反向）。
Other important graphs include: a vs. x (straight line with negative slope −ω²), v² vs. x² (linear relation from energy), and kinetic energy vs. displacement (parabolic). Also, damping graphs show an exponential decay envelope.
其他重要图像包括：a 对 x 图（斜率为负的直线 −ω²），v² 对 x² 图（由能量得出的线性关系），以及动能对位移图（抛物线）。阻尼图像则显示指数衰减的包络线。
When plotting experimental data, such as T² vs. L for a pendulum, the gradient provides an indirect measurement of g. You must include uncertainty bars where appropriate and calculate gradient uncertainty for full marks.
在绘制实验数据时，例如单摆的 T² 对 L 图，斜率可用于间接测量 g。你需要适当地添加误差棒，并计算斜率的不确定度以获取满分。
10. Experimental Methods | 实验方法
Measuring g using a simple pendulum: Vary the length L, measure the period T for small oscillations (θ < 10°), timing for 10–20 oscillations to reduce random error. Plot T² against L, find gradient = 4π²/g, thus g = 4π²/gradient. Repeat and calculate a mean.
用单摆测量 g：改变摆长 L，在小角度摆动（θ < 10°）下测量周期 T，计时 10–20 次振动以减少随机误差。绘制 T²-L 图，斜率 = 4π²/g，因此 g = 4π²/斜率。重复实验并求平均值。
To determine the spring constant k: Use Hooke’s law in static mode (add masses, measure extension, slope of F-x graph = k). Dynamic method: measure T for different masses, plot T² vs. m, slope = 4π²/k. Both methods have sources of uncertainty, such as parallax, timing reaction, and spring’s own mass.
测量弹簧劲度系数 k：静态法使用胡克定律（加砝码，测伸长量，F-x 图斜率 = k）。动态法：测量不同质量下的周期，绘制 T² 对 m 图，斜率 = 4π²/k。两种方法都有误差来源，如视差、计时反应时间和弹簧自身质量。
CCEA practical questions often require you to describe how to reduce uncertainties, e.g., timing from equilibrium position, using a fiducial marker, and measuring L to the centre of the bob. Always discuss repeat readings and appropriate data handling.
CCEA 实验题常要求你描述如何减小不确定度，例如从平衡位置开始计时、使用标记线、测量摆长至小球中心。务必讨论重复读数及合理的数据处理。
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