Simple Harmonic Motion | 简谐运动

📚 Simple Harmonic Motion | 简谐运动

Simple harmonic motion (SHM) is one of the most fundamental periodic motions in physics. It appears in mass–spring systems, simple pendulums, and many real-world oscillations. In CIE A-Level Physics, SHM is essential because it links kinematics, dynamics, and energy transfer in a single oscillating system. This article summarises definitions, equations, graphs, energy changes, damping, and resonance with a focus on exam-style understanding.

简谐运动（SHM）是物理学中最基本的周期运动之一。它出现在弹簧振子、单摆以及许多现实世界的振动中。在 CIE A-Level 物理中，简谐运动至关重要，因为它将运动学、动力学和能量转移整合在一个振动系统中。本文概述定义、方程、图像、能量变化、阻尼和共振，并聚焦考试导向的理解。

1. Defining SHM | 简谐运动的定义

Simple harmonic motion is defined as oscillatory motion in which the acceleration of the object is directly proportional to its displacement from a fixed equilibrium position and is always directed towards that equilibrium position.

简谐运动定义为：物体的加速度与它相对固定平衡位置的位移成正比，并且始终指向该平衡位置。

Mathematically, this condition is written as:

a = −ω²x

The minus sign shows that acceleration and displacement are in opposite directions. The constant ω is called the angular frequency and determines how rapidly the oscillation occurs.

负号表示加速度与位移方向相反。常数 ω 称为角频率，它决定振动发生的快慢。

For an object to be in SHM, the resultant force must also be a restoring force proportional to displacement: F = −kx. This is the physical origin of the acceleration expression.

物体要做简谐运动，合外力必须也是与位移成正比的回复力：F = −kx。这是加速度表达式的物理来源。

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2. Key Terms and Quantities | 关键术语与物理量

In a full cycle of SHM, the main quantities are amplitude, period, frequency, angular frequency, and phase. The amplitude x₀ is the maximum displacement from equilibrium. The period T is the time for one complete oscillation.

在简谐运动的一个完整周期中，主要物理量有振幅、周期、频率、角频率和相位。振幅 x₀ 是离开平衡位置的最大位移。周期 T 是完成一次完整振动所需的时间。

Frequency f is the number of cycles per second, and angular frequency ω is related by:

ω = 2πf = 2π / T

频率 f 是每秒振动的次数，角频率 ω 与它的关系为：ω = 2πf = 2π / T。

Phase describes the stage of the oscillation within a cycle. Phase difference between two oscillating objects is measured in radians and shows whether one motion is ahead of or behind another.

相位描述振动在一个周期内所处的阶段。两个振动物体之间的相位差以弧度为单位，表示一个运动超前或落后于另一个运动。

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3. Equations of Motion | 运动方程

If timing starts when the object passes through equilibrium moving in the positive direction, displacement can be written as:

如果从物体经过平衡位置向正方向运动时开始计时，位移可写为：

x = x₀ sin(ωt)

If timing starts at maximum displacement, the displacement is instead:

如果从最大位移处开始计时，位移则为：

x = x₀ cos(ωt)

Velocity in SHM can be expressed in terms of displacement:

简谐运动中的速度可以用位移表示：

v = ±ω√(x₀² − x²)

The maximum speed occurs at x = 0 and is v_max = ωx₀. The acceleration is always a = −ω²x, so maximum acceleration occurs at x = x₀ and has magnitude ω²x₀.

最大速度出现在 x = 0 处，为 v_max = ωx₀。加速度始终为 a = −ω²x，因此最大加速度出现在 x = x₀ 处，大小为 ω²x₀。

Displacement, velocity, and acceleration graphs versus time are sinusoidal. Velocity leads displacement by π/2, and acceleration is π out of phase with displacement.

位移、速度、加速度随时间变化的图像都是正弦曲线。速度超前位移 π/2，加速度与位移反相，相位差为 π。

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4. Mass–Spring System | 弹簧振子

A mass m attached to a spring of stiffness k obeys Hooke’s law F = −kx. Because the restoring force is proportional to displacement, the motion is simple harmonic.

质量为 m 的物体连接在劲度系数为 k 的弹簧上，满足胡克定律 F = −kx。由于回复力与位移成正比，该运动是简谐运动。

Using Newton’s second law, ma = −kx, and comparing with a = −ω²x gives:

利用牛顿第二定律 ma = −kx，并与 a = −ω²x 比较可得：

ω² = k / m

Therefore the period of a mass–spring system is:

因此弹簧振子的周期为：

T = 2π√(m / k)

This result is independent of amplitude for ideal springs. A stiffer spring gives a smaller period, while a larger mass gives a larger period.

对于理想弹簧，该结果与振幅无关。弹簧越硬，周期越小；质量越大，周期越大。

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5. Simple Pendulum | 单摆

A simple pendulum consists of a point mass suspended by a light, inextensible string. For small angular displacements (typically less than about 10°), the restoring force is approximately proportional to displacement, so the motion is simple harmonic.

单摆由一根轻质且不可伸长的细线悬挂一个质点组成。当角位移较小（通常小于约 10°）时，回复力近似与位移成正比，因此运动近似为简谐运动。

For a small displacement x along the arc, the restoring force is F = −mg sinθ ≈ −mg(x/l). Comparing with F = −kx gives an effective spring constant k = mg/l, hence the period is:

对于沿圆弧的小位移 x，回复力为 F = −mg sinθ ≈ −mg(x/l)。与 F = −kx 比较可得等效劲度系数 k = mg/l，因此周期为：

T = 2π√(l / g)

where l is the length of the pendulum and g is the gravitational field strength. The mass of the bob does not appear in the formula, so it does not affect the period.

其中 l 是摆长，g 是重力场强度。摆球的质量不出现在公式中，因此它不影响周期。

Increasing the length increases the period; increasing g decreases the period. For large angles, the approximation fails and the motion is not perfectly simple harmonic.

增加摆长会使周期增大；增大 g 会使周期减小。对于大角度，该近似不再成立，运动不是严格简谐运动。

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6. Energy in SHM | 简谐运动中的能量

In SHM, energy continuously changes between kinetic energy and potential energy, but the total energy remains constant if there is no damping.

在简谐运动中，能量在动能和势能之间不断转化，但如果没有阻尼，总能量保持不变。

At displacement x, kinetic energy and potential energy can be written as:

在位移 x 处，动能和势能可写为：

KE = ½ mω²(x₀² − x²)

PE = ½ mω

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