📚 GCSE Edexcel Physics: Simple Harmonic Motion – Key Concepts | GCSE Edexcel 物理：简谐运动 考点精讲
Simple harmonic motion (SHM) is a special type of periodic motion found in many natural and engineered systems. In GCSE Edexcel Physics, understanding SHM helps explain how springs, pendulums, and even sound waves work. This article covers the essential concepts, equations, graphs, and real-world examples you need for your exam.
简谐运动（SHM）是一种特殊的周期性运动，广泛存在于自然和工程系统中。在 GCSE Edexcel 物理中，理解 SHM 有助于解释弹簧、单摆乃至声波的工作原理。本文涵盖考试必备的基本概念、方程、图像和实际例子。
1. What is Simple Harmonic Motion? | 什么是简谐运动？
An object undergoes simple harmonic motion when its acceleration is directly proportional to its displacement from a fixed equilibrium position and is always directed towards that position. The motion is periodic and can be described by a sine or cosine function.
当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时，该物体就做简谐运动。这种运动是周期性的，可以用正弦或余弦函数描述。
2. The Defining Equation | 定义方程
The mathematical condition for SHM is expressed using acceleration a, angular frequency ω, and displacement x:
简谐运动的数学条件用加速度 a、角频率 ω 和位移 x 表示为：
a = – ω² x
The negative sign indicates that acceleration always acts in the opposite direction to displacement, pulling the object back towards equilibrium. Since ω = 2πf, we can also write a = – (2πf)² x.
负号表示加速度始终与位移方向相反，将物体拉回平衡位置。由于 ω = 2πf，方程也可写成 a = – (2πf)² x。
3. Key Terms: Amplitude, Period, Frequency | 关键术语：振幅、周期、频率
Amplitude (A) is the maximum displacement from the equilibrium position, measured in metres.
振幅（A）是物体离开平衡位置的最大位移，单位为米（m）。
Period (T) is the time taken for one complete oscillation, measured in seconds.
周期（T）是完成一次全振动所需的时间，单位为秒（s）。
Frequency (f) is the number of complete oscillations per second, measured in hertz (Hz).
频率（f）是每秒完成全振动的次数，单位为赫兹（Hz）。
Angular frequency ω relates to frequency by ω = 2πf. It is useful for calculations involving phase and circular motion analogies.
角频率 ω 与频率的关系为 ω = 2πf，在涉及相位和圆周运动类比时非常有用。
The relationships among these quantities are summarised in the table below.
这些量之间的关系总结于下表中。
| T = 1/f | f = 1/T | ω = 2πf = 2π/T |
4. SHM as a Model for Oscillations | 简谐运动作为振动模型
SHM is an idealised model that describes systems where the restoring force is directly proportional to displacement. For a mass–spring system, Hooke’s Law gives F = –k x, where k is the spring constant. Applying Newton’s second law yields a = – (k/m) x, which matches the SHM condition with ω² = k/m.
简谐运动是一个理想化模型，描述恢复力与位移成正比的系统。对于弹簧振子，胡克定律给出 F = –k x，其中 k 为弹簧劲度系数。应用牛顿第二定律可得 a = – (k/m) x，符合 SHM 条件，且 ω² = k/m。
Similarly, a simple pendulum approximates SHM for small angles (typically less than about 10°), where the restoring force component is proportional to displacement along the arc.
类似地，单摆在小角度（通常小于约 10°）时近似为简谐运动，此时恢复力的分量与弧位移成正比。
5. Displacement–Time Graphs | 位移–时间图
The displacement–time graph for an object released from maximum positive displacement (+A) is a cosine curve. It starts at +A, passes through equilibrium (x = 0) at T/4, reaches –A at T/2, returns through equilibrium at 3T/4, and completes one cycle at T.
物体从最大正向位移（+A）处释放时，其位移–时间图为余弦曲线。它从 +A 开始，在 T/4 时经过平衡位置（x = 0），在 T/2 时到达 –A，在 3T/4 时再次经过平衡位置，并在 T 时完成一个循环。
If the object starts from equilibrium and is given an initial push, the graph is a sine curve, crossing the time axis at t = 0 and reaching +A at T/4.
若物体从平衡位置开始并受到初始推动，图形则为正弦曲线，在 t = 0 时通过时间轴，在 T/4 时到达 +A。
Key features to identify on such graphs are amplitude A and period T. The frequency can be calculated from f = 1/T.
识别这类图像的关键特征是振幅 A 和周期 T。频率可由 f = 1/T 计算得出。
6. Velocity in SHM | 简谐运动的速度
The speed of the oscillating object is greatest when it passes through the equilibrium position (x = 0). The speed is zero at the extreme positions (x = ±A) because the object momentarily stops before changing direction.
振动物体通过平衡位置（x = 0）时速率最大。在极端位置（x = ±A）时速率为零，因为物体在改变方向前会瞬间静止。
The magnitude of the velocity at any displacement x can be found using:
任意位移 x 处的速度大小可用下式计算：
v = ω √(A² – x²)
In terms of kinetic energy, the maximum speed vmax = ωA occurs at x = 0. The direction of velocity changes each half-cycle.
就动能而言，最大速率 vmax = ωA 出现在 x = 0 处。速度方向每半个周期改变一次。
7. Acceleration and Displacement Relationship | 加速度与位移的关系
The defining equation a = – ω² x shows that acceleration is always opposite in sign to displacement. A graph of acceleration against displacement is a straight line through the origin with a negative gradient equal to –ω². This linear relationship is a classic test for SHM.
定义方程 a = – ω² x 表明加速度的符号总是与位移相反。加速度–位移图是一条过原点、斜率为 –ω² 的直线。这种线性关系是检验 SHM 的经典方法。
When x = 0, a = 0; when x = ±A, the acceleration reaches its maximum magnitude a_max = ω² A.
当 x = 0 时，a = 0；当 x = ±A 时，加速度达到最大量值 a_max = ω² A。
8. Energy Changes in SHM | 简谐运动的能量变化
In an undamped SHM system, total mechanical energy remains constant, continuously changing between kinetic and potential forms. At maximum displacement (x = ±A), the kinetic energy is zero and potential energy is maximum. At equilibrium (x = 0), kinetic energy is maximum and potential energy is minimum.
在无阻尼的 SHM 系统中，总机械能保持不变，在动能和势能之间不断转换。在最大位移（x = ±A）处，动能为零，势能最大；在平衡位置（x = 0）处，动能最大，势能最小。
For a mass–spring system, the elastic potential energy is (1/2)kx² and kinetic energy is (1/2)mv². The total energy is:
对于弹簧振子，弹性势能为 (1/2)kx²，动能为 (1/2)mv²。总能量为：
E_total = ½ k A² = ½ m ω² A²
This energy conservation explains why amplitude remains constant in the absence of external forces or damping.
这种能量守恒解释了在没有外力或阻尼作用时，振幅为何保持恒定。
9. Example: Mass–Spring System | 实例：弹簧振子
A mass m attached to a horizontal spring of constant k executes SHM when displaced from equilibrium. The angular frequency is ω = √(k/m), and the period is given by:
质量为 m 的物体与劲度系数为 k 的水平弹簧相连，当偏离平衡位置后做简谐运动。角频率为 ω = √(k/m)，周期为：
T = 2π √(m/k)
The period depends only on mass and spring constant, not on amplitude. Experimentally, measuring T for different masses allows determination of the spring constant.
周期仅取决于质量和劲度系数，与振幅无关。在实验中，通过测量不同质量对应的周期可以确定劲度系数。
If the spring hangs vertically, the equilibrium position is shifted by gravity but the SHM characteristics remain valid with the same period formula.
若弹簧竖直悬挂，重力会使平衡位置下移，但 SHM 特性依然成立，周期公式相同。
10. Example: Simple Pendulum | 实例：单摆
A simple pendulum consists of a point mass suspended by a light, inextensible string. For small angular displacements (θ < about 10°), the motion approximates SHM. The period is independent of mass and amplitude and is given by:
单摆由一个质点通过轻质、不可伸长的细绳悬挂而成。当角位移很小（θ < 约 10°）时，运动近似为简谐运动。周期与质量和振幅无关，表达式为：
T = 2π √(l/g)
Here l is the length of the pendulum and g is the acceleration due to gravity. By measuring T and l, students can calculate an experimental value for g.
其中 l 为摆长，g 为重力加速度。通过测量 T 和 l，学生可以计算 g 的实验值。
Note that the period formula remains accurate only for small amplitudes; at larger angles, the motion is no longer simple harmonic.
注意，周期公式仅在小振幅下精确成立；角度较大时，运动将不再是简谐运动。
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