A-Level Physics: Simple Harmonic Motion (SHM) — Complete Guide | A-Level物理：简谐运动完全指南

Introduction

Simple Harmonic Motion (SHM) is one of the most fundamental concepts in A-Level Physics. It describes a special type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. From the swinging of a pendulum to the vibration of atoms in a crystal lattice, SHM provides the mathematical framework for understanding oscillatory systems across all scales of physics.

简谐运动（Simple Harmonic Motion，简称 SHM）是 A-Level 物理中最基础的概念之一。它描述了一种特殊的周期性运动：回复力与偏离平衡位置的位移成正比，且方向始终指向平衡位置。从钟摆的摆动到晶格中原子的振动，简谐运动为理解各个尺度的振荡系统提供了数学框架。

1. Defining Simple Harmonic Motion

1.1 The Fundamental Condition

For an object to be in simple harmonic motion, the resultant force acting on it must satisfy:

F = −kx

where:

F is the restoring force (N)
k is the force constant or spring constant (N m⁻¹)
x is the displacement from equilibrium (m)
The negative sign indicates that the force always opposes the displacement

This linear relationship between force and displacement is the hallmark of SHM. When plotted on a graph, F vs x yields a straight line passing through the origin with a negative gradient of −k.

简谐运动的定义条件：物体所受的合力必须满足 F = −kx，其中 F 为回复力，k 为力常数（劲度系数），x 为偏离平衡位置的位移，负号表示力的方向始终与位移方向相反。这种力与位移之间的线性关系是简谐运动的标志。

1.2 Acceleration in SHM

Applying Newton’s Second Law (F = ma) to the SHM condition gives the acceleration equation:

a = −ω²x

where ω (omega) is the angular frequency, related to the time period T by:

ω = 2π / T or ω = 2πf

This is perhaps the most useful form of the SHM definition for problem-solving. It tells us that acceleration is always proportional to displacement but in the opposite direction, and the constant of proportionality is ω².

将牛顿第二定律 (F = ma) 代入简谐运动条件，得到加速度方程：a = −ω²x。其中 ω 为角频率，与周期 T 的关系为 ω = 2π/T = 2πf。这是解题时最有用的 SHM 表达形式——加速度始终与位移成正比且方向相反。

2. Mathematical Description of SHM

2.1 Displacement as a Function of Time

The displacement of an object in SHM varies sinusoidally with time. Starting from maximum displacement (amplitude A) at t = 0:

x = A cos(ωt)

If the object starts from equilibrium at t = 0, we use:

x = A sin(ωt)

More generally, including a phase constant φ:

x = A cos(ωt + φ)

where A is the amplitude (maximum displacement, in metres) and φ is the phase constant (in radians).

简谐运动中位移随时间按正弦（或余弦）规律变化。若从最大位移处开始计时：x = A cos(ωt)；若从平衡位置开始计时：x = A sin(ωt)。更一般的形式为 x = A cos(ωt + φ)，其中 A 为振幅，φ 为初相位。

2.2 Velocity in SHM

Velocity is the rate of change of displacement. Differentiating x = A cos(ωt):

v = dx/dt = −Aω sin(ωt)

The maximum speed occurs as the object passes through equilibrium (x = 0):

v_max = Aω

A very useful relationship links velocity and displacement without involving time:

v = ± ω √(A² − x²)

This equation tells us the speed at any displacement x. At x = ±A (the extremes), v = 0; at x = 0 (equilibrium), v = ±Aω.

速度是位移对时间的导数。由 x = A cos(ωt) 求导得 v = −Aω sin(ωt)。最大速度出现在经过平衡位置时：v_max = Aω。速度与位移的关系式 v = ± ω√(A² − x²) 不显含时间，非常实用——在极端位置速度为零，在平衡位置速度最大。

2.3 Acceleration in SHM

Differentiating velocity gives acceleration:

a = dv/dt = −Aω² cos(ωt) = −ω²x

The maximum acceleration occurs at the extremes of motion (x = ±A):

a_max = Aω²

At equilibrium (x = 0), acceleration is zero.

对速度求导得加速度：a = −Aω² cos(ωt) = −ω²x。最大加速度出现在极端位置：a_max = Aω²。在平衡位置处加速度为零。

3. Graphical Representation

When we plot displacement, velocity, and acceleration against time on the same axes, several important patterns emerge:

The displacement graph is a cosine wave (if starting from maximum displacement)
The velocity graph is a negative sine wave — it lags displacement by π/2 (90°)
The acceleration graph is a negative cosine wave — it is exactly out of phase with displacement (phase difference of π or 180°)

Key relationships visible on the graphs:

When displacement is maximum, velocity is zero and acceleration is maximum (negative)
When displacement is zero, velocity is maximum and acceleration is zero
Velocity leads displacement by π/2
Acceleration is always opposite in sign to displacement

将位移、速度和加速度随时间变化的图像绘制在同一坐标系上，可以观察到：位移为余弦波、速度滞后位移 π/2（即 90°）、加速度与位移反相（相位差 π 或 180°）。当位移最大时速度为零、加速度最大（反向）；当位移为零时速度最大、加速度为零。

4. Energy in Simple Harmonic Motion

4.1 Kinetic and Potential Energy

During SHM, energy continuously transforms between kinetic and potential forms:

Kinetic Energy:

KE = ½mv² = ½mω²(A² − x²)

Potential Energy:

PE = ½kx² = ½mω²x²

Total Mechanical Energy:

E_total = KE + PE = ½mω²A² = ½kA²

4.2 Energy Distribution

Position / 位置	KE	PE	Total / 总能量
x = 0 (Equilibrium)	½mω²A² (max)	0	½mω²A²
x = ±A/2	¾ × ½mω²A²	¼ × ½mω²A²	½mω²A²
x = ±A (Extremes)	0	½mω²A² (max)	½mω²A²

The total energy of an SHM system is proportional to the square of the amplitude (E ∝ A²). This is a crucial result — doubling the amplitude quadruples the total energy.

在简谐运动中，能量在动能和势能之间不断转换。总机械能守恒：E_total = ½mω²A² = ½kA²。一个重要结论是：总能量与振幅的平方成正比 (E ∝ A²)，振幅翻倍则总能量翻四倍。

5. The Mass-Spring System

5.1 Horizontal Mass-Spring Oscillator

A mass m attached to a spring with spring constant k on a frictionless horizontal surface is the simplest SHM system. The time period is given by:

T = 2π √(m/k)

This equation reveals two important physical insights:

The period depends only on the mass and spring constant — not on the amplitude (isochronous nature of SHM)
Increasing mass increases the period (slower oscillation)
Increasing spring stiffness decreases the period (faster oscillation)

The angular frequency for a mass-spring system is:

ω = √(k/m)

质量为 m 的物体连接劲度系数为 k 的弹簧，在光滑水平面上振动，是最简单的 SHM 系统。其周期为 T = 2π√(m/k)。周期仅取决于质量和劲度系数，与振幅无关——这体现了简谐运动的等时性。质量越大周期越长，弹簧越硬周期越短。

5.2 Vertical Mass-Spring Oscillator

When a mass hangs vertically from a spring, gravity shifts the equilibrium position but does not affect the period. The extension at equilibrium is:

mg = ke → e = mg/k

where e is the equilibrium extension. The period formula remains:

T = 2π √(m/k)

This is a common exam trick — students sometimes try to include gravity in the period equation, but it cancels out in the derivation.

竖直悬挂的弹簧振子中，重力仅改变平衡位置，不影响振动周期。平衡时满足 mg = ke，但周期公式仍为 T = 2π√(m/k)。这是考试中常见的陷阱——重力最终在推导中消去，不影响周期。

6. The Simple Pendulum

6.1 Small-Angle Approximation

A simple pendulum consists of a point mass (bob) suspended by a light, inextensible string. For small angular displacements (θ less than about 10°), the motion approximates SHM with period:

T = 2π √(L/g)

where L is the length of the pendulum (m) and g is the acceleration due to gravity (9.81 m s⁻²).

The restoring force on the bob is the tangential component of gravity: F = −mg sin θ. For small angles, sin θ ≈ θ (in radians), giving F ≈ −mgθ = −(mg/L)x, which matches the SHM condition F = −kx.

单摆由轻质不可伸长的细绳悬挂一个质点（摆球）构成。在小角度摆动（θ 约小于 10°）时，运动近似为简谐运动，周期 T = 2π√(L/g)。回复力为重力的切向分量，利用小角近似 sin θ ≈ θ 即可得到 SHM 形式。

6.2 Factors Affecting Pendulum Period

Starting from T = 2π√(L/g), we observe:

Length (L): Period increases with √L. Doubling length increases period by √2 ≈ 1.41×.
Gravity (g): Period decreases in stronger gravitational fields. A pendulum runs slower on the Moon.
Mass: The period is independent of bob mass — a key result of the equivalence of gravitational and inertial mass.
Amplitude: For small angles (isochronous), period is independent of amplitude. For larger angles, the period increases slightly.

从 T = 2π√(L/g) 可以看出：周期与摆长的平方根成正比；重力加速度越大周期越短；周期与摆球质量无关（引力质量与惯性质量等效的结果）；在小角度范围内周期与振幅无关（等时性）。

7. Damping in SHM

7.1 Types of Damping

In real oscillatory systems, dissipative forces (friction, air resistance) cause the amplitude to decrease over time. This is called damping. There are three regimes:

Light damping (underdamping): The system oscillates with gradually decreasing amplitude. The frequency is slightly less than the natural frequency. Most real systems exhibit light damping.
Critical damping: The system returns to equilibrium in the shortest possible time without oscillating. This is the design goal for car suspension systems, door closers, and seismometers.
Heavy damping (overdamping): The system returns to equilibrium slowly without oscillation. The damping force is so large that the system creeps back to equilibrium.

在实际振动系统中，耗散力（摩擦、空气阻力）导致振幅随时间减小，称为阻尼。三种阻尼类型：欠阻尼（振幅逐渐减小的振荡）、临界阻尼（最快回到平衡位置而不振荡，应用于汽车悬挂和地震仪）、过阻尼（缓慢回到平衡位置）。

7.2 Exponential Decay of Amplitude

For light damping, the amplitude decreases exponentially:

A(t) = A₀ e^−bt/2m

where b is the damping coefficient and m is the mass. The larger the damping coefficient, the faster the amplitude decays.

对于欠阻尼，振幅按指数衰减：A(t) = A₀ e^−bt/2m，其中 b 为阻尼系数。

8. Resonance

8.1 Forced Oscillations

When a periodic driving force is applied to an oscillatory system, the system vibrates at the driving frequency (not its natural frequency). This is called a forced oscillation.

8.2 Resonance Condition

Resonance occurs when the driving frequency equals the natural frequency of the system. At resonance:

The amplitude of oscillation reaches a maximum
The system absorbs energy most efficiently from the driver
The phase difference between driver and oscillator is π/2 (90°)

Famous examples of resonance include:

The collapse of the Tacoma Narrows Bridge (1940) due to wind-induced resonance
The shattering of a wine glass by a singer hitting its resonant frequency
MRI machines using nuclear magnetic resonance for medical imaging
Tuning a radio to match the resonant frequency of an LC circuit

当周期性驱动力作用于振动系统时，系统以驱动频率振动（而非固有频率），称为受迫振动。当驱动频率等于系统固有频率时发生共振：振幅达到最大值，系统从驱动源吸收能量的效率最高，驱动与振动之间的相位差为 π/2。著名例子包括塔科马海峡大桥坍塌、酒杯共振破碎、核磁共振成像（MRI）和收音机调谐。

9. Exam Tips and Common Mistakes

9.1 Calculation Checklist

Always convert displacement to metres before substituting into equations
Ensure angular frequency ω is in rad s⁻¹, not Hz
When using v = ±ω√(A² − x²), the ± sign means direction — use positive for motion away from equilibrium, negative for motion toward it (depending on your sign convention)
Remember that ω = 2π/T = 2πf — many students confuse ω with f
For the pendulum, g is taken as 9.81 m s⁻² unless stated otherwise

9.2 Common Pitfalls

Confusing ω with f: ω is in rad s⁻¹, f is in Hz. Always check which one a question is asking for.
Forgetting the negative sign: a = −ω²x means acceleration is opposite to displacement. If x is positive, a is negative.
Applying T = 2π√(L/g) to large amplitudes: This formula is only valid for small angles (θ < 10°).
Including gravity in the mass-spring period: Gravity affects the equilibrium position, not the period.
Energy conservation: In undamped SHM, total energy is constant. In damped SHM, total energy decreases.

9.3 常见错误提醒

混淆角频率 ω 和频率 f：ω 的单位是 rad/s，f 的单位是 Hz
忘记负号：a = −ω²x 中负号表示加速度与位移反向
大角度使用单摆公式：T = 2π√(L/g) 仅适用于小角度 (θ < 10°)
将重力加入弹簧振子周期公式：重力只影响平衡位置，不影响周期
能量守恒理解：无阻尼 SHM 总能量守恒；有阻尼 SHM 总能量递减

10. Worked Example

Question: A 0.50 kg mass attached to a spring oscillates with an amplitude of 0.12 m. The spring constant is 80 N m⁻¹. Calculate:

(a) The angular frequency and period of oscillation
(b) The maximum speed of the mass
(c) The total energy of the system
(d) The speed of the mass when it is 0.060 m from equilibrium

Solution:

(a) ω = √(k/m) = √(80/0.50) = √160 = 12.6 rad s⁻¹
T = 2π/ω = 2π/12.6 = 0.50 s

(b) v_max = Aω = 0.12 × 12.6 = 1.51 m s⁻¹

(d) v = ω√(A² − x²) = 12.6 × √(0.12² − 0.060²) = 12.6 × √(0.0144 − 0.0036) = 12.6 × √0.0108 = 12.6 × 0.1039 = 1.31 m s⁻¹

例题：一个 0.50 kg 的质量块连接劲度系数为 80 N/m 的弹簧，振幅为 0.12 m。求：(a) 角频率和周期 (b) 最大速度 (c) 总能量 (d) 距离平衡位置 0.060 m 时的速度。

解答：(a) ω = 12.6 rad/s, T = 0.50 s (b) v_max = 1.51 m/s (c) E_total = 0.576 J (d) v = 1.31 m/s

Summary

Simple Harmonic Motion is defined by the relationship a = −ω²x. The key systems you need to master are the mass-spring oscillator (T = 2π√(m/k)) and the simple pendulum (T = 2π√(L/g)). Understanding the energy transformations, graphical representations, and the effects of damping and resonance will serve you well in A-Level Physics examinations.

简谐运动的核心是 a = −ω²x 这一关系。需要熟练掌握的两大系统是弹簧振子 (T = 2π√(m/k)) 和单摆 (T = 2π√(L/g))。理解能量转换、图像表示以及阻尼和共振的影响，将为你的 A-Level 物理考试打下坚实基础。

This article provides a comprehensive overview of Simple Harmonic Motion for A-Level Physics students. Practice with past paper questions to reinforce these concepts, and remember that SHM appears not only in mechanics but also in wave theory, alternating current circuits, and quantum physics.

本文全面介绍了 A-Level 物理中的简谐运动。建议通过历年真题练习巩固这些概念。简谐运动不仅出现在力学中，还广泛应用于波动理论、交流电路和量子物理等领域。