📚 Simple Harmonic Motion (SHM) – IB Physics C.1 | 简谐运动 (SHM) – IB物理C.1
Welcome to our comprehensive revision guide on IB Physics Topic C.1: Simple Harmonic Motion. This article covers the defining features of SHM, its kinematic and dynamic equations, energy considerations, graphical analysis, and applications to mass–spring systems and pendulums. Whether you are studying at Standard or Higher Level, mastering these concepts is essential for exam success and deeper understanding of oscillatory phenomena.
欢迎阅读 IB 物理 C.1 简谐运动的全面复习指南。本文涵盖 SHM 的定义特征、运动学和动力学方程、能量分析、图解表示，以及在质量–弹簧系统和单摆中的应用。无论你是标准水平还是高水平课程，掌握这些概念对考试成功和深入理解振动现象至关重要。
1. Defining Simple Harmonic Motion | 简谐运动的定义
Simple harmonic motion (SHM) is a special type of periodic oscillation in which the restoring force is directly proportional to the displacement from the equilibrium position and always acts towards that equilibrium point.
简谐运动（SHM）是一种特殊的周期振动，其回复力与偏离平衡位置的位移成正比，且总是指向平衡位置。
²x, where a is the instantaneous acceleration, x is the displacement, and ω is the angular frequency (in rad s⁻¹). The negative sign indicates that acceleration is always opposite in direction to the displacement.
SHM的定义方程为 a = −ω²x，其中 a 为瞬时加速度，x 为位移，ω 为角频率（单位 rad s⁻¹）。负号表示加速度方向始终与位移方向相反。
ny system that obeys this linear restoring force law is undergoing simple harmonic motion.
由于 ω = 2πf = 2π/T，也可写成 a = −(2πf)²x 或 a = −(4π²/T²)x。任何遵循这一线性回复力定律的系统都在做简谐运动。
2. The Restoring Force and SHM Condition | 回复力与简谐运动条件
directed towards equilibrium, making F = −kx a sufficient condition for SHM, with k = mω².
对于质量为 m 的物体做简谐运动，牛顿第二定律给出回复力 F = ma = −mω²x。力总与位移成正比且指向平衡位置，因此 F = −kx 是简谐运动的充分条件，其中 k = mω²。
ncy depends only on the physical parameters of the system, not on the amplitude of motion.
常量 k 称为劲度系数或弹性系数。由 ω² = k/m 可知，角频率仅取决于系统的物理参数，与运动振幅无关。
inciple allows us to identify SHM in unfamiliar systems: if the net force takes the form F ∝ −x, then the motion is simple harmonic and we can immediately write ω = √k/m.
这一原理使我们能在陌生系统中识别 SHM：若合力取 F ∝ −x 形式，则运动为简谐运动，并可立即写出 ω = √k/m。
3. Kinematics of SHM: Displacement Equation | SHM运动学：位移方程
s are x = x₀ sin(ωt) and x = x₀ cos(ωt), where x₀ is the amplitude (maximum displacement).
简谐运动中质点的位移可表示为时间的正弦函数。最常见的形式为 x = x₀ sin(ωt) 和 x = x₀ cos(ωt)，其中 x₀ 为振幅（最大位移）。
= x₀ cos(ωt). A general form is x = x₀ sin(ωt + φ), where φ is the initial phase constant.
选择正弦或余弦取决于初始条件。从平衡位置开始（t=0 时 x=0）用 x = x₀ sin(ωt)；从最大正向位移开始用 x = x₀ cos(ωt)。更一般形式为 x = x₀ sin(ωt + φ)，φ 为初相常数。
SHM can be visualised as the projection of uniform circular motion onto a diameter. The radius equals the amplitude x₀, the angular velocity of the reference circle equals ω, and the projection coordinate varies as x₀ sin(ωt) or x₀ cos(ωt).
SHM 可看作匀速圆周运动在直径上的投影。参考圆半径等于振幅 x₀，参考圆角速度等于 ω，投影坐标按 x₀ sin(ωt) 或 x₀ cos(ωt) 变化。
4. Velocity and Acceleration in SHM | SHM中的速度和加速度
x/dt = ωx₀ cos(ωt). The velocity can also be expressed in terms of displacement as v = ± ω √x₀² − x².
对位移求时间导数得到速度。对于 x = x₀ sin(ωt)，v = dx/dt = ωx₀ cos(ωt)。速度也可用位移表示为 v = ± ω √x₀² − x²。
The maximum speed occurs when the object passes through equilibrium (x = 0), and is vmax = ωx₀. At the utmost displacements (x = ±x₀), the speed is zero.
物体通过平衡位置（x = 0）时速度最大，vmax = ωx₀；在最大位移处（x = ±x₀）速度为零。
is confirms the SHM definition. The magnitude of acceleration is maximum at the extremes, amax = ω²x₀, and zero at equilibrium.
加速度为二阶导数：a = dv/dt = −ω² x₀ sin(ωt) = −ω²x，这确证了 SHM 的定义。加速度大小在最大位移处最大，amax = ω²x₀，在平衡位置为零。
In terms of phase, velocity leads displacement by π/2 rad (90°), and acceleration leads velocity by another π/2 rad, which means acceleration and displacement are in anti‑phase (π rad difference).
就相位而言，速度比位移超前 π/2 rad (90°)，加速度又比速度超前 π/2 rad，因此加速度与位移反相（相差 π rad）。
5. The Time Period and Frequency | 周期与频率
ency f is the number of oscillations per second: f = 1/T = ω/(2π), measured in hertz (Hz).
周期 T 是一次完整振动所需时间，与角频率关系为 T = 2π/ω。频率 f 是每秒振动次数：f = 1/T = ω/(2π)，单位赫兹 (Hz)。
For any SHM system, ω depends only on the intrinsic properties of the system (mass, spring constant, length, etc.). Consequently, T and f are independent of the amplitude, a property known as isochronism.
对任何 SHM 系统，ω 只取决于系统的固有属性（质量、劲度系数、摆长等）。因此 T 和 f 与振幅无关，这一特性称为等时性。
Experimentally, the period can be measured by timing many oscillations and then dividing by the number of cycles to reduce uncertainty. For a spring–mass system, a graph of T² against mass m yields a straight line whose slope can be used to find the spring constant k.
实验上，可通过
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