A2 Physics: Simple Harmonic Motion Key Points Explored | A2物理：简谐运动考点精讲

📚 A2 Physics: Simple Harmonic Motion Key Points Explored | A2物理：简谐运动考点精讲

Simple Harmonic Motion (SHM) is a cornerstone of A2 Physics, appearing in topics from mechanics to wave behaviour. Mastering its equations, energy transformations, and graphical analysis is essential for tackling both numerical and theoretical questions, and will enable you to link oscillations to a wide range of physical systems.

简谐运动是A2物理的核心内容，涉及力学和波动等多个主题。掌握其方程、能量转化和图像分析，对解答计算题和概念题至关重要，并能帮助你把振动与各种物理系统联系起来。

1. Definition and Conditions for SHM | 简谐运动的定义与条件

Simple Harmonic Motion occurs when the net force (and therefore the acceleration) acting on an object is directly proportional to its displacement from a fixed equilibrium position and is always directed towards that point. Mathematically, this is written as a ∝ −x.

当物体所受合力（从而加速度）与偏离固定平衡位置的位移成正比，且方向始终指向该平衡位置时，物体做简谐运动。数学上表示为 a ∝ −x。

a = −ω²x

where ω is the angular frequency (rad s⁻¹), x is the displacement, and the minus sign indicates the restoring nature of the motion.

其中 ω 是角频率（单位 rad s⁻¹），x 是位移，负号反映了运动的回复性。

The angular frequency is linked to the period T and the ordinary frequency f by:

ω = 2πf = 2π/T

This implies that for a system to be SHM, the acceleration–displacement graph must be a straight line through the origin with a gradient of –ω².

这意味着，要满足简谐运动，加速度-位移图必须是一条通过原点、斜率为 –ω² 的直线。

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2. The Defining Equation a = −ω²x | 特征方程 a = −ω²x

= 0). The constant ω² is always positive and determines how quickly the system oscillates.

方程 a = −ω²x 是简谐运动的特征标志。它告诉我们，加速度的大小在最大位移处（x = ±A）最大，在平衡位置（x = 0）为零。常数 ω² 恒为正值，决定了系统振动的快慢。

provided the oscillation is about a stable equilibrium. The constant must be independent of displacement; this is what yields the straight-line acceleration–displacement graph and sinusoidal time variation.

任何系统，只要其运动方程能写成 a = −(常数) × x 的形式，且振动发生在稳定平衡位置附近，就在做简谐运动。常数必须与位移无关，这样才能得到直线的加速度-位移图和正弦式的时间变化。

In practical experiments, you can verify SHM by measuring the period, plotting a graph of a against x, or checking that the oscillation is isochronous (period independent of amplitude, for small amplitudes).

在实验中，可以通过测量周期、作 a-x 图或检验等时性（小振幅下周期与振幅无关）来验证简谐运动。

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3. Displacement, Velocity and Acceleration in SHM | 简谐运动中的位移、速度和加速度

mum displacement (x = A at t = 0), the displacement can be described by a cosine function:

x = A cos(ωt)

where A is the amplitude. The velocity is the time derivative:

v = −Aω sin(ωt)

and the acceleration is the second derivative or, from the defining equation:

a = −Aω² cos(ωt) = −ω²x

ng in the positive direction, use x = A sin(ωt); then v = Aω cos(ωt) and a = −Aω² sin(ωt).

若从最大位移处开始计时（t=0时 x = A），位移可用余弦函数表示：x = A cos(ωt)。速度是其导数：v = −Aω sin(ωt)，加速度是二阶导数或由特征方程得出：a = −Aω² cos(ωt) = −ω²x。若从平衡位置正向通过时开始计时，则用 x = A sin(ωt)，此时 v = Aω cos(ωt)，a = −Aω² sin(ωt)。

A useful relationship between speed and displacement, independent of time, is:

v = ± ω √A² − x²

From this, the maximum speed occurs at x = 0 (v_max = ωA) and the maximum acceleration at x = ±A (a_max = ω²A).

速度与位移间一个不显含时间的关系为 v = ± ω √A² − x²。由此可知，最大速度在 x = 0 处（v_max = ωA），最大加速度在 x = ±A 处（a_max = ω²A）。

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4. Graphical Representation of SHM | 简谐运动的图像表示

The x–t, v–t and a– or SHM are sinusoidal. For an oscillation starting at maximum displacement (x = A cos ωt):

• Displacement x–t is a cosine curve with amplitude A.
• Velocity v–t is a negative sine curve with amplitude ωA; it leads displacement by π/2 rad (90°).
• Acceleration a–t is a negative cosine curve with amplitude ω²A; it is in antiphase (π rad or 180° out of phase) with the displacement.

对于从最大位移开始的简谐运动：位移–时间图是振幅为 A 的余弦曲线；速度–时间图是振幅为 ωA 的负正弦曲线，相位超前位移 π/2；加速度–时间图是振幅为 ω²A 的负余弦曲线，与位移反相（相位差 π）。

These phase relationships can be summarised in a phasor diagram: three vectors of lengths A, ωA and ω²A rotating with angular speed ω, each offset by 90°. The projection onto the vertical or horizontal axis gives the instantaneous values.

这些相位关系可用旋转矢量图概括：长度分别为 A、ωA 和 ω²A 的三个矢量以角速度 ω 旋转，彼此相差 90°。投影到竖直或水平轴即得瞬时值。

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

In SHM, the total mechanical energy is conserved (if no damping). It is the sum of kinetic energy E_k and potential energy E_p. For a spring–mass system, E_p = ½ k x²; for a simple pendulum, it is gravitational potential energy.

在无阻尼的情况下，简谐运动的总机械能守恒，为动能 E_k 与势能 E_p 之和。对于弹簧振子，E_p = ½ k x²；对于单摆，则为重力势能。

Using the velocity–displacement relation, the kinetic energy can be expressed as:

E_k = ½ m v² = ½ m ω² (A² − x²)

while the total energy is entirely kinetic at equilibrium and entirely potential at the extremes:

E_total = ½ m ω² A² = ½ k A²

利用速度–位移关系，动能可表示为 E_k = ½ m v² = ½ m ω² (A² − x²)。总能量在平衡位置全为动能，在最大位移处全为势能：E_total = ½ m ω² A² = ½ k A²。

Energy bar charts or energy–displacement graphs show that E_k and E_p exchange continuously. At any displacement, E_k + E_p ant. This conservation is a powerful tool for solving problems when forces are not needed.

能量柱状图或能量–位移图表明动能和势能不断相互转化。在任意位移处，E_k + E_p 为常数。这一守恒特性是解决某些问题的有力工具，无需考虑受力细节。

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6. The Spring-Mass System (Horizontal and Vertical) | 弹簧振子（水平与竖直）

on a frictionless horizontal surface, Hooke’s law gives F = −kx. From Newton’s second law:

−kx = ma → a = −(k/m) x

Thus the motion is SHM with angular frequency ω = √k/m and period:

T = 2π √m/k

cillation about that new equilibrium is still SHM with the same period, as the restoring force is still −kx (with x measured from the new equilibrium).

对于连接在劲度系数为 k 的弹簧上、在光滑水平面上的质量 m，由胡克定律 F = −kx 和牛顿第二定律得 −kx = ma，即 a = −(k/m) x。因此运动为简谐运动，角频率 ω = √k/m，周期 T = 2π √m/k。在竖直弹簧振子中，重力只是使平衡位置下移 x₀ = mg/k，围绕新平衡点的振动仍是简谐运动，周期相同，因为回复力仍为 −kx（x 从新平衡点量起）。

Always remember to use the mass m in kg, stiffness k in N m⁻¹, and that the period is independent of amplitude for small oscillations. The period increases with mass and decreases with stiffer springs.

务必注意质量 m 用 kg，劲度系数 k 用 N m⁻¹，周期在小振幅下与振幅无关。质量越大周期越长，弹簧越硬周期越短。

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7. The Simple Pendulum | 单摆

−mgθ. Using the arc displacement x = Lθ, the acceleration a = −(g/L) x, which is SHM with:

ω = √g/L and T = 2π √L/g

单摆由轻质不可伸长的细线悬挂一个质点构成。在小角度（θ < 约 10°）下，回复力为 −mg sinθ ≈ −mgθ。利用弧位移 x = Lθ，得加速度 a = −(g/L) x，属于简谐运动，角频率 ω = √g/L，周期 T = 2π √L/g。

Notice that the period does not depend on the mass of the bob – only on the length L and the local gravitational field strength g. This is a classic exam point. For larger amplitudes, the motion is still periodic but no longer SHM; the period increases slightly with amplitude.

Published by TutorHao | Physics Revision Series | aleveler.com

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