IB Physics: Characteristics and Mathematical Representation of Simple Harmonic Motion | IB物理：简谐运动特征与数学表达

📚 IB Physics: Characteristics and Mathematical Representation of Simple Harmonic Motion | IB物理：简谐运动特征与数学表达

Simple harmonic motion (SHM) is one of the most fundamental and elegant topics in physics, describing any oscillatory system where the restoring force is directly proportional to displacement and acts in the opposite direction. This article provides a comprehensive review of the defining features of SHM and its complete mathematical formulation, tailored to the IB Physics syllabus and Edexcel examination requirements.

简谐运动（SHM）是物理学中最基础且最优美的主题之一，它描述的是恢复力与位移成正比且方向相反的振荡系统。本文基于IB物理教学大纲与爱德思考试要求，系统梳理简谐运动的判断特征及其完整的数学表达体系。

1. Definition and Physical Origin of SHM | 简谐运动的定义与物理起源

A particle undergoes simple harmonic motion if its acceleration is proportional to its displacement from a fixed equilibrium position, and is always directed towards that equilibrium. This implies the acceleration and displacement are oppositely directed, which is the signature of all oscillatory restoring systems.

如果一个质点的加速度与其相对于固定平衡位置的位移成正比，且始终指向平衡位置，则该质点做简谐运动。这意味着加速度与位移方向相反，这是所有振荡恢复系统的共同特征。

The physical origin lies in a restoring force that obeys a linear relationship with displacement — commonly arising from elastic materials, gravitational pendulums at small angles, or buoyant forces in fluids.

其物理根源在于一种恢复力与位移呈线性关系—通常来自弹性材料、小角度下的重力摆或流体中的浮力。

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2. The Defining Acceleration–Displacement Relation | 核心判据：加速度—位移关系

The most concise criterion for SHM is the differential equation that links acceleration \(a\) to displacement \(x\):

简谐运动最简洁的判据是连接加速度a与位移x的微分方程：

a = −ω²·x

where ω is the angular frequency (unit: rad s⁻¹). The negative sign guarantees that the acceleration is always directed towards the equilibrium position. This single equation completely characterizes SHM and can be used to test whether any physical system qualifies as a simple harmonic oscillator.

其中ω是角频率（单位：rad s⁻¹）。负号保证了加速度始终指向平衡位置。这一方程完整地定义了简谐运动，可用于检验任何物理系统是否为简谐振荡器。

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3. Solution of the SHM Equation: Displacement as a Function of Time | 简谐方程的解：位移随时间的变化

Solving the second-order differential equation d²x/dt² = −ω²x yields a sinusoidal time dependence. The general solution can be written in either cosine or sine form:

求解二阶微分方程d²x/dt² = −ω²x，得到正弦形式的时间依赖关系。通解可写作余弦或正弦形式：

x(t) = A·cos(ωt + φ)

Here, A (metres) is the amplitude — the maximum displacement from equilibrium — and φ (radians) is the phase constant, determined by the initial position and velocity of the oscillator. The quantity (ωt + φ) is called the phase of the motion.

其中A（米）是振幅，即离开平衡位置的最大位移；φ（弧度）是初相位，由振荡的初始位置和初速度决定。量(ωt + φ)称为运动的相位。

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4. Velocity and Acceleration Equations | 速度与加速度方程

Differentiating the displacement equation once gives the velocity, and differentiating again gives the acceleration:

对位移方程求一次导数得到速度，求两次导数得到加速度：

v(t) = −Aω·sin(ωt + φ)
a(t) = −Aω²·cos(ωt + φ)

Notice that the acceleration equation simplifies to a = −ω²x, confirming consistency with the defining relation. The velocity is maximum at the equilibrium position and zero at the extremities.

注意加速度表达式可化简为a = −ω²x，印证了其与定义关系的一致性。速度在平衡位置最大，在两端为零。

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5. Phase Relationships Among x, v and a | 位移、速度与加速度的相位关系

It is crucial to understand how the three kinematic quantities are shifted relative to each other in time:

理解三个运动学量之间的相位差至关重要：

• If displacement is a cosine function, velocity is a negative sine, which leads the displacement by 90° (π/2 rad).

  如果位移为余弦函数，速度则为负正弦，比位移超前90°（π/2 rad）。

• Acceleration is a negative cosine, which is 180° (π rad) out of phase with displacement.

  加速度为负余弦，与位移反相180°（π rad）。

• Velocity is 90° out of phase with acceleration; when acceleration is maximum, velocity is zero and vice versa.

  速度与加速度相位差90°；当加速度最大时速度为零，反之亦然。

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6. Energy in Simple Harmonic Motion | 简谐运动的能量

A key feature of ideal SHM is the continuous transformation between kinetic and potential energy, while the total mechanical energy remains constant.

理想简谐运动的一个关键特征是与势能之间不断相互转化，而总机械能保持不变。

For a mass-spring system with spring constant k, the potential energy stored in the spring and the kinetic energy of the mass are:

对于劲度系数为k的弹簧—质量系统，弹簧储存的势能与质量的动能分别为：

E_p = ½kx² = ½kA²·cos²(ωt + φ)
E_k = ½mv² = ½kA²·sin²(ωt + φ)

The total energy is E_total = ½kA², and it remains constant. At the equilibrium position, all energy is kinetic; at the extremes, all energy is potential.

总能量E_total = ½kA²，保持恒定。在平衡位置全部能量为动能；在两端点全部能量为势能。

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7. Angular Frequency, Period and Frequency | 角频率、周期与频率

Angular frequency ω is related to the period T and linear frequency f by:

角频率ω与周期T和频率f的关系为：

ω = 2π/T = 2πf

For a mass–spring system, the period depends only on the mass and the spring constant:

对于弹簧—质量系统，周期仅取决于质量与劲度系数：

T = 2π√(m/k)

For a simple pendulum oscillating with small amplitudes, the period is independent of mass:

对于小幅振荡的单摆，周期与质量无关：

T = 2π√(L/g)

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

The x–t graph is a cosine curve; the v–t graph is a negative sine curve; the a–t graph is a negative cosine curve. When sketching graphs for exams, pay attention to the following features:

x–t图像为余弦曲线；v–t图像为负正弦曲线；a–t图像为负余弦曲线。在考试作图时，注意以下特征：

• On the x–t graph, the slope at any instant equals the instantaneous velocity, so zero slope at the extremes and maximum slope at equilibrium.

  在x–t图像中，任意时刻的斜率等于瞬时速度，因此在两端斜率为零，在平衡位置斜率最大。

• On the v–t graph, the slope gives acceleration; check the signs: when displacement is positive, acceleration is negative.

  在v–t图像中，斜率给出加速度；注意符号：当位移为正时，加速度为负。

• All three graphs share the same period and frequency; only their phases differ.

  三条图像具有相同的周期和频率；仅相位不同。

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9. Initial Conditions and Phase Constant | 初始条件与初相位

The phase constant φ is determined by where the oscillator starts at t = 0. Some common cases:

初相位φ由t = 0时刻振荡器的位置决定。常见情况如下：

Initial condition | 初始条件	φ value | φ值
Starts at maximum positive displacement | 从最大正位移开始	0
Starts at equilibrium moving in positive direction | 从平衡位置向正方向运动	−π/2
Starts at maximum negative displacement | 从最大负位移开始	π
Starts at equilibrium moving in negative direction | 从平衡位置向负方向运动	+π/2

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10. SHM as a Projection of Uniform Circular Motion | 简谐运动：匀速圆周运动的投影

SHM can be elegantly interpreted as the projection of uniform circular motion onto a diameter. A particle moving in a circle of radius A with constant angular speed ω has coordinates x = A·cos(ωt + φ) and y = A·sin(ωt + φ); the x-coordinate alone exhibits perfect SHM.

简谐运动可以理解为匀速圆周运动在直径上的投影。一个以恒定角速度ω沿半径为A的圆周运动的质点，其坐标为x = A·cos(ωt + φ)、y = A·sin(ωt + φ)；仅x坐标本身就呈现完美的简谐运动。

This analogy is extremely helpful for solving problems involving phase differences and determining velocities at given positions.

这一类比对于解决相位差问题及求特定位置的速度极为有帮助。

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11. Energy–Displacement Graphs and the Relation v = ±ω√(A² − x²) | 能量—位移图像与速度—位移关系

The relationship between velocity and displacement in SHM is obtained from the conservation of energy:

由能量守恒可得到简谐运动中速度与位移的关系：

v = ±ω√(A² − x²)

The positive sign corresponds to motion in the positive direction, and the negative sign to motion in the negative direction. When x = ±A, v = 0; when x = 0, v = ±ωA, the maximum speed.

正号对应正方向的运动，负号对应负方向的运动。当x = ±A时，v = 0；当x = 0时，v = ±ωA，即为最大速度。

The energy–displacement graph is a parabola for potential energy opening upwards and an inverted parabola for kinetic energy. The total energy is a horizontal straight line, marking the sum of the two curves.

能量—位移图像中，势能曲线为开口向上的抛物线，动能曲线为开口向下的抛物线。总能量为水平直线，表示两条曲线之和。

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12. Common Examination Traps and Key Formulae Summary | 考试常见陷阱与核心公式总结

Candidates frequently lose marks for four reasons: forgetting the negative sign in a = −ω²x; confusing linear frequency f with angular frequency ω; using degrees instead of radians for phase; and assuming SHM for a pendulum at large amplitudes where the small-angle approximation fails. Always verify with the defining equation before classifying any motion as SHM.

考生常因四个原因失分：忘记a = −ω²x中的负号；混淆频率f与角频率ω；用角度制而非弧度制计算相位；以及在摆角较大、小角近似不成立时仍将单摆视为简谐运动。在判定任何运动为简谐运动之前，务必用定义方程进行验证。

Essential formula list:

核心公式清单：

• a = −ω²x (defining equation | 定义方程)

• x(t) = A·cos(ωt + φ), v(t) = −Aω·sin(ωt + φ), a(t) = −Aω²·cos(ωt + φ)

• v = ±ω√(A² − x²) (energy–displacement relation | 速度—位移关系)

• E_k = ½·m·ω²·(A² − x²), E_p = ½·m·ω²·x², E_total = ½·m·ω²·A² = ½·k·A²

• T = 2π·√(m/k) for mass–spring | 弹簧—质量系统

• T = 2π·√(L/g) for small-angle pendulum | 小角度单摆

Mastering the features and mathematics of SHM is essential not only for exam success but also as a gateway to understanding waves, oscillations, resonance and quantum mechanical harmonic oscillators.

掌握简谐运动的特征与数学表达不仅对考试至关重要，更是理解波动、共振与量子力学中的谐振子的基石。

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