📚 Simple Harmonic Motion: CIE GCSE Physics Exam Guide | GCSE CIE 物理：简谐运动 考点精讲
Simple harmonic motion (SHM) is a fundamental type of oscillation that appears across the CIE GCSE Physics specification. Grasping its defining rule – an acceleration always proportional and opposite to displacement – along with real systems like pendulums and spring-mass oscillators, is vital for top marks. This bilingual guide walks you through every examined concept, with clear explanations, diagrams described in words, and paired English–Chinese notes.
简谐运动是 CIE GCSE 物理中一类基础的振动，其核心是加速度始终与位移成正比且方向相反。掌握单摆、弹簧振子等实例中的动力学规律、图像特征和能量转化，是考试得分的关键。本文以中英对照形式梳理全部考点，配合术语解析，帮助你透彻理解每一个要点。
1. What Is Simple Harmonic Motion? | 什么是简谐运动
An object is said to perform simple harmonic motion when its acceleration is directly proportional to its displacement from a fixed equilibrium position, and the acceleration is always directed towards that equilibrium point.
当物体的加速度与其相对于某一固定平衡位置的位移成正比，且加速度的方向始终指向该平衡点时，该物体就在做简谐运动。
In mathematical terms, this condition is expressed as a ∝ −x. The negative sign indicates that acceleration and displacement are in opposite directions. This restoring behaviour is what makes SHM a regular, repeating oscillation.
数学上，这一条件可表示为 a ∝ −x。负号表明加速度与位移方向相反，这种始终指向平衡点的恢复特性使得简谐运动成为规则且可重复的振动。
2. The Key Equation: a = −ω²x | 简谐运动基本方程
The proportionality in SHM is written with a constant term as:
a = −ω² x
Here, ω (omega) is the angular frequency of the oscillation. It links the linear motion to circular motion concepts. The quantity ω is related to the period T and frequency f by:
ω = 2πf and T = 1 / f
简谐运动的比例关系以常数形式给出：a = −ω² x。其中 ω 为角频率，它将直线振动与圆周运动联系起来。角频率与周期 T、频率 f 满足 ω = 2πf，且 T = 1/f。
The equation tells us that the magnitude of acceleration is greatest at maximum displacement (the amplitude) and zero as the object passes through equilibrium. The minus sign ensures the acceleration is always a restoring effect.
该方程表明，加速度的大小在最大位移处（振幅处）最大，在经过平衡点时为零。负号确保加速度始终为恢复力作用的结果。
3. Key Quantities: Amplitude, Period, Frequency | 关键物理量：振幅、周期、频率
- Amplitude (A): the maximum displacement from the equilibrium position (measured in metres).
- Period (T): the time taken for one complete oscillation (seconds).
- Frequency (f): the number of oscillations per unit time (hertz, Hz).
- Angular frequency (ω): the rate of change of phase, ω = 2πf.
- 振幅 (A)：物体离平衡位置的最大位移（单位：米）。
- 周期 (T)：完成一次全振动所需的时间（单位：秒）。
- 频率 (f)：单位时间内振动的次数（单位：赫兹 Hz）。
- 角频率 (ω)：相位的变化率，ω = 2πf。
Neither the period nor the frequency of an ideal SHM system depends on the amplitude – this property is called isochronism. It holds true for small oscillations of a pendulum and for a mass–spring system within the elastic limit.
理想简谐运动的周期和频率均与振幅无关——这一性质称为等时性。单摆在小角度摆动以及弹簧振子在弹性限度内均表现出等时性。
4. Displacement–Time Graph | 位移–时间图像
The displacement–time (x–t) graph for SHM is a sinusoidal wave. If time measurement begins when the object is at the equilibrium position moving in the positive direction, the curve is a sine wave starting from zero. If timing starts at maximum positive displacement, it is a cosine wave.
简谐运动的位移–时间 (x–t) 图像是一条正弦曲线。若计时从物体在平衡位置且向正方向运动开始，图像为从零开始的正弦波；若从正向最大位移处开始计时，则为余弦波。
The graph oscillates smoothly between +A and −A with a well-defined period T. The steepest slope occurs when the object passes through equilibrium, indicating maximum speed. The slope is zero at the extreme displacements where the object momentarily stops.
图像在 +A 与 −A 之间圆滑振荡，周期固定为 T。物体经过平衡点时曲线斜率最大，表明速度最大；在最大位移处斜率为零，物体瞬时静止。
5. Velocity and Acceleration Graphs | 速度与加速度图像
The velocity–time (v–t) graph is also sinusoidal but is phase‑shifted relative to displacement. Velocity is zero at the extremes and reaches its maximum magnitude at the equilibrium position. On a v–t graph, the peaks occur one‑quarter of a period before the displacement peaks when describing motion that starts at the equilibrium.
速度–时间 (v–t) 图像同样是正弦曲线，但与位移之间存在相位差。速度在端点处为零，在平衡位置达到最大值。在 v–t 图上，速度峰值比位移峰值提前四分之一周期出现（若从平衡位置开始计时）。
The acceleration–time (a–t) graph is exactly opposite in sign to the displacement graph. When displacement is maximum positive, acceleration is maximum in the negative direction. As the object moves through equilibrium, acceleration is zero. This is because a = −ω²x, so acceleration is a scaled, inverted version of the displacement curve.
加速度–时间 (a–t) 图像与位移图像符号完全相反。位移正向最大时，加速度负向最大；经过平衡点时加速度为零。这是因为 a = −ω²x，加速度曲线是位移曲线按比例缩放且反相得到的。
6. Restoring Force and Hooke’s Law | 恢复力与胡克定律
In a mass–spring system, the restoring force that drives SHM is provided by the spring. According to Hooke’s law, the force F exerted by a spring is proportional to its extension or compression x:
F = −k x
where k is the spring constant (stiffness). The negative sign shows the force acts to return the mass to equilibrium.
在弹簧振子中，驱动简谐运动的恢复力来自弹簧。根据胡克定律，弹簧施加的力 F 与其伸长量或压缩量 x 成正比：F = −k x，其中 k 为劲度系数，负号表示力的方向始终指向平衡位置。
Using Newton’s second law (F = ma) we find ma = −kx, leading to a = −(k/m)x. Comparing this with a = −ω²x gives ω² = k/m. This directly explains how the mass and spring stiffness determine the oscillation period.
结合牛顿第二定律 F = ma 可得 ma = −kx，即 a = −(k/m)x。与 a = −ω²x 对比得到 ω² = k/m，这直接说明了质量和劲度系数如何决定振动周期。
7. The Mass–Spring System | 弹簧振子
For a mass m attached to a spring of spring constant k, the period of oscillation is given by:
T = 2π √(m / k)
This formula tells us that:
- Increasing the mass increases the period (slower oscillation).
- A stiffer spring (larger k) decreases the period (faster oscillation).
- The period does not depend on the amplitude of motion, as long as the spring obeys Hooke’s law.
对于质量为 m、劲度系数为 k 的弹簧振子，其振动周期为：T = 2π √(m/k)。此式表明：增大质量会使周期变长（振动变慢）；弹簧越硬（k 越大）周期越短（振动变快）；只要弹簧遵守胡克定律，周期与振幅无关。
In an experiment, one can plot T² against m to obtain a straight line through the origin, verifying the relationship and allowing calculation of k from the gradient.
实验中可绘制 T²–m 图像，得到一条过原点的直线，以此验证关系式并利用斜率求出劲度系数 k。
8. The Simple Pendulum | 单摆
A simple pendulum consists of a small mass (bob) suspended by a light, inextensible string. For small angular displacements (typically less than about 10°), the motion approximates SHM. The period is:
T = 2π √(l / g)
where l is the length of the pendulum and g is the acceleration due to gravity.
单摆由一根轻质不可伸长的细线悬挂一个小球构成。当摆角很小（通常小于约 10°）时，其运动近似为简谐运动。周期公式为：T = 2π √(l/g)，其中 l 为摆长，g 为重力加速度。
Crucially, the period of a simple pendulum is independent of the mass of the bob and the amplitude (provided the angle is small). This property is used in pendulum clocks and in experiments to determine g by measuring T and l.
关键一点：单摆的周期与摆球的质量以及振幅（小角度下）无关。这一特性被用于摆钟以及通过测量 T 和 l 来测定重力加速度 g 的经典实验中。
9. Energy in Simple Harmonic Motion | 简谐运动中的能量
Throughout SHM, energy is continuously exchanged between kinetic and potential forms, but the total mechanical energy stays constant if there is no damping. At the extremes of the motion, the displacement is maximum (x = A), speed is zero, so all the energy is stored as potential energy. At the equilibrium position, the potential energy is zero (for a spring) or minimum (for a pendulum), and the kinetic energy reaches its maximum.
在简谐运动过程中，动能与势能不断相互转化，但在无阻尼条件下总机械能守恒。在振动端点，位移最大 (x = A)，速度为零，所有能量以势能形式储存。在平衡位置，弹簧的弹性势能为零（单摆的势能为最小），动能达到最大值。
For a mass–spring system, the total energy can be expressed as:
Etotal = ½ k A²
At any intermediate displacement x, the kinetic energy is Ek = ½ k (A² − x²) and the potential energy is Ep = ½ k x². Energy–displacement graphs therefore show a parabolic distribution: potential energy is a parabola opening upwards, kinetic energy is an inverted parabola, and the sum is a flat line.
对于弹簧振子，总能量可表示为 Etotal = ½ k A²。在任意位移 x 处，动能 Ek = ½ k (A² − x²)，势能 Ep = ½ k x²。因此能量–位移图像呈抛物线分布：势能为开口向上的抛物线，动能为倒置抛物线，两者之和为一条水平直线，反映能量守恒。
10. Free and Damped Oscillations | 自由振动与阻尼振动
A free oscillation occurs when no external force acts on the system other than the restoring force, and no energy is lost. The amplitude remains constant forever in an idealised model.
除了恢复力外无其他外力作用且无能量损耗时，系统进行自由振动，理想情况下振幅永不衰减。
In real systems, damping forces such as air resistance or internal friction remove energy, causing the amplitude to decrease gradually over time. This is called a damped oscillation. Light damping reduces amplitude slowly, while heavy damping may stop the motion without oscillation.
实际系统中总存在空气阻力或内摩擦等阻尼力，它们消耗能量，使振幅随时间逐渐减小，这就是阻尼振动。轻微阻尼下振幅缓慢减小；严重阻尼时物体可能不经振动就回到平衡位置。
Importantly, the period of a lightly damped oscillator remains nearly the same as that of the free oscillation, but the amplitude decays exponentially. Damping is deliberately introduced in many practical devices such as car shock absorbers to avoid excessive bouncing.
值得注意的是，轻微阻尼振荡器的周期与自由振动时几乎相同，但振幅呈指数衰减。许多实际装置（如汽车减震器）特意引入阻尼，以避免持续弹跳。
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