Periodicity of Physical Properties | 物理性质的周期性

📚 Periodicity of Physical Properties | 物理性质的周期性

In Cambridge International AS & A Level Physics, periodicity describes how physical quantities repeat at regular intervals in time or space. Oscillations, waves and alternating currents all show periodic behaviour, and the concepts of period, frequency, phase and wavelength provide a common language for analysing them.

在剑桥国际 AS 与 A Level 物理中，周期性描述物理量如何按规律的时间或空间间隔重复出现。振动、波动和交流电都表现出周期性行为，而周期、频率、相位和波长等概念为分析它们提供了共同的语言。

1. What is Periodicity in Physics? | 什么是物理学中的周期性

In physics, a periodic property is one that returns to the same value after a fixed interval. If the interval is measured in time, it is called the period T. If it is measured along a direction of travel, it is called the wavelength λ.

在物理学中，周期性性质是指经过固定间隔后恢复到相同数值的性质。若间隔按时间测量，称为周期 T；若沿着传播方向测量，则称为波长 λ。

Typical temporal periodicities include the swinging of a pendulum, the oscillation of a mass on a spring and the alternating current in a circuit. Spatial periodicity appears in water waves, sound waves and electromagnetic waves.

典型的时间周期性包括单摆摆动、弹簧振子振动和电路中的交流电。空间周期性出现在水波、声波和电磁波中。

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2. Describing Periodic Motion: Period and Frequency | 描述周期性运动：周期与频率

The period T and frequency f are inversely related. Measuring these quantities allows physicists to compare different oscillating systems and to predict future motion.

周期 T 与频率 f 互为倒数。测量这些物理量使物理学家能够比较不同的振动系统并预测未来的运动。

T = 1/f

ω = 2πf = 2π/T

The unit of period is the second (s), frequency is the hertz (Hz), and angular frequency is the radian per second (rad s⁻¹).

周期的单位是秒（s），频率的单位是赫兹（Hz），角频率的单位是弧度每秒（rad s⁻¹）。

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3. Simple Harmonic Motion as the Basic Periodic Model | 简谐运动作为基本周期模型

Simple harmonic motion (SHM) is the most fundamental periodic motion in the Cambridge A Level course. It occurs when the restoring force is proportional to displacement from equilibrium and acts towards equilibrium.

简谐运动（SHM）是剑桥 A Level 课程中最基本的周期运动。当回复力与偏离平衡位置的位移成正比并指向平衡位置时，就会产生简谐运动。

a = −ω²x

The solutions x = A sin(ωt) or x = A cos(ωt) show that displacement repeats sinusoidally. The amplitude A is the maximum displacement from equilibrium.

解 x = A sin(ωt) 或 x = A cos(ωt) 表明位移呈正弦式重复。振幅 A 是偏离平衡位置的最大位移。

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4. Displacement, Phase and Graphs | 位移、相位与图像

The phase of an oscillator indicates its position within the cycle. Two oscillations of the same frequency can have a phase difference φ. When φ = 0 they are in phase; when φ = π radians they are in antiphase.

振子的相位表示它在循环中所处的位置。频率相同的两个振动可以具有相位差 φ。当 φ = 0 时它们同相；当 φ = π 弧度时它们反相。

x = A sin(ωt + φ)

An x-t graph for SHM is a sine or cosine curve. The horizontal spacing between two identical points on the graph is one period T.

简谐运动的 x-t 图像是正弦或余弦曲线。图像上两个相同点之间的水平间隔为一个周期 T。

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5. Velocity and Acceleration in SHM | 简谐运动中的速度与加速度

The velocity and acceleration of an oscillator also vary periodically. Velocity leads displacement by π/2 radians, and acceleration is always opposite to displacement.

振子的速度和加速度也呈周期性变化。速度比位移超前 π/2 弧度，加速度始终与位移方向相反。

v = ±ω√(A² − x²)

a = −ω²x

At the equilibrium position x = 0, speed is maximum v_max = ωA. At the extremes x = A, acceleration magnitude is maximum a_max = ω²A.

在平衡位置 x = 0 处，速率最大 v_max = ωA。在端点 x = A 处，加速度大小最大 a_max = ω²A。

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6. Energy Periodicity in Oscillations | 振动中的能量周期性

In an ideal undamped oscillator, total mechanical energy remains constant. However, kinetic energy and potential energy each oscillate with time, and their sum is constant.

在理想的无阻尼振子中，总机械能保持不变。但动能和势能各自随时间振荡，它们的总和保持不变。

E_total = ½ mω²A²

E_k = ½ mω²(A² − x²)

E_p = ½ mω²x²

Energy periodicity means the system is never ‘using up’ energy; it simply exchanges energy between two forms at twice the oscillation frequency.

能量的周期性意味着系统从未耗尽能量；它只是以两倍于振动频率的速率在两种形式之间交换能量。

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7. Damping and Periodic Decay | 阻尼与周期性衰减

Real systems experience resistive forces such as air resistance or friction. These remove energy and cause the amplitude to decay with time. Light damping preserves an approximately periodic motion.

实际系统会受到空气阻力或摩擦力等阻力。这些力消耗能量并使振幅随时间衰减。弱阻尼下运动仍近似保持周期性。

The frequency of a damped oscillator is slightly lower than its natural undamped frequency. Heavy damping may prevent oscillation altogether.

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