IB Physics: Simple Harmonic Motion Rules and Graphs | IB物理：简谐运动规律与图像

📚 IB Physics: Simple Harmonic Motion Rules and Graphs | IB物理：简谐运动规律与图像

Simple Harmonic Motion (SHM) is a fundamental concept in physics that describes the periodic motion of an object about an equilibrium position. In IB Physics, you are expected to recognise the defining equation, derive and interpret the graphs of displacement, velocity and acceleration, and apply energy conservation to solve problems.

简谐运动（SHM）是物理学中的一个基本概念，描述了物体在平衡位置附近的周期性运动。在IB物理中，你不仅需要识别其定义方程，还要能够推导并解释位移、速度和加速度的图像，并运用能量守恒定律来解决问题。

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1. What is SHM? | 什么是简谐运动？

Simple Harmonic Motion is a special type of periodic motion where the restoring force is directly proportional to the displacement from equilibrium and acts in the opposite direction. This gives rise to a sinusoidal oscillation that repeats with a constant period and amplitude, assuming no energy loss.

简谐运动是一种特殊的周期性运动：回复力与物体偏离平衡位置的位移成正比，且方向始终指向平衡位置。这使得物体以正弦（或余弦）规律振荡，在无能量损失的情况下具有恒定的周期和振幅。

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2. Conditions and Equations of Motion | 简谐运动的条件与运动方程

The defining condition for SHM is that the net restoring force F acting on the object is directly proportional to the displacement x from equilibrium and opposite in direction: F = -kx, where k is the force constant. Using Newton’s second law, this leads to the acceleration equation:

简谐运动的定义条件是：物体所受的回复力 F 与它相对平衡位置的位移 x 成正比，且方向相反，即 F = -kx，其中 k 为力常数。利用牛顿第二定律，可以得到加速度方程：

a = -ω²x

where ω = √(k/m) is the angular frequency. The negative sign indicates that acceleration always points towards the equilibrium position.

其中 ω = √(k/m) 为角频率。负号表示加速度总是指向平衡位置。

The conditions for SHM are as follows:

简谐运动必须满足以下条件：

• The acceleration is always directed towards the equilibrium position.
• Acceleration is proportional to displacement.
• The motion is periodic and can be described by sinusoidal functions.

• 加速度始终指向平衡位置。
• 加速度与位移成正比。
• 运动是周期性的，且可用正弦或余弦函数描述。

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3. Kinematic Equations: Displacement, Velocity, Acceleration | 运动学方程：位移、速度与加速度

The solution to the SHM equation gives the displacement as a function of time. If the motion starts from maximum displacement, we use the cosine form:

简谐运动方程的解给出了位移随时间变化的函数。若从最大位移处开始计时，则使用余弦形式：

x = A cos(ωt + φ)

where A is the amplitude, ω is the angular frequency, and φ is the phase constant. The velocity and acceleration are obtained by differentiating the displacement with respect to time:

其中 A 为振幅，ω 为角频率，φ 为初相位。速度和加速度可通过位移对时间求导得到：

v = -Aω sin(ωt + φ)

a = -Aω² cos(ωt + φ) = -ω²x

The table below summarises the values of x, v, and a at key positions in the oscillation:

下表总结了振动中关键位置处 x、v 和 a 的取值：

Position	Displacement x	Velocity v	Acceleration a
Equilibrium	0	±Aω (maximum)	0
Extreme displacement	±A (maximum)	0	∓Aω² (maximum)

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4. Graphs of SHM | 简谐运动的图像

The graphs of displacement, velocity and acceleration against time reveal their phase relationships. Displacement follows a cosine curve; velocity is a negative sine curve, and acceleration is a negative cosine curve. This means that velocity leads displacement by π/2, and acceleration leads displacement by π (i.e. they are in anti-phase).

位移、速度和加速度随时间变化的图像揭示了它们之间的相位关系。位移曲线为余弦曲线；速度为负的正弦曲线；加速度为负的余弦曲线。这意味着速度比位移超前 π/2，而加速度比位移超前 π（即与位移反相）。

Graphical relationships are directly connected by differentiation:

这些图像之间的关系通过求导直接联系起来：

• The gradient of the displacement-time graph gives the velocity.
• The gradient of the velocity-time graph gives the acceleration.
• The acceleration-time graph is proportional to the negative of the displacement-time graph.

• 位移-时间图像的斜率等于速度。
• 速度-时间图像的斜率等于加速度。
• 加速度-时间图像与位移-时间图像的负值成正比。

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5. Energy in SHM | 简谐运动中的能量

In SHM, energy is continuously exchanged between kinetic energy (KE) and potential energy (PE). The total mechanical energy E remains constant (ignoring friction) and is given by E = ½kA². At displacement x, the kinetic energy is KE = ½mω²(A² − x²) and the potential energy is PE = ½kx².

在简谐运动中，动能和势能之间不断相互转化。忽略摩擦时，总机械能 E 保持不变，且 E = ½kA²。在位移为 x 时，动能 KE = ½mω²(A² − x²)，势能 PE = ½kx²。

Total energy: E = ½kA² = ½mω²A²

When plotted against displacement, KE forms an inverted parabola and PE forms an upright parabola, with their sum constant. When plotted against time, both KE and PE oscillate with twice the frequency of the SHM, but remain in phase with each other (they peak twice every cycle) while always summing to E.

若以位移为横轴，动能曲线为开口向下的抛物线，势能曲线为开口向上的抛物线，二者之和为常数。若以时间为横轴，动能和势能都以简谐运动频率的2倍振荡，但始终一起达到最大值（每个周期内各出现两次），并且总是相加为总能量 E。

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6. Period and Frequency | 周期与频率

The period T of SHM depends on the physical system. For a mass-spring system, the period is given by T = 2π√(m/k), where m is the mass and k is the spring constant. For a simple pendulum swinging with small amplitude, the period is T = 2π√(L/g), where L is the length of the pendulum and g is the gravitational field strength.

简谐运动的周期 T 由系统的物理性质决定。对于弹簧振子，周期为 T = 2π√(m/k)，其中 m 为质量，k 为劲度系数。对于小角度摆动的单摆，周期为 T = 2π√(L/g)，其中 L 为摆长，g 为重力场强度。

Mass-spring: T = 2π√(m/k)

Simple pendulum: T = 2π√(L/g)

Key point: in ideal SHM, the period is independent of the amplitude. This property is called isochronism and is a central assumption in many IB exam problems.

关键点：在理想简谐运动中，周期与振幅无关。这一性质称为等时性，也是IB考题中常见的隐含条件。

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7. Phase and Initial Conditions | 相位与初始条件

The phase constant φ is determined by the initial position and velocity of the oscillator. For example, if the object starts at maximum displacement x = A, then φ = 0. If it starts at the equilibrium position x = 0 and moves in the positive direction, then φ = −π/2. These initial conditions allow you to write the complete equation of motion.

初相位 φ 由振子的初始位置和初始速度决定。例如，若物体从最大位移 x = A 处开始运动，则 φ = 0；若从平衡位置 x = 0 开始并沿正方向运动，则 φ = −π/2。通过初始条件，你可以写出完整的运动方程。

Phase differences between the three kinematic quantities are fixed:

三个运动学量之间的相位差是固定的：

• Velocity leads displacement by π/2 (quarter cycle).
• Acceleration leads velocity by π/2.
• Acceleration is in anti-phase with displacement (phase difference π).

• 速度超前位移 π/2（四分之一周期）。
• 加速度超前速度 π/2。
• 加速度与位移反相（相位差为 π）。

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8. Exam Tips and Common Misconceptions | 考试技巧与常见误区

To score well in IB Physics on SHM, remember these key exam tips: at the maximum displacement (amplitude), velocity is zero and acceleration is maximum; at the equilibrium position, velocity is maximum and acceleration is zero. This immediately allows you to sketch and interpret graphs without lengthy derivation.

要在IB物理简谐运动部分取得好成绩，请牢记以下考试技巧：在最大位移（振幅）处，速度为零，加速度最大；在平衡位置处，速度为最大，加速度为零。利用这一规律，可以快速绘制或解读图像，无需冗长推导。

Common misconceptions include thinking that the period of a pendulum depends on its mass or amplitude. For ideal SHM with small angles, the period depends only on the length of the pendulum and the gravitational field. Also, students sometimes confuse the graphs of x, v, and a; using the gradient relationships will help you verify the correct shapes and phase offsets.

常见误区包括认为单摆的周期与质量或振幅有关。实际上，在小角度理想简谐运动中，周期只取决于摆长和重力场强度。此外，学生常混淆 x、v 和 a 的图像；利用斜率关系可以帮助你验证正确的曲线形状和相位偏移。

Finally, always check energy conservation in numerical problems: total energy E = ½kA² must equal the sum of kinetic and potential energy at any position. This provides a powerful cross-check for answers involving speed or amplitude.

最后，在数值计算中务必检查能量守恒：总能量 E = ½kA² 必须等于任意位置处动能与势能之和。这是检验速度或振幅相关答案最有效的方法。

Published by TutorHao | IB Physics Revision Series | aleveler.com

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