Observing Oscillations | 观察振动

📚 Observing Oscillations | 观察振动

Oscillations appear throughout the CIE A-Level Physics syllabus, from simple pendulums to resonance in bridges and electrical circuits. Observing oscillations accurately means measuring displacement, period, frequency and energy changes, and using those observations to test the equations of simple harmonic motion.

振动贯穿 CIE A-Level 物理课程，从单摆到桥梁和电路中的共振。准确观察振动意味着测量位移、周期、频率和能量变化，并利用这些观察来检验简谐运动方程。

1. What Is an Oscillation? | 什么是振动

An oscillation is a repeated back-and-forth motion about a fixed equilibrium position. After a system is displaced from equilibrium and released, a restoring force pulls it back, causing it to move through the equilibrium position to the opposite side.

振动是围绕固定平衡位置的重复往返运动。当系统偏离平衡位置并被释放后，回复力将其拉回，使其穿过平衡位置到达另一侧。

Common examples include a pendulum swinging, a mass bouncing on a spring, a vibrating tuning fork, and alternating current in an electrical circuit.

常见例子包括摆动的单摆、弹簧上弹跳的质量块、振动的音叉以及电路中的交流电。

When a system oscillates without an external periodic driving force, it is called a free oscillation, and it occurs at the natural frequency f₀ of the system.

当系统在没有外部周期性驱动力的情况下振动时，称为自由振动，它以系统的固有频率 f₀ 发生。

2. Terms Used to Describe Oscillations | 描述振动的术语

Displacement x is the distance from the equilibrium position at any instant. Amplitude A is the maximum displacement from equilibrium.

位移 x 是任意时刻距离平衡位置的距离。振幅 A 是离开平衡位置的最大位移。

Period T is the time for one complete oscillation, while frequency f is the number of complete oscillations per second: f = 1/T. Angular frequency ω is related by ω = 2πf = 2π/T.

周期 T 是一次完整振动所需的时间，而频率 f 是每秒完整振动的次数：f = 1/T。角频率 ω 满足 ω = 2πf = 2π/T。

Quantity	Symbol	Unit	Meaning
Displacement	x	m	distance from equilibrium
Amplitude	A	m	maximum displacement
Period	T	s	time for one cycle
Frequency	f	Hz	cycles per second
Angular frequency	ω	rad s⁻¹	ω = 2πf = 2π/T

3. Simple Harmonic Motion: The Defining Equation | 简谐运动：定义方程

Simple harmonic motion, SHM, occurs when the acceleration of an object is directly proportional to its displacement from equilibrium, but always directed towards equilibrium. This is written as:

当物体的加速度与其离开平衡位置的位移成正比，且始终指向平衡位置时，就发生简谐运动。这可以写成：

a = −ω²x

The negative sign shows that acceleration and displacement are always in opposite directions. If this condition is satisfied, the displacement varies sinusoidally with time.

负号表示加速度与位移始终方向相反。如果满足此条件，位移将随时间呈正弦变化。

Two standard solutions are x = A cos(ωt) and x = A sin(ωt), depending on whether the oscillation starts at maximum displacement or at equilibrium.

两个标准解为 x = A cos(ωt) 和 x = A sin(ωt)，取决于振动是从最大位移处还是从平衡位置开始。

4. Observing a Mass-Spring System | 用弹簧-质量系统观察振动

A mass hanging from a spring is a convenient system for observing SHM in the laboratory. For small displacements, the period is independent of amplitude and is given by:

弹簧下悬挂质量块是实验室观察简谐运动的方便系统。对于小位移，周期与振幅无关，并由下式给出：

T = 2π√(m/k)

To observe the relationship, suspend a spring vertically, add a mass, displace it by a small amount, release it, and time at least 10 complete oscillations. Repeat for several different masses.

为了观察这一关系，将弹簧竖直悬挂，加上质量块，将其小幅拉离后释放，并至少计时 10 次完整振动。对多个不同质量重复实验。

A graph of T² against m should be a straight line through the origin with gradient 4π²/k.

T² 对 m 的图像应为一条过原点的直线，斜率为 4π²/k。

T² = (4π²/k)m

The straight-line graph shows that T² is proportional to m, which is a key prediction for the mass-spring oscillator.

这条直线图像表明 T² 与 m 成正比，这是弹簧振子的关键预测。

5. Observing a Simple Pendulum | 用单摆观察振动

For a simple pendulum with small angular amplitude, the oscillation approximates simple harmonic motion. The period depends only on length L and gravitational field strength g:

对于小角度振幅的单摆，其振动近似为简谐运动。周期只取决于摆长 L 和重力场强度 g：

T = 2π√(L/g)

Keep the angular amplitude below about 10° so that sin θ ≈ θ remains a good approximation. Measure the length L from the pivot to the centre of mass of the bob.

将角振幅保持在约 10° 以下，使 sin θ ≈ θ 仍为良好近似。从悬挂点量到摆球质心测量摆长 L。

Time 10 or 20 complete swings and divide by the number of swings to reduce reaction-time uncertainty. Plot T² against L; the gradient is 4π²/g.

计时 10 或 20 次完整摆动并除以摆动次数，以减小反应时间不确定度。绘制 T² 对 L 的图像；其斜率为 4π²/g。

T² = (4π²/g)L

This allows g to be determined experimentally as g = 4π²/gradient.

由此可以通过实验测定 g，即 g = 4π²/斜率。

6. Using Sensors and Data Loggers | 使用传感器与数据记录仪

Modern oscillation observations often use motion sensors, force sensors, or light gates connected to a data logger. A position sensor can record displacement at a high sampling rate, producing a smooth x-t graph.

现代振动观察通常使用连接到数据记录仪的运动传感器、力传感器或光门。位置传感器可以以高采样率记录位移，生成平滑的 x-t 图像。

The data logger reduces human reaction-time error and allows the software to calculate velocity and acceleration by numerical differentiation of the displacement data.

数据记录仪减少了人的反应时间误差，并允许软件通过对位移数据进行数值微分来计算速度和加速度。

Use a sampling rate at least ten times the oscillation frequency — 至少使用振动频率十倍的采样率。
Check that the sensor does not interfere with the motion — 检查传感器不会干扰运动。
Record several complete cycles to check repeatability — 记录多个完整周期以检查可重复性。

7. Graphical Observation of Oscillations | 振动的图像观察

For an oscillator described by x = A sin(ωt), the velocity and acceleration curves are also sinusoidal but with important phase differences.

对于由 x = A sin(ωt) 描述的振子，速度和加速度曲线也是正弦形，但具有重要的相位差。

x = A sin(ωt) v = ωA cos(ωt) a = −ω²A sin(ωt)

The velocity is π/2 ahead of displacement, while acceleration is π radians out of phase with displacement. In other words, when displacement is zero, acceleration is zero; when displacement is maximum, acceleration is maximum in the opposite direction.

速度比位移超前 π/2，而加速度与位移反相 π 弧度。换句话说，当位移为零时，加速度为零；当位移最大时，加速度在相反方向达到最大。

An acceleration-displacement graph is the most direct way to test for SHM: it should be a straight line through the origin with a negative gradient equal to −ω².

加速度-位移图像是检验简谐运动最直接的方法：它应为一条过原点、斜率为 −ω² 的负斜率直线。

8. Energy Changes During Oscillations | 振动过程中的能量变化

In undamped SHM, energy is continuously exchanged between kinetic energy and potential energy. Kinetic energy is maximum at equilibrium, while potential energy is maximum at the extremes of motion.

在无阻尼简谐运动中，能量在动能和势能之间不断转换。动能最大处为平衡位置，势能最大处为运动端点。

For a mass-spring oscillator, the total energy remains constant and is given by:

对于弹簧振子，总能量保持恒定，并由下式给出：

E = ½mω²A²

Observing energy changes can be done by video analysis or by measuring speed at the equilibrium position with a light gate. The maximum speed v_max = ωA occurs at equilibrium.

可以通过视频分析或用光门测量平衡位置处的速度来观察能量变化。最大速度 v_max = ωA 出现在平衡位置。

9. Damping and Its Observation | 阻尼及其观察

Real oscillations lose energy to air resistance or internal friction, so the amplitude decreases gradually. This is called damping.

实际振动会因空气阻力或内摩擦而损失能量，因此振幅逐渐减小。这称为阻尼。

In light damping, the oscillator still completes many cycles and the amplitude decays approximately exponentially. The period remains almost unchanged.

在轻度阻尼下，振子仍能完成许多次振动，振幅近似指数衰减。周期几乎保持不变。

A = A₀e^(−bt/2m)

Critical damping brings the system back to equilibrium in the shortest possible time without oscillating. Heavy damping also avoids oscillation but returns more slowly.

临界阻尼使系统在最短时间内回到平衡位置且不发生振动。重阻尼也不发生振动，但返回速度更慢。

Light damping: exponential decay of amplitude — 轻度阻尼：振幅指数衰减。
Critical damping: fastest return without oscillation — 临界阻尼：无振动的最快返回。
Heavy damping: slow return without oscillation — 重阻尼：无振动的缓慢返回。

10. Forced Oscillations and Resonance | 受迫振动与共振

When an external periodic force drives a system, the system oscillates at the driving frequency rather than its natural frequency. This is a forced oscillation.

当外部周期性驱动力作用于系统时，系统以驱动力频率而非其固有频率振动。这就是受迫振动。

Resonance occurs when the driving frequency equals the natural frequency of the system. At resonance, the amplitude of oscillation becomes very large, and the system absorbs energy most efficiently.

当驱动力频率等于系统的固有频率时，就会发生共振。在共振时，振动幅度变得很大，系统最有效地吸收能量。

The phase difference between driver and oscillator is about π/2 at resonance. Increased damping reduces the maximum amplitude and broadens the resonance peak.

共振时驱动力与振子的相位差约为 π/2。增大阻尼会降低最大振幅并使共振峰变宽。

Resonance can be observed using Barton’s pendulums or a vibrating string driven by a signal generator. Everyday examples include radio tuning, microwave heating and the oscillation of bridges in strong winds.

共振可以用巴顿摆或由信号发生器驱动的振动弦来观察。日常例子包括无线电调谐、微波加热以及强风中桥梁的振动。

11. Practical Precautions and Uncertainty | 实验注意事项与不确定度

Whenever you observe oscillations in the laboratory, small changes in method can significantly reduce uncertainty.

在实验室观察振动时，方法的微小改进都能显著降低不确定度。

Time at least 10 oscillations, not one — 至少计时 10 次振动，而不是一次。
Keep the amplitude small so SHM equations apply — 保持小振幅，使简谐运动方程适用。
Measure length from the pivot to the centre of the bob for a pendulum — 单摆要从悬挂点量到摆球质心。
Use a fiduciary marker at the equilibrium position to improve timing consistency — 在平衡位置设置参考标记，提高计时一致性。
Repeat readings and calculate mean periods — 重复读数并计算平均周期。

For a pendulum, the percentage uncertainty in T is the same as the percentage uncertainty in the total timed interval, because the number of swings is counted exactly.

对于单摆，T 的百分比不确定度与总计时区间的百分比不确定度相同，因为摆动次数是精确计数的。

12. Exam Tips for Oscillation Questions | 振动题考试技巧

CIE exam questions often ask you to describe an experiment to investigate SHM. A strong answer should state that you would use a motion sensor to record displacement, plot acceleration against displacement, and check for a straight line through the origin with a negative gradient.

CIE 考试题常要求描述一个研究简谐运动的实验。完整答案应说明使用运动传感器记录位移，绘制加速度-位移图像，并检验是否为过原点且斜率为负的直线。

Do not claim that a pendulum of any amplitude is SHM; specify that the angle must be small. Do not confuse frequency f with angular frequency ω.

不要声称任何振幅的单摆都是简谐运动；必须说明角度很小。不要混淆频率 f 与角频率 ω。

When reading a displacement-time graph, remember that the gradient gives velocity, and the gradient of the velocity-time graph gives acceleration.

阅读位移-时间图像时，记住斜率给出速度，速度-时间图像的斜率给出加速度。

Published by TutorHao | Physics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Observing Oscillations | 观察振动