IB Physics: Simple Harmonic Motion – Key Concepts and Model Analysis | IB物理：简谐运动考点与模型精讲

📚 IB Physics: Simple Harmonic Motion – Key Concepts and Model Analysis | IB物理：简谐运动考点与模型精讲

Simple Harmonic Motion (SHM) is one of the most important topics in IB Physics, appearing in both Standard Level and Higher Level examinations. It describes any oscillation where the net restoring force is proportional to the negative displacement, leading to sinusoidal motion in time.

简谐运动是IB物理最重要的考点之一，出现在标准级别和高级别的考试中。它描述了净回复力与位移的负值成正比的振荡形式，从而在时间上呈现正弦规律。

---

1. Definition and Conditions | 定义与条件

The defining mathematical condition of SHM is a = -ω²x, where a is acceleration, x is displacement from equilibrium, and ω is the angular frequency. This equation shows that acceleration is always opposite to displacement and proportional to it.

简谐运动的定义式是 a = -ω²x，其中 a 是加速度，x 是相对平衡位置的位移，ω 是角频率。该式表明加速度始终与位移方向相反，并且大小与位移成正比。

For a physical system to undergo SHM, two conditions must be met: there must be a restoring force following Hooke’s law, and the system must have inertia that carries it past equilibrium. Common examples are a mass on a spring and a small-angle pendulum.

物理系统要做简谐运动，必须满足两个条件：存在遵循胡克定律的回复力，并且系统具有使它能越过平衡位置的惯性。常见例子有弹簧振子和小角度单摆。

• Equilibrium position is the point where the net force is zero.

  平衡位置是合力为零的位置。

• Amplitude A is the maximum displacement from equilibrium.

  振幅 A 是离开平衡位置的最大位移。

• Period T is the time taken for one complete cycle of oscillation.

  周期 T 是完成一次全振动所需的时间。

• Frequency f is the number of oscillations per second, with f = 1/T.

  频率 f 是每秒振动的次数，f = 1/T。

---

2. Kinematic Equations | 运动学方程

Taking displacement as x = A cos(ωt + φ), where φ is the phase constant, the velocity and acceleration are obtained by differentiation: v = -Aω sin(ωt + φ) and a = -Aω² cos(ωt + φ).

若位移表达式为 x = A cos(ωt + φ)，其中 φ 是初相位，则对时间求导可得速度 v = -Aω sin(ωt + φ) 和加速度 a = -Aω² cos(ωt + φ)。

The phase angle (ωt + φ) determines the instantaneous state of motion, such as position, velocity direction, and acceleration direction. Two SHMs with the same frequency can have a constant phase difference.

相位角 (ωt + φ) 决定了运动的瞬时状态，包括位置、速度方向和加速度方向。两个同频率的简谐运动可以具有恒定的相位差。

At the equilibrium position, x = 0 and speed is maximum: v_max = Aω. At the extreme positions, x = ±A, speed is zero and acceleration magnitude is maximum: a_max = Aω².

在平衡位置，x = 0，速度最大：v_max = Aω；在极端位置，x = ±A，速度为零，而加速度大小最大：a_max = Aω²。

Initial conditions are used to find A and φ. For example, if the oscillator starts from maximum displacement, then φ = 0; if it starts from equilibrium moving in the positive direction, then φ = -π/2.

初始条件用于确定 A 和 φ。例如，若振子从最大位移处开始，则 φ = 0；若从平衡位置向正方向运动开始，则 φ = -π/2。

---

3. Dynamics and Restoring Force | 动力学与回复力

For a mass attached to a spring, Hooke’s law gives F = -kx. Applying Newton’s second law, ma = -kx, so a = -(k/m)x. Comparing with a = -ω²x gives ω² = k/m.

对于连接在弹簧上的物体，胡克定律给出 F = -kx。由牛顿第二定律 ma = -kx，可得 a = -(k/m)x。与 a = -ω²x 比较，得到 ω² = k/m。

The direction of the restoring force is always towards equilibrium. Its magnitude grows linearly with displacement, which is the dynamic hallmark of SHM.

回复力的方向始终指向平衡位置。它的大小随位移线性增大，这是简谐运动的动力学特征。

In the vertical spring-mass system, gravity shifts the equilibrium position to x₀ = mg/k. The force law relative to this new equilibrium is still F = -kx’, where x’ is displacement from the new equilibrium.

在竖直弹簧振子中，重力使平衡位置移动到 x₀ = mg/k。相对这个新平衡位置，力的规律依然是 F = -kx’，其中 x’ 是相对新平衡位置的位移。

---

4. Period and Frequency | 周期与频率

The angular frequency is related to the period by ω = 2π/T. For a spring-block system, the period is T = 2π√(m/k), independent of amplitude.

角频率与周期的关系为 ω = 2π/T。对于弹簧振子，周期 T = 2π√(m/k)，与振幅无关。

For a simple pendulum with small amplitude, the period is T = 2π√(L/g), where L is the length of the pendulum and g is gravitational field strength.

对于小角度单摆，周期 T = 2π√(L/g)，其中 L 是摆长，g 是重力场强度。

Key facts to remember: the spring-mass period increases with mass and decreases with spring constant; the pendulum period increases with length and decreases with g, but does not depend on mass or amplitude (for small angles).

需要记住的关键结论：弹簧振子的周期随质量增大而增大，随劲度系数增大而减小；单摆周期随摆长增大而增大，随 g 增大而减小，但与质量和小振幅无关。

In experiments, the period is often determined by timing multiple oscillations, say 20 cycles, then dividing by the number of cycles to reduce timing error.

在实验中，通常通过计时多个振荡周期（例如20次全振动）再除以次数来测量周期，以减小计时误差。

---

5. Energy in SHM | 简谐运动的能量

The total mechanical energy in SHM is constant: E = ½ kA² = ½ mω²A². It is conserved if no damping exists.

简谐运动中的总机械能守恒：E = ½ kA² = ½ mω²A²。在没有阻尼的情况下保持不变。

At any displacement x, the kinetic energy is K = ½ mω²(A² – x²) and the potential energy is U = ½ kx². The sum always equals E.

在任意位移 x 处，动能 K = ½ mω²(A² – x²)，势能 U = ½ kx²。两者之和恒等于 E。

At the equilibrium position, all energy is kinetic: K_max = ½ mω²A². At the extremes, all energy is potential: U_max = ½ kA².

在平衡位置，能量全部为动能：K_max = ½ mω²A²；在极端位置，能量全部为势能：U_max = ½ kA²。

Energy conversion can also be visualized on a graph of K and U versus time: both oscillate at twice the frequency of the motion, while the total energy remains a horizontal line.

动能和势能随时间变化的图像显示它们以运动频率的两倍振荡，而总能量则保持水平直线。

---

6. Graphical Representation | 图像分析

The x-t graph is a cosine curve, the v-t graph is a negative sine curve, and the a-t graph is a negative cosine curve. Velocity leads displacement by π/2, and acceleration is in antiphase with displacement.

x-t 图像是余弦曲线，v-t 图像是负正弦曲线，a-t 图像是负余弦曲线。速度比位移超前 π/2，加速度与位移反相。

On these graphs, the slope of x-t gives velocity, and the slope of v-t gives acceleration. Make sure to check units and scale before reading any values.

在图像中，x-t 图的斜率表示速度，v-t 图的斜率表示加速度。读取数据前务必确认单位与标度。

Energy-time graphs show K and U oscillating between 0 and E, each completing two cycles of oscillation during one displacement cycle. The frequency of energy oscillation is 2f.

能量-时间图像显示 K 和 U 在 0 与 E 之间振荡，每个位移周期内各完成两个循环。能量振荡频率为 2f。

---

7. Typical Models: Spring and Pendulum | 典型模型：弹簧振子与单摆

The horizontal spring-mass model is the simplest: the equilibrium is at the natural length of the spring, and motion is purely horizontal. All SHM equations apply directly.

水平弹簧振子模型最简单：平衡位置在弹簧原长处，运动纯水平，所有简谐运动方程直接适用。

The vertical spring-mass model has gravity shifting the equilibrium downward by x₀ = mg/k. If you measure displacement from this new equilibrium, the motion is identical to the horizontal case.

竖直弹簧振子模型中，重力使平衡位置下移 x₀ = mg/k。若从新平衡位置测量位移，其运动与水平情形完全一致。

The simple pendulum is SHM only for small angular displacements, where θ < about 10° (0.17 rad). The restoring force is the tangential component of weight: F = -mg sinθ ≈ -mgθ = -(mg/L)x.

单摆仅在小角度（θ 约小于10°，即0.17 rad）下才做简谐运动。回复力是重力的切向分量：F = -mg sinθ ≈ -mgθ = -(mg/L)x。

For a pendulum, the mass of the bob and the amplitude do not affect the period, but the length does. This is exploited in pendulum clocks and in measuring g.

对于单摆，摆球质量和振幅不影响周期，但摆长影响周期。这被用于摆钟设计和测量 g 的实验。

---

8. Damping, Forced Oscillations and Resonance | 阻尼、受迫振动与共振

Damping is the loss of energy from an oscillating system due to resistive forces such as friction and air resistance. As a result, the amplitude decays exponentially with time.

阻尼是由摩擦、空气阻力等阻力导致系统能量损失的现象。因此，振幅随时间呈指数衰减。

In lightly damped systems, the period is slightly larger than the undamped period, but for small damping the change is often negligible. The total energy decreases continuously, converting into thermal energy.

在弱阻尼系统中，周期略大于无阻尼周期，但小阻尼时变化常可忽略。总能量不断减少，转化为内能。

A forced oscillation occurs when an external periodic driving force is applied. The system then oscillates at the driving frequency, which may differ from its natural frequency.

受迫振动发生在施加周期性外部驱动力时。此时系统以驱动频率振动，该频率可能与固有频率不同。

Resonance is the condition when the driving frequency equals the natural frequency. The amplitude reaches a maximum, and energy transfer into the system is most efficient.

共振是指驱动频率等于系统固有频率的状态。此时振幅达到最大，能量输入效率也最高。

Resonance can be useful, such as in tuning a radio or in microwave heating, but it can also be destructive, as in the famous Tacoma Narrows bridge collapse. Understanding resonance helps in designing safe structures.

共振可以是有益的，如收音机

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

更多咨询请联系16621398022（同微信）