📚 C.1 Simple Harmonic Motion | C.1 简谐运动
Simple harmonic motion (SHM) forms the foundation for understanding oscillatory systems in physics, from the vibrations of atoms to the motion of a pendulum. This topic introduces the key characteristics, mathematical descriptions, and energy transformations of SHM. At higher level (HL), we delve deeper into the differential equation governing SHM, explore damped and forced oscillations, and examine the phenomenon of resonance. By mastering these concepts, IB Physics students gain a powerful toolkit for analysing a wide range of physical systems.
简谐运动（SHM）是理解物理学中振动系统的基础，从原子的振动到摆的运动。本主题介绍简谐运动的关键特性、数学描述和能量转换。在高级水平（HL）中，我们深入探讨控制简谐运动的微分方程，研究阻尼振动和受迫振动，并考察共振现象。通过掌握这些概念，IB物理学生将获得分析各种物理系统的强大工具。
1. Introduction to Periodic Motion and SHM | 周期运动与简谐运动简介
c motion is a special type of periodic motion in which the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. This leads to a motion that is sinusoidal in time, making SHM one of the most mathematically tractable forms of oscillation.
周期运动是任何以固定时间间隔重复的运动。完成一个完整循环所需的时间称为周期T，单位时间内循环的次数称为频率f，满足f = 1/T。简谐运动是一种特殊的周期运动，其中回复力与离开平衡位置的位移成正比，且方向相反。这导致运动在时间上呈正弦曲线，使简谐运动成为数学上最容易处理的振动形式之一。
Common examples of systems undergoing SHM include a mass attached to a spring (ignoring friction), a simple pendulum with small angular displacement, and the oscillation of charge in certain LC circuits. In all these cases, the defining feature is that the acceleration a is proportional to the negative of the displacement x.
常见的简谐运动系统包括弹簧连接的物块（忽略摩擦）、小角度摆动的单摆以及某些LC电路中的电荷振荡。在所有这些情况下，关键特征是加速度a与位移x的负值成正比。
2. Defining Simple Harmonic Motion | 简谐运动的定义
The fundamental definition of SHM is expressed by the equation: a ∝ ω is the angular frequency, a constant that depends on the physical properties of the system. This condition means that the acceleration is always directed towards the equilibrium position and increases in magnitude with distance from it. Any system that satisfies this condition for small displacements will execute simple harmonic motion.
简谐运动的基本定义由方程表示：a ∝ −x，或更精确地，a = −ω²x。这里ω是角频率，是一个取决于系统物理性质的常数。这个条件意味着加速度始终指向平衡位置，且其大小随离开平衡位置的距离而增加。任何在小位移下满足此条件的系统都会做简谐运动。
be a sinusoidal function. The standard solutions are x = A sin(ωt + φ) or x = A cos(ωt + φ' itude (maximum displacement) and φ is the phase constant determined by initial conditions.
从这个定义方程，我们可以推导出位移x作为时间t的函数必定是正弦函数。标准解为x = A sin(ωt + φ) 或 x = A cos(ωt + φ')，其中A是振幅（最大位移），φ是由初始条件决定的相位常数。
2π/T. The unit of ω is radian per second (rad s⁻¹). It is important to note that ω is not the same as the instantaneous angular velocity of a circular motion analogy; rather, it is a measure of how rapidly the oscillation cycles occur.
角频率ω与周期T和频率f的关系为ω = 2πf = 2π/T。ω的单位是弧度每秒（rad s⁻¹）。需要强调的是，ω不同于圆周运动类比中的瞬时角速度；它衡量的是振动循环发生的快慢。
3. Displacement, Velocity, and Acceleration in SHM | 简谐运动中的位移、速度和加速度
onships show that velocity is a quarter cycle (π/2 radians) ahead of displacement, and acceleration is a half cycle (π radians) ahead of displacement.
如果取位移方程为x = A cos(ωt)（为简单起见假设相位常数为零），那么速度v是时间导数：v = dx/dt = −Aω sin(ωt)。加速度a是速度的导数：a = dv/dt = −Aω² cos(ωt) = −ω²x。这些关系表明速度超前位移四分之一周期（π/2弧度），加速度超前位移半个周期（π弧度）。
aximum speed occurs when the object passes through equilibrium (x=0), and it is given by vmax he maximum acceleration occurs at the extreme positions (x = ±A), where its magnitude is amax any arbitrary displacement, the speed can be expressed in terms of displacement as v = ±ω√A² − x².
最大速度出现在物体通过平衡位置时（x=0），其大小为vmax = ωA。最大加速度出现在极端位置（x = ±A），其大小为amax = ω²A。对于任意位移，速度可以用位移表示为v = ±ω√A² − x²。
These equations are valid for any SHM system, provided the displacement is measured from equilibrium. They allow us to determine position and velocity at any time if amplitude, angular frequency, and initial conditions are known.
这些方程适用于任何简谐运动系统，前提是位移从平衡位置量起。它们使我们能够在已知振幅、角频率和初始条件的情况下确定任意时刻的位置和速度。
4. The Spring-Mass System | 弹簧振子
ictionless horizontal surface. The restoring force exerted by the spring is given by Hooke's law: F = −kx. Using Newton' = −(k/m)x. Comparing with a = −ω²x, we identify ω² = k/m, so the angular frequency is ω = √k/m.
简谐运动的一个经典例子是质量为m的物块连接在劲度系数为k的轻弹簧上，在无摩擦水平面上运动。弹簧施加的回复力由胡克定律给出：F = −kx。利用牛顿第二定律F = ma，我们得到ma = −kx，可整理为a = −(k/m)x。与a = −ω²x比较，可得ω² = k/m，因此角频率为ω = √k/m。
The period of oscillation for the spring-mass system is T = 2π/ω = 2π√m/k. Note that the period depends on the mass and the spring constant but is independent of the amplitude, a property known as isochronism. This holds true as long as Hooke's law is valid and the motion is undamped.
弹簧振子的振动周期为T = 2π/ω = 2π√m/k。注意，周期取决于质量和弹簧劲度系数，但与振幅无关，这一性质称为等时性。只要胡克定律成立且运动无阻尼，等时性就成立。
When the spring is oriented vertically, the equilibrium position shifts due to gravity, but the restoring force for displacements from the new equilibrium is still −kx, so the SHM equations remain valid. The angular frequency and period are unchanged.
当弹簧竖直放置时，由于重力，平衡位置会移动，但对于偏离新平衡位置的位移，回复力仍为−kx，因此简谐运动方程仍然有效。角频率和周期不变。
5. The Simple Pendulum | 单摆
α = −(g/L)θ, which has the same form as a = −ω²x. Hence, the angular frequency for a simple pendulum is ω = √g/L, and the period is T = 2π√L/g.
单摆由一个悬挂在长为L的轻质不可伸长的细线上的质点（摆锤）组成。对于小角位移θ（通常小于约10°，此时以弧度为单位的sin θ ≈ θ），回复力矩与−θ成正比。运动方程导出角加速度α = −(g/L)θ，这与a = −ω²x的形式相同。因此，单摆的角频率为ω = √g/L，周期为T = 2π√L/g。
Like the spring-mass system, the period of a simple pendulum is independent of the amplitude for small oscillations. It depends only on the length of the pendulum and the acceleration due to gravity. This property makes the pendulum useful for timekeeping and for measuring g.
与弹簧振子系统一样，单摆在小幅度摆动时的周期与振幅无关。它仅取决于摆长和重力加速度。这一性质使单摆可用于计时和测量g值。
, and the restoring force along the arc is −mg sin θ ≈ −mgθ, leading to the SHM condition.
在小角度近似中使用弧度很重要。以弧度为单位，沿弧线的位移为s = Lθ，沿弧线的回复力为−mg sin θ ≈ −mgθ，从而得到简谐运动条件。
6. Energy in Simple Harmonic Motion | 简谐运动中的能量
In SHM, energy continuously interchanges between kinetic energy (Ek) and potential energy (Ep), while the total mechanical energy (Etotal) remains constant (assuming no damping). For a horizontal spring-mass system, the potential energy stored in the spring is Ep = ½kx² = ½mω²x². The kinetic energy is Ek = ½mv² = ½mω²(A² − x²). Adding these gives Etotal = ½mω²A² = ½kA².
在简谐运动中，能量在动能（Ek）和势能（Ep）之间不断转换，而总机械能（Etotal）保持恒定（假设无阻尼）。对于水平弹簧振子系统，储存在弹簧中的势能为Ep = ½kx² = ½mω²x²。动能为Ek = ½mv² = ½mω²(A² − x²)。两者相加得Etotal = ½mω²A² = ½kA²。
y potential, and the speed is zero. The energy remains constant, proportional to the square of the amplitude and the square of the angular frequency.
在平衡位置（x=0），能量完全是动能，速度最大。在极端位置（x = ±A），能量完全是势能，速度为零。能量保持恒定，与振幅的平方和角频率的平方成正比。
For a simple pendulum, the potential energy is gravitational: Ep elative to the lowest point. For small angles, h can be approximated as ½Lθ², leading to Ep = ½mgLθ², which again has the form ½mω²x² if we use x = Lθ.
对于单摆，势能是重力势能：Ep = mgh，其中h是相对于最低点的竖直高度。对于小角度，h可近似为½Lθ²，得到Ep = ½mgLθ²，若使用x = Lθ，此式同样可写成½mω²x²的形式。
The concept of energy conservation in SHM allows the calculation of speed as a function of position without needing to solve differential equations. It also provides a powerful method for deriving equations of motion in more complex systems.
简谐运动中的能量守恒概念允许在不求解微分方程的情况下计算速度随位置的变化。它也为推导更复杂系统中的运动方程提供了一种强有力的方法。
7. Graphical Representations of SHM | 简谐运动的图像表示
se by π radians (a = −ω²x), while v leads x by π/2 radians. Energy-time graphs show that Ek and Ep are both non-negative and oscillate with twice the frequency of the displacement, each varying between 0 and the total energy. Their sum is a horizontal line representing constant total energy.
简谐运动中位移x、速度v和加速度a随时间t变化的图像是具有相同频率的正弦函数。通常，x和a的相位相差π弧度（a = −ω²x），而v超前x π/2弧度。能量-时间图像显示，Ek和Ep均为非负量，并以位移频率的两倍振荡，各自在0到总能量之间变化。它们的总和是一条水平线，代表恒定的总能量。
Graphs of kinetic and potential energy against displacement are parabolic: Ep = ½k x², and Ek These graphical representations are invaluable for visualising the dynamic nature of SHM.
动能和势能随位移变化的图像是抛物线：Ep = ½k x²，Ek = ½k (A² − x²)。这清楚地显示了能量交换以及x = ±A处的转折点。这些图像表示对于可视化简谐运动的动态特性极为宝贵。
In IB examinations, students are often asked to sketch or interpret these graphs, identifying key points such as amplitude, period, turning points, and regions of maximum speed or acceleration. Understanding the phase relationships is critical.
在IB考试中，经常要求学生绘制或解释这些图像，识别关键点如振幅、周期、转折点以及最大速度或加速度区域。理解相位关系至关重要。
8. HL: Differential Equation of SHM | HL: 简谐运动的微分方程
nd-order linear differential equation: d²x/dt² + ω²x = 0. This equation arises from Newton' standard form. The general solution is x = A sin(ωt + φ) or equivalently x = A cos(ωt + φ'), where A and φ are arbitrary constants determined by initial displacement and velocity.
在高级水平中，简谐运动由二阶线性微分方程定义：d²x/dt² + ω²x = 0。该方程源自牛顿第二定律，结合回复力F = −kx，给出m d²x/dt² = −kx，整理即得标准形式。通解为x = A sin(ωt + φ) 或等价地x = A cos(ωt + φ')，其中A和φ是由初始位移和速度决定的任意常数。
ed to find φ: if x(0) = x₀ and v(0) = v₀, then φ = tan⁻¹(x₀ω/v₀) and A = √(x₀² + (v₀/ω)²).
为验证该解，对x = A sin(ωt + φ)求二阶导：v = Aω cos(ωt + φ)，a = −Aω² sin(ωt + φ) = −ω²x，确认其满足微分方程。t=0时的速度可用于求φ：若x(0) = x₀且v(0) = v₀，则φ = tan⁻¹(x₀ω/v₀)，A = √(x₀² + (v₀/ω)²)。
xhibit SHM with angular frequency ω, regardless of the physical origin of the force. This unifies the mathematical treatment of mechanical, electrical, and other oscillators.
微分方程方法非常强大，因为它表明任何服从d²x/dt² + ω²x = 0的系统都将以角频率ω做简谐运动，无论力的物理来源如何。这统一了机械、电气和其他振荡器的数学处理方法。
9. HL: Damped Harmonic Motion | HL: 阻尼简谐运动
In real systems, dissipative forces such as friction or air resistance cause the amplitude of oscillation to decrease over time, a phenomenon called damping. For a damping force proportional to velocity, Fdamp 0, or d²x/dt² + 2γ dx/dt + ω₀²x = 0, where γ = b/(2m) is the damping coefficient and ω₀ = √k/m is the natural angular frequency.
在实际系统中，耗散力如摩擦或空气阻力导致振幅随时间减小，这一现象称为阻尼。对于与速度成正比的阻尼力Fdamp = −bv（b为阻尼常数），运动方程变为：m d²x/dt² + b dx/dt + kx = 0，或d²x/dt² + 2γ dx/dt + ω₀²x = 0，其中γ = b/(2m)为阻尼系数，ω₀ = √k/m为固有角频率。
s with gradually decreasing amplitude, described by x = A e⁻ᵞᵗ cos(ω_d t + φ), where ω_d = √ω₀² − γ² returns to equilibrium in the shortest possible time without oscillating. (c) Heavy damping (overdamped, γ > ω₀): the system returns to equilibrium without oscillation but more slowly than in the critical case.
根据γ相对于ω₀的大小，会出现三种类型的阻尼：（a）轻阻尼（欠阻尼，γ < ω₀）：系统以逐渐减小的振幅振荡，由x = A e⁻ᵞᵗ cos(ω_d t + φ)描述，其中ω_d = √ω₀² − γ²为阻尼角频率。（b）临界阻尼（γ = ω₀）：系统以最短时间返回平衡位置而不发生振荡。（c）重阻尼（过阻尼，γ > ω₀）：系统无振荡地返回平衡位置，但比临界情况更慢。
s the damped period. This provides a convenient method for measuring the damping constant experimentally.
在轻阻尼中，振幅按指数衰减，A(t) = A₀ e⁻ᵞᵗ。对数衰减量δ定义为相隔一个周期两个连续振幅比值的自然对数：δ = ln(A₁/A₂) = γT，其中T为阻尼周期。这为实验测量阻尼常数提供了便利的方法。
10. HL: Forced Oscillations and Resonance | HL: 受迫振动与共振
cy and the damping. The equation of motion becomes: m d²x/dt² + b dx/dt + kx = F₀ sin(ωt).
当外部周期性驱动力F = F₀ sin(ωt)施加到阻尼振子上时，系统最终以驱动频率ω振动，而非其固有频率ω₀。受迫振动的振幅取决于驱动频率和阻尼。运动方程变为：m d²x/dt² + b dx/dt + kx = F₀ sin(ωt)。
becomes very large, especially for lightly damped systems. The resonant frequency ω_res = √ω₀² − 2γ², which reduces to ω₀ for very small damping.
稳态解具有形式x = A sin(ωt − φ)，其中振幅A由A = F₀ / (m √((ω₀² − ω²)² + (2γω)²))给出。当驱动频率接近固有频率时，即ω ≈ ω₀，发生共振，此时振幅变得非常大，特别是对于轻阻尼系统。共振频率ω_res = √ω₀² − 2γ²，在阻尼极小时趋近于ω₀。
A sharp resonance peak is observed for low damping, whereas increased damping broadens the peak and reduces the maximum amplitude. The phase difference φ between the driving force and the displacement varies from 0 at low frequencies to π at high frequencies, passing through π/2 at resonance. Resonance is important in many applications, from tuning radios to designing earthquake-resistant structures, but it can also lead to catastrophic failures if not controlled.
对于小阻尼，可观察到尖锐的共振峰，而增大阻尼会使峰变宽并降低最大振幅。驱动力与位移之间的相位差φ从低频率时的0变化到高频率时的π，在共振时经过π/2。共振在许多应用中很重要，从调谐收音机到设计抗震结构，但如果不加以控制，也可能导致灾难性破坏。
11. HL: Phase Difference and Initial Conditions | HL: 相位差与初始条件
of sine or cosine function is a matter of convenience, as sin(ωt + φ) = cos(ωt + φ − π/2).
位移方程x = A sin(ωt + φ)中的相位常数φ决定了振子的初始状态。它表示运动相对于从x=0开始并正向移动的参考正弦函数超前或滞后的周期部分。选择正弦或余弦函数是方便问题，因为sin(ωt + φ) = cos(ωt + φ − π/2)。
In systems involving two oscillators or two variables (e.g., displacement and velocity), the phase difference describes the relative timing. For SHM, velocity leads displacement by π/2 rad, and acceleration leads displacement by π rad. These phase relationships are independent of amplitude and frequency and are a direct consequence of the differential equation.
在涉及两个振子或两个变量（如位移和速度）的系统中，相位差描述了相对时序。对于简谐运动，速度超前位移π/2 rad，加速度超前位移π rad。这些相位关系与振幅和频率无关，是微分方程的直接结果。
When two SHM systems with the same frequency but different phase constants are combined, superposition leads to interference effects, which are studied in wave topics. Understanding initial conditions and phase allows students to correctly predict the motion from given starting displacement and velocity, an essential skill for problem-solving in HL Physics.
当两个频率相同但相位常数不同的简谐运动系统组合时，叠加会导致干涉效应，这将在波动专题中学习。理解初始条件和相位使学生能够根据给定的起始位移和速度正确预测运动，这是HL物理中解决问题的基本技能。
12. Summary and Key Formulas | 总结与关键公式
The study of simple harmonic motion provides a unified framework for analysing oscillatory behaviour across different areas of physics. The key definitions and equations are summarised in the table below. Mastering these formulas and their derivations equips students to tackle a wide range of problems, from basic oscillations to advanced resonance phenomena.
对简谐运动的学习提供了一个统一的框架，用于分析物理学不同领域中的振动行为。关键定义和方程总结于下表中。掌握这些公式及其推导使学生能够应对从基本振动到高级共振现象的各种问题。
| Quantity / 量 | Formula / 公式 | Notes / 注释 |
|---|---|---|
| Defining equation for SHM | a = −ω²x | True for any SHM system |
| Displacement (typical) | x = A sin(ωt + φ) | A: amplitude; φ: phase constant |
| Velocity | v = ±ω√A² − x² | vmax = ωA at x=0 |
| Acceleration | a = −ω²x | amax = ω²A at x=±A |
| Angular frequency ω | ω = 2πf = 2π/T | f: frequency; T: period |
| Spring-mass ω | ω = √k/m | Period T = 2π√m/k |
| Simple pendulum ω | ω = √g/L (small θ) | Period T = 2π√L/g |
| Kinetic energy | Ek = ½mω²(A² − x²) | Maximum at equilibrium |
| Potential energy (spring) | Ep = ½mω²x² = ½kx² | Maximum at extremes |
| Total energy | Etotal = ½mω²A² = ½kA² | Constant in undamped SHM |
| Damped SHM (HL, light) | x = A₀ e⁻ᵞᵗ cos(ω_d t + φ) | γ = b/(2m); ω_d = √ω₀²−γ² |
| Resonance amplitude (HL) | A = F₀/(m√((ω₀²−ω²
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