AS Physics: Simple Harmonic Motion | AS 物理：简谐运动考点精讲

📚 AS Physics: Simple Harmonic Motion | AS 物理：简谐运动考点精讲

Simple Harmonic Motion (SHM) is the foundation for understanding oscillations in physics. From a mass on a spring to the swing of a pendulum, SHM models periodic motion with a restoring force proportional to displacement. This article comprehensively covers the key concepts, equations, energy transformations, practical examples, and common pitfalls that AS Physics students must master.

简谐运动（SHM）是理解物理振动的基石。从弹簧振子到单摆，简谐运动用与位移成正比的回复力来描述周期性运动。本文全面梳理 AS 物理中必须掌握的核心概念、方程、能量转换、实际案例以及常见易错点。

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

Simple Harmonic Motion is defined as oscillatory motion in which the acceleration is directly proportional to the displacement from equilibrium and always directed towards that equilibrium position.

简谐运动定义为一种振动，其加速度与相对于平衡位置的位移成正比，且方向始终指向该平衡位置。

The defining condition is a ∝ −x, which leads to a = −ω²x, where a is acceleration, x is displacement, and ω is the angular frequency.

其定义条件是 a ∝ −x，由此得出 a = −ω²x，其中 a 是加速度，x 是位移，ω 是角频率。

In SHM, there is a stable equilibrium point, and the motion is periodic and sinusoidal. The restoring force F = −kx is the physical origin for mechanical oscillators like springs.

在简谐运动中，存在一个稳定的平衡点，运动是周期性的且呈正弦曲线变化。回复力 F = −kx 是弹簧等机械振动体的物理根源。

2. The Basic Equations of SHM | 简谐运动的基本方程

Displacement as a function of time can be written as:

位移作为时间的函数可以写作：

x = A cos(ωt) or x = A sin(ωt)

The choice between sine and cosine depends on the initial conditions. If starting from maximum displacement, use cosine; if from equilibrium, use sine.

选用正弦或余弦取决于初始条件。若从最大位移处开始，用余弦；若从平衡位置开始，用正弦。

Velocity and acceleration are derived by differentiation:

速度和加速度通过求导得出：

v = −Aω sin(ωt) (or v = Aω cos(ωt) if x uses sine)

a = −Aω² cos(ωt) = −ω²x

The maximum speed is vₘₐₓ = ωA, and the maximum acceleration is aₘₐₓ = ω²A.

最大速率 vₘₐₓ = ωA，最大加速度 aₘₐₓ = ω²A。

3. Displacement, Velocity and Acceleration | 位移、速度和加速度

The three kinematic quantities are linked through phase relationships. Velocity leads displacement by π/2 (90°) and acceleration leads velocity by another π/2, meaning acceleration is π (180°) out of phase with displacement.

这三个运动学量通过相位关系联系起来。速度领先位移 π/2(90°)，加速度再领先速度 π/2，也就是说加速度与位移反相(相差 π)。

The velocity at any displacement can be found using:

任意位移处的速度可用下式求得：

v² = ω²(A² − x²) or v = ±ω√(A² − x²)

You must remember that velocity is zero at the extreme positions (x = ±A) and maximum at equilibrium (x = 0).

必须记住，在两端极限位置 (x = ±A) 速度为零，在平衡位置 (x = 0) 速度最大。

4. Energy in SHM | 简谐运动的能量

Energy in SHM continuously converts between kinetic and potential forms while the total mechanical energy stays constant (in an undamped system).

简谐运动中的能量在动能与势能之间持续转换，而总机械能保持不变（无阻尼系统）。

Kinetic energy: Eₖ = ½ m v² = ½ m ω² (A² − x²)

动能：Eₖ = ½ m v² = ½ m ω² (A² − x²)

Potential energy for a spring system: Eₚ = ½ k x², and since k = m ω², this becomes Eₚ = ½ m ω² x².

弹簧系统的势能：Eₚ = ½ k x²，又因 k = m ω²，此式化为 Eₚ = ½ m ω² x²。

Total energy: Eₜₒₜ = ½ k A² = ½ m ω² A², which is constant.

总能量：Eₜₒₜ = ½ k A² = ½ m ω² A²，保持不变。

Energy graphs show that Eₖ and Eₚ vary with displacement, each being maximum when the other is zero.

能量图显示，动能和势能随位移变化，一者最大时另一者为零。

5. The Mass-Spring System | 弹簧 – 质量系统

For a mass m attached to a spring of stiffness (spring constant) k, the time period T is given by:

对于连接在劲度系数为 k 的弹簧上的质量块 m，其周期 T 由下式给出：

T = 2π √(m/k)

This shows that T depends only on the mass and the spring constant, not on the amplitude – a key feature of isochronous oscillations.

这表明 T 仅取决于质量和弹簧常数，与振幅无关——这是等时振动的一个关键特征。

The angular frequency is ω = √(k/m), and the motion remains SHM as long as Hooke’s law (F = −kx) applies.

角频率 ω = √(k/m)，只要胡克定律 (F = −kx) 成立，运动就是简谐运动。

In a vertical mass-spring system, the equilibrium position shifts due to gravity, but the SHM equations remain identical after measuring displacement from the new equilibrium position.

在竖直弹簧振子中，重力会改变平衡位置，但只要从新的平衡位置开始测量位移，简谐运动方程依然不变。

6. The Simple Pendulum | 单摆

A simple pendulum consists of a point mass suspended by a light, inextensible string. For small angles (< ~10°), the motion approximates SHM with:

单摆由轻质、不可伸长的绳子悬挂质点构成。在小角度（约 <10°）下，运动近似为简谐运动，其周期为：

T = 2π √(l/g)

The restoring force arises from the component of weight tangential to the arc: F = −mg sinθ ≈ −mgθ. The angular frequency ω = √(g/l).

回复力来自重力沿圆弧切线方向的分量：F = −mg sinθ ≈ −mgθ。角频率 ω = √(g/l)。

Note that the period does not depend on the mass of the bob, only on length and gravitational field strength.

注意，周期与摆球质量无关，只取决于摆长和重力场强度。

7. Resonance | 共振

When a system undergoing SHM is driven by an external periodic force, resonance occurs if the driving frequency matches the natural frequency of the system.

当一个做简谐运动的系统受到周期性外力的驱动时，若驱动频率等于系统的固有频率，就会发生共振。

At resonance, the amplitude of oscillation becomes very large, which can be useful (e.g., musical instruments) or destructive (e.g., bridge collapse).

共振时，振幅变得非常大，这可能是有益的（如乐器），也可能是破坏性的（如桥梁坍塌）。

The sharpness of resonance is described by the Q-factor; light damping leads to a sharp peak, while heavy damping broadens and reduces the peak.

共振的尖锐度用品质因数 Q 描述；轻阻尼导致尖锐的峰，而重阻尼会使峰变宽并降低幅度。

8. Damping | 阻尼

Damping removes energy from an oscillating system, causing the amplitude to gradually decrease. Three main types are: light (underdamped), critical, and heavy (overdamped).

阻尼从振动系统中消耗能量，使振幅逐渐减小。主要分为三类：弱阻尼（欠阻尼）、临界阻尼和过阻尼。

In light damping, the period stays nearly constant but amplitude decays exponentially. In critical damping, the system returns to equilibrium in the shortest possible time without oscillating.

弱阻尼时，周期几乎不变，但振幅按指数衰减。临界阻尼时，系统在最短时间内回到平衡位置而不振荡。

Damping also affects forced oscillations by reducing the resonant amplitude and shifting the peak slightly to a lower frequency (in the case of velocity resonance).

阻尼还会通过降低共振振幅并将峰值略微移向更低频率（在速度共振情况下）来影响受迫振动。

9. Graphical Analysis | 图形分析

Three key graphs for SHM are displacement-time, velocity-time, and acceleration-time. Drawing and interpreting these graphs is a common exam task.

简谐运动的三个关键图像是位移 – 时间图、速度 – 时间图和加速度 – 时间图。绘图和解读这些图像是常见的考试任务。

The x-t graph is a cosine (or sine) wave. The v-t graph is the gradient of x-t and thus leads by T/4. The a-t graph is the gradient of v-t and is inverted with respect to x-t.

x-t 图是余弦（或正弦）波形。v-t 图是 x-t 的梯度，因此超前 T/4。a-t 图是 v-t 的梯度，与 x-t 反相。

Energy-time graphs show total energy as a horizontal line, while kinetic and potential energies oscillate like sin² and cos² functions, both with a period half that of the motion.

能量 – 时间图中，总能量是一条水平线，而动能和势能如同 sin² 和 cos² 函数一样振荡，两者的周期均为运动周期的一半。

10. Key Exam Points and Common Mistakes | 必考要点与常见错误

The restoring force condition a ∝ −x is the only definition of SHM – do not rely on “sinusoidal” alone.

回复力条件 a ∝ −x 是简谐运动的唯一定义——不要仅凭“正弦形式”来判断。
Confusing angular frequency ω with ordinary frequency f. Remember ω = 2πf and T = 1/f.

混淆角频率 ω 与普通频率 f。记住 ω = 2πf 且 T = 1/f。
Forgetting the sign in a = −ω²x, which indicates direction opposite to displacement.

忘记 a = −ω²x 中的负号，它表示方向与位移相反。
Using degrees in radian formulas; phase relationships and circular functions must use radian measure.

在弧度制公式中使用角度；相位关系和三角函数必须使用弧度。
Assuming amplitude changes in energy calculations for an undamped free oscillation; total energy stays constant.

在无阻尼自由振动的能量计算中误认为振幅变化；总能量是常数。

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AS Physics: Simple Harmonic Motion | AS 物理：简谐运动 考点精讲