📚 Equations of Simple Harmonic Motion | 简谐运动方程
The equations of simple harmonic motion (SHM) form the mathematical backbone for understanding many oscillatory systems in physics—from mass–spring setups to pendulums and even alternating currents. Mastering these equations allows you to describe displacement, velocity, acceleration, and energy at any point in the cycle. This article covers the complete set of SHM equations required for the Cambridge International A-Level Physics syllabus, with clear derivations and practical examples.
简谐运动（SHM）方程构成了理解物理学中许多振荡系统（从弹簧振子到单摆乃至交流电）的数学基础。掌握这些方程能让你描述循环中任意点的位移、速度、加速度和能量。本文涵盖了剑桥国际 A-Level 物理大纲所需的全部 SHM 方程，并附有清晰的推导与实例。
1. What is Simple Harmonic Motion? | 什么是简谐运动？
Simple harmonic motion is a type of periodic motion where the acceleration of an object is directly proportional to its displacement from a fixed equilibrium position and is always directed towards that equilibrium. This condition is expressed mathematically as a ∝ -x, which leads to the fundamental equation of SHM:
a = -ω²x
简谐运动是一种周期性运动，物体的加速度与其偏离固定平衡位置的位移成正比，且方向始终指向平衡位置。这一条件在数学上表示为 a ∝ -x，从而得出简谐运动的基本方程：a = -ω²x。任何满足此关系的运动都将产生正弦式的时间变化规律。
ons to this equation are sinusoidal functions, meaning the displacement can be written using sine or cosine. Thus, SHM is often called sinusoidal oscillation.
由于加速度是位移对时间的二阶导数，我们得到微分方程 d²x/dt² = -ω²x。该方程的解为正弦函数，这意味着位移可用正弦或余弦表示。因此，简谐运动常被称为正弦振荡。
2. The Defining Equation: a = -ω²x | 定义方程：a = -ω²x
ure of SHM. The constant ω is called the angular frequency, and it determines how quickly the motion oscillates. The negative sign indicates that acceleration always points opposite to the displacement, acting as a restoring influence. Whenever you encounter a system where the net force (and thus acceleration) is proportional to the negative of displacement, you can model its motion using SHM equations.
方程 a = -ω²x 是简谐运动的标志。常数 ω 称为角频率，它决定了振荡的快慢。负号表示加速度始终指向位移的反方向，起到回复作用。每当你遇到一个系统，其合力（因而加速度）与负位移成正比，你就可以用简谐运动方程来模拟它的运动。
is the amplitude (maximum displacement) and φ is the phase constant. Both expressions are equivalent through a phase shift of π/2 rad.
由 a = -ω²x 出发，结合 a = dv/dt = d²x/dt²，可得 d²x/dt² + ω²x = 0。其最一般的解为 x = A sin(ωt + φ) 或 x = A cos(ωt + φ)，其中 A 为振幅（最大位移），φ 为初相常数。两种表达式通过 π/2 rad 的相位移动相互等价。
3. Displacement Equation: x = A sin(ωt + φ) | 位移方程：x = A sin(ωt + φ)
The displacement of a particle executing SHM varies sinusoidally with time. Depending on the initial conditions, one of the following forms is used:
x = A sin(ωt + φ)
x = A cos(ωt + φ')
简谐运动中质点的位移随时间按正弦规律变化。根据初始条件的不同，可采用以下形式之一：x = A sin(ωt + φ) 或 x = A cos(ωt + φ')。
). If at t = 0 it is at the maximum positive displacement (x = +A), the cosine form with φ' quantity (ωt + φ) is called the phase and determines the oscillatory state at any instant.
若在 t = 0 时质点位于平衡位置 (x = 0) 并向正方向运动，则正弦形式可取 φ = 0，即 x = A sin(ωt)。若在 t = 0 时质点在最大正向位移 (x = +A) 处，则余弦形式取 φ' = 0 较为方便，即 x = A cos(ωt)。量值 (ωt + φ) 称为相位，决定了任意时刻的振荡状态。
4. Velocity in SHM | 简谐运动中的速度
e of change of displacement. Differentiating x = A sin(ωt + φ) with respect to time gives:
v = dx/dt = ωA cos(ωt + φ)
速度是位移的变化率。对 x = A sin(ωt + φ) 求时间导数可得：v = ωA cos(ωt + φ)。
sin(ωt + φ), we obtain the extremely useful expression for speed in terms of displacement:
v = ± ω √A² − x²
利用三角恒等式 cos(ωt + φ) = ±√[1 − sin²(ωt + φ)] 并代入 x = A sin(ωt + φ)，我们得到用位移表示的速率表达式：v = ± ω √A² − x²。正负号代表运动方向。
n the particle passes through the equilibrium position (x = 0). Substituting x = 0 gives vmax ±A), the speed is zero because the particle momentarily stops before reversing direction.
最大速率出现在质点通过平衡位置 (x = 0) 的时刻。代入 x = 0 得 vmax = ωA。在极端位置 (x = ±A)，速率为零，因为质点在转向之前瞬间静止。
5. Acceleration in SHM | 简谐运动中的加速度
Acceleration is the derivative of velocity. Differentiating v = ωA cos(ωt + φ) yields:
