📚 Edexcel A Level Physics Topic 13: Oscillations and Simple Harmonic Motion | 爱德思 A Level 物理 第13讲：振动与简谐运动
Oscillations are a central part of Edexcel A Level Physics Topic 13. From a mass bouncing on a spring to a pendulum swinging in a clock, many systems move back and forth about an equilibrium position. Simple harmonic motion (SHM) is the idealised model that describes the smoothest form of this repeating motion, and it provides the mathematical tools needed for understanding energy, waves, and resonance.
振动是爱德思 A Level 物理第 13 讲的核心内容。从弹簧上弹跳的质量块到钟摆的摆动，许多系统都会围绕平衡位置往复运动。简谐运动（SHM）是描述这类重复运动最理想化的模型，它提供了理解能量、波动和共振所需的数学工具。
1. Describing Oscillations: Amplitude, Period, Frequency and Angular Frequency | 描述振动：振幅、周期、频率和角频率
Any oscillation can be described by a few key quantities. The amplitude A is the maximum displacement from the equilibrium position, measured in metres. The period T is the time taken for one complete cycle, measured in seconds. The frequency f is the number of complete cycles per second, measured in hertz (Hz), and is related to the period by f = 1/T.
任何振动都可以用几个关键物理量来描述。振幅 A 是距离平衡位置的最大位移，单位是米。周期 T 是完成一次完整循环所需的时间，单位是秒。频率 f 是每秒钟完成的完整循环次数，单位是赫兹（Hz），它与周期的关系为 f = 1/T。
In SHM it is often more useful to work with the angular frequency ω, measured in radians per second. It is related to the ordinary frequency by ω = 2πf or equivalently ω = 2π/T. The term angular frequency makes it easy to connect circular motion with oscillating motion, because one complete cycle corresponds to an angle of 2π radians.
在简谐运动中，使用角频率 ω 通常更方便，单位是弧度每秒。它与普通频率的关系为 ω = 2πf，或者等价地 ω = 2π/T。角频率这一术语使得圆周运动与振动之间很容易建立联系，因为一个完整周期对应 2π 弧度的角度。
2. Simple Harmonic Motion: Definition and Conditions | 简谐运动：定义与条件
Simple harmonic motion occurs when the resultant force on an object is directly proportional to its displacement from a fixed point, and always acts towards that fixed point. That fixed point is the equilibrium position. Mathematically the condition can be written as F = -kx, where k is the force constant and x is the displacement. The negative sign shows that the force is a restoring force, always pointing back towards equilibrium.
当作用在物体上的合力与其偏离某一固定点的位移成正比，并且始终指向该固定点时，物体就做简谐运动。这个固定点就是平衡位置。数学上该条件可以写成 F = -kx，其中 k 是力常数，x 是位移。负号表示该力是一个回复力，始终指向平衡位置。
Equivalently, in terms of acceleration, SHM satisfies the defining equation a = -ω²x. This arises because F = ma and the ratio k/m equals ω². If a graph of acceleration against displacement gives a straight line through the origin with a negative gradient, then the motion is definitely SHM.
等价地，用加速度表示时，简谐运动满足定义方程 a = -ω²x。这是因为 F = ma，且比值 k/m 等于 ω²。如果加速度对位移的图像是一条过原点且斜率为负的直线，那么该运动一定是简谐运动。
3. Displacement-Time Graphs and Phase Difference | 位移-时间图像与相位差
For an object released from maximum displacement at t = 0, the displacement-time relationship is x = A cos(ωt). If the object starts at equilibrium and moves in the positive direction, the relationship is x = A sin(ωt). Both graphs are sinusoidal, repeating every period T = 2π/ω.
对于 t = 0 时从最大位移处释放的物体，位移-时间关系为 x = A cos(ωt)。如果物体从平衡位置开始并向正方向运动，则关系为 x = A sin(ωt)。两者的图像都是正弦曲线，每隔周期 T = 2π/ω 重复一次。
x = A cos(ωt) or x = A sin(ωt)
Phase difference measures how far one oscillation leads or lags another, expressed in radians. In SHM, displacement and velocity have a phase difference of π/2 rad, velocity and acceleration also differ by π/2 rad, and displacement and acceleration differ by π rad, meaning they are in antiphase.
相位差用来衡量一个振动比另一个振动超前或落后多少，用弧度表示。在简谐运动中，位移与速度的相位差为 π/2 弧度，速度与加速度也相差 π/2 弧度，而位移与加速度相差 π 弧度，即它们处于反相。
4. Velocity and Acceleration in SHM | 简谐运动中的速度与加速度
Velocity is the rate of change of displacement. If x = A cos(ωt), then differentiating gives v = -Aω sin(ωt). Acceleration is the rate of change of velocity, so a = -Aω² cos(ωt). Since x = A cos(ωt), this confirms that a = -ω²x.
速度是位移的变化率。如果 x = A cos(ωt)，对其求导可得 v = -Aω sin(ωt)。加速度是速度的变化率，因此 a = -Aω² cos(ωt)。由于 x = A cos(ωt)，这就证实了 a = -ω²x。
v = ±ω√(A² – x²)
The speed as a function of position is given by v = ±ω√(A² – x²). The maximum speed occurs at equilibrium, x = 0, and is v_max = ωA. The maximum acceleration occurs at the extreme positions, x = ±A, and is a_max = ω²A.
速度随位置变化的关系为 v = ±ω√(A² – x²)。最大速度出现在平衡位置，即 x = 0 处，v_max = ωA。最大加速度出现在极端位置，即 x = ±A 处，a_max = ω²A。
5. The Defining Equation a = -ω²x | 定义方程 a = -ω²x
For a mass-spring system, the restoring force is F = -kx. Using Newton’s second law F = ma gives ma = -kx, so a = -(k/m)x. Comparing this with the SHM defining equation a = -ω²x shows that ω² = k/m.
对于弹簧-质量系统，回复力为 F = -kx。使用牛顿第二定律 F = ma 可得 ma = -kx，因此 a = -(k/m)x。将其与简谐运动定义方程 a = -ω²x 进行比较，可得 ω² = k/m。
The significance of a = -ω²x is that acceleration is always proportional to displacement and in the opposite direction. This means the acceleration is not constant, so the SUVAT equations cannot be used for SHM. The defining equation is used to test whether a given motion is simple harmonic.
a = -ω²x 的意义在于，加速度始终与位移成正比，且方向相反。这意味着加速度不是恒定的，因此不能对简谐运动使用 SUVAT 方程。该定义方程可用于检验某一给定运动是否为简谐运动。
6. Mass-Spring Systems and the Simple Pendulum | 弹簧-质量系统与单摆
For a mass m attached to a spring of force constant k, the period is independent of amplitude and is given by T = 2π√(m/k). A larger mass increases the period, while a stiffer spring decreases the period.
对于与力常数为 k 的弹簧相连的质量 m，周期与振幅无关，并由 T = 2π√(m/k) 给出。质量越大，周期越长；弹簧越硬，周期越短。
T = 2π√(m/k)
For a simple pendulum of length l in a gravitational field g, the period for small angles is T = 2π√(l/g). This assumes the amplitude is small, typically less than about 10°, so that the motion approximates SHM. The period does not depend on the mass of the pendulum bob.
对于重力场为 g、摆长为 l 的单摆，小角度条件下的周期为 T = 2π√(l/g)。该公式假设振幅很小，通常小于约 10°，从而运动近似为简谐运动。周期与摆锤的质量无关。
T = 2π√(l/g)
| System | Period formula | Independent of |
|---|---|---|
| Mass-spring | T = 2π√(m/k) | Amplitude |
| Simple pendulum | T = 2π√(l/g) | Mass, amplitude (small) |
7. Energy Changes in SHM | 简谐运动中的能量变化
In SHM, energy is continuously exchanged between kinetic energy and potential energy, but the total energy remains constant if there is no damping. The kinetic energy is E_k = ½mv² = ½mω²(A² – x²). The potential energy for a mass-spring system is E_p = ½kx² = ½mω²x².
在简谐运动中，能量不断在动能和势能之间交换，但如果没有阻尼，总能量保持不变。动能为 E_k = ½mv² = ½mω²(A² – x²)。对于弹簧-质量系统，势能为 E_p = ½kx² = ½mω²x²。
E_total = ½mω²A² = ½kA²
At equilibrium, all energy is kinetic; at maximum displacement, all energy is potential. The total energy is proportional to A², so doubling the amplitude quadruples the stored energy. Graphs of E_k and E_p against displacement are parabolas, while their sum is a horizontal line.
在平衡位置，所有能量都是动能；在最大位移处，所有能量都是势能。总能量与 A² 成正比，因此振幅加倍会使储存的能量变为原来的四倍。动能和势能随位移变化的图像为抛物线，而它们的总和是一条水平线。
8. Free and Forced Oscillations | 自由振动与受迫振动
A free oscillation occurs when a system is displaced and released, oscillating at its natural frequency f₀ without any external driving force. In the absence of damping, the amplitude remains constant. In practice small damping causes the oscillation to decay slowly.
自由振动是指系统被移动后释放，在没有任何外部驱动力的情况下以其固有频率 f₀ 振动。在没有阻尼的情况下，振幅保持不变。实际中微小的阻尼会使振动逐渐衰减。
A forced oscillation occurs when an external periodic force is applied continuously. The system then vibrates at the driving frequency, not its natural frequency. The amplitude of the forced oscillation depends on how close the driving frequency is to f₀.
受迫振动是指持续施加外部周期性力时发生的振动。系统此时以驱动力频率振动，而不是其固有频率。受迫振动的振幅取决于驱动力频率与 f₀ 的接近程度。
9. Damping: Light, Heavy, and Critical | 阻尼：轻阻尼、重阻尼与临界阻尼
Damping is the removal of energy from an oscillating system by a resistive force such as friction or air resistance. Light damping reduces the amplitude gradually, and the period remains approximately equal to the natural period. Heavy damping is so large that the system returns to equilibrium without oscillating.
阻尼是指通过摩擦力或空气阻力等阻力将能量从振动系统中带走。轻阻尼使振幅逐渐减小，而周期大致保持等于固有周期。重阻尼非常大，以至于系统不再振动，而是缓慢地回到平衡位置。
Critical damping is the minimum amount of damping needed to return the system to equilibrium in the shortest possible time without oscillating. It is important in car shock absorbers, door closers, and galvanometers, where quick and smooth return is desired.
临界阻尼是使系统在尽可能短的时间内回到平衡位置且不发生振动所需的最小阻尼。它在汽车减震器、门闭合器和电流计等设备中非常重要，这些场合需要快速而平稳地复位。
10. Resonance and Its Applications | 共振及其应用
Resonance occurs when the driving frequency equals the natural frequency of a system. At resonance, the system absorbs energy most efficiently, and the amplitude of oscillation becomes very large. The resonance peak is sharp when damping is light and broader when damping is heavy.
当驱动力频率等于系统的固有频率时，就会发生共振。在共振时，系统最有效地吸收能量，振动振幅变得非常大。阻尼越轻，共振峰越尖锐；阻尼越重，共振峰越宽。
Resonance has many applications, including tuning a radio to a particular station, warming food in a microwave, and the production of musical notes in instruments. It can also be dangerous: buildings and bridges can be damaged if their natural frequencies match wind or seismic vibrations.
共振有许多应用，包括将收音机调谐到特定电台、用微波炉加热食物以及乐器产生乐音。共振也可能带来危险：如果建筑物或桥梁的固有频率与风或地震振动相匹配，就可能导致损坏。
11. Experimental Methods and Data Analysis | 实验方法与数据分析
A common Edexcel practical is to determine the acceleration of free fall g using a simple pendulum. The period T is measured for a range of lengths l, and then T is plotted against √l or T² is plotted against l. The gradient of T² against l is 4π²/g, so g can be calculated.
爱德思常见的实验是用单摆测定自由落体加速度 g。测量不同摆长 l 下的周期 T，然后绘制 T 对 √l 或 T² 对 l 的图像。T² 对 l 图像的斜率为 4π²/g，因此可以计算 g。
T² = (4π²/g) × l
For a mass-spring system, the spring constant can be found by measuring T for different masses. Plotting T² against m gives a straight line with gradient 4π²/k. Uncertainties should be considered for length, time, and mass, and repeated readings should be taken to reduce random errors.
对于弹簧-质量系统，可以通过测量不同质量下的周期 T 来求得弹簧劲度系数。绘制 T² 对 m 的图像会得到斜率为 4π²/k 的直线。应考虑长度、时间和质量的不确定度，并进行重复测量以减少随机误差。
12. Exam Tips: Common Pitfalls in Edexcel Topic 13 | 考试技巧：爱德思第 13 讲常见易错点
Students often confuse frequency f with angular frequency ω. Remember that ω = 2πf, so a frequency of 2 Hz corresponds to ω ≈ 12.6 rad s⁻¹, not 2 rad s⁻¹. Always use radians for phase differences and angular quantities, not degrees.
学生经常混淆频率 f 与角频率 ω。请记住 ω = 2πf，因此 2 Hz 的频率对应的角频率约为 12.6 rad s⁻¹，而不是 2 rad s⁻¹。相位差和角度量始终使用弧度，而不是度。
Another common mistake is forgetting the negative sign
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