Energy Changes in Simple Harmonic Motion | 简谐运动中的能量变化

📚 Energy Changes in Simple Harmonic Motion | 简谐运动中的能量变化

In simple harmonic motion, also abbreviated as s.h.m., energy is continuously exchanged between kinetic energy and potential energy. Although the displacement, velocity and acceleration of the oscillator all change with time, the total mechanical energy of an undamped system remains constant. This article summarises the energy equations, energy-displacement graphs, energy-time graphs and damping effects required for CIE A-Level Physics.

在简谐运动中，能量在动能和势能之间不断转换。虽然振子的位移、速度和加速度都随时间变化，但无阻尼系统的总机械能保持不变。本文总结 CIE A-Level 物理中要求的能量方程、能量-位移图、能量-时间图以及阻尼效应。

1. What is Simple Harmonic Motion? | 什么是简谐运动？

Simple harmonic motion occurs when the resultant force F on an object is directly proportional to its displacement x from equilibrium and always acts towards the equilibrium position. This condition is written as F = −kx, where k is the force constant. The negative sign indicates that the force is a restoring force.

当物体所受合力 F 与其离开平衡位置的位移 x 成正比，并且总指向平衡位置时，物体做简谐运动。该条件写作 F = −kx，其中 k 为力常数。负号表示该力为回复力。

Any system obeying F = −kx has a natural angular frequency ω given by ω² = k/m. The displacement can be described by x = x₀ sin(ωt) or x = x₀ cos(ωt), where x₀ is the amplitude and t is time.

任何满足 F = −kx 的系统都具有固有角频率 ω，且 ω² = k/m。位移可表示为 x = x₀ sin(ωt) 或 x = x₀ cos(ωt)，其中 x₀ 为振幅，t 为时间。

F = −kx, ω² = k/m, x = x₀ sin(ωt)

The period T = 2π/ω is independent of amplitude for an ideal simple harmonic oscillator. This property is used to define timing in pendulum clocks and many vibration sensors.

理想简谐振子的周期 T = 2π/ω 与振幅无关。这一特性被用于摆钟和许多振动传感器中的计时。

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2. Relating Displacement, Velocity and Acceleration | 位移、速度与加速度的关系

Velocity in s.h.m. is not constant. It is greatest at the equilibrium position and zero at the amplitude positions. Using calculus, the velocity is v = ±ω√(x₀² − x²), where the sign depends on the direction of motion.

简谐运动中的速度不是恒定的。速度在平衡位置最大，在振幅处为零。利用微积分可得速度 v = ±ω√(x₀² − x²)，正负号取决于运动方向。

v = ±ω√(x₀² − x²)

Acceleration is proportional to displacement but directed towards equilibrium, so a = −ω²x. The acceleration has maximum magnitude ω²x₀ at x = ±x₀ and becomes zero at x = 0.

加速度与位移成正比，但方向指向平衡位置，即 a = −ω²x。加速度在 x = ±x₀ 处最大，量值为 ω²x₀；在 x = 0 处为零。

a = −ω²x

These relationships are essential for understanding energy changes, because kinetic energy depends on v² and potential energy depends on x².

这些关系对于理解能量变化至关重要，因为动能取决于 v²，势能取决于 x²。

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3. Kinetic Energy in S.H.M. | 简谐运动中的动能

Kinetic energy arises from the motion of the oscillating mass. If the displacement is x and the amplitude is x₀, the kinetic energy is Eₖ = ½ m v² = ½ m ω² (x₀² − x²).

动能由振动质量的运动产生。如果位移为 x、振幅为 x₀，则动能为 Eₖ = ½ m v² = ½ m ω² (x₀² − x²)。

Eₖ = ½ m v² = ½ m ω² (x₀² − x²)

At the equilibrium position x = 0, the kinetic energy reaches its maximum value of ½ m ω² x₀². At the amplitude positions x = ±x₀, the kinetic energy is zero because the oscillator is momentarily at rest.

在平衡位置 x = 0 处，动能达到最大值 ½ m ω² x₀²。在振幅位置 x = ±x₀ 处，动能为零，因为振子瞬时静止。

The kinetic energy is always non-negative. Its variation with displacement is an inverted parabola, showing that the oscillator moves fastest through equilibrium and slows down towards the extremes.

动能始终为非负值。它随位移的变化是一条倒抛物线，表明振子在经过平衡位置时运动最快，在接近端点时减慢。

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4. Potential Energy in S.H.M. | 简谐运动中的势能

Potential energy is stored by the restoring force. For a mass-spring system, the potential energy is Eₚ = ½ k x² = ½ m ω² x², measured relative to the equilibrium position.

势能由回复力储存。对于弹簧振子系统，势能为 Eₚ = ½ k x² = ½ m ω² x²，该值是相对平衡位置测量的。

Eₚ = ½ k x² = ½ m ω² x²

Potential energy is zero at x = 0 and reaches a maximum of ½ m ω² x₀² at the amplitude positions. The form ½ m ω² x² also applies to the simple pendulum for small angular displacements.

势能在 x = 0 处为零，在振幅位置达到最大值 ½ m ω² x₀²。对于小角度摆动的单摆，½ m ω² x² 的形式同样适用。

This quadratic dependence on displacement is a direct consequence of the linear restoring force F = −kx. The stiffer the oscillator, the more rapidly potential energy increases with displacement.

势能对位移的二次依赖是线性回复力 F = −kx 的直接结果。振子越刚硬，势能随位移增加得越快。

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5. Total Energy and Conservation | 总能量与能量守恒

In an undamped simple harmonic oscillator no mechanical energy is lost to the surroundings. Therefore the total energy E_total = Eₖ + Eₚ is constant. Substituting the kinetic and potential energy expressions gives E_total = ½ m ω² x₀² = ½ k x₀².

在无阻尼简谐振子中，没有机械能散失到周围环境。因此总能量 E_total = Eₖ + Eₚ 保持不变。代入动能和势能表达式可得 E_total = ½ m ω² x₀² = ½ k x₀²。

E_total = Eₖ + Eₚ = ½ m ω² x₀² = ½ k x₀²

The total energy depends only on the mass, angular frequency and amplitude. It does not depend on displacement or time. During oscillation, energy is simply transferred back and forth between the kinetic and potential stores.

总能量仅取决于质量、角频率和振幅。它与

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