Oscillations & Simple Harmonic Motion
A-Level Physics Topic 17 15:13 English narration · English + 中文 subtitles burned in
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In 1940, a brand-new bridge opened across the Tacoma Narrows — long, elegant, and, as it turned out, dangerous.
1940年，美国塔科马海峡上一座新桥落成——它修长、优雅，但事实证明，它很危险。
Within months, an ordinary wind set it heaving and twisting, harder and harder, until people ran for their lives.
几个月后，一阵普通的风让它起伏、扭动，越来越剧烈，人们不得不仓皇逃命。
How could a gentle wind do that?
为什么和缓的风能做到这一点？
The wind was never strong.
风其实并不强。
It simply pushed at just the right rhythm, again and again, matching the bridge's own natural rhythm — and so the swaying grew, and grew.
它只是以恰好合适的节奏一次又一次地推， 与桥自身的固有节奏一致——于是摇晃越来越大。
<slow>Until the whole bridge tore itself apart.</slow> No new force ever arrived — only rhythm.
直到整座桥把自己撕裂。 没有新的力出现，只有节奏。
That is the power of oscillations.
这就是振动的力量。
And it all begins with one simple, repeating motion.
而这一切都始于一种简单的往复运动。
Today we lay the foundation of oscillations: simple harmonic motion.
今天我们来打好振动的基础——简谐运动。
The one rule that defines it, the words you need, and the graphs the exam loves.
定义它的那一条规则、你需要的术语， 以及考试最爱考的图像。
Let's begin.
让我们开始吧。
An oscillation is any motion that repeats back and forth about a central point.
振动是指围绕一个中心点来回重复的运动。
We call that centre the equilibrium position.
这个中心叫做平衡位置。
Pull the object away, and something always pulls it back. Watch these three pictures.
把物体拉开，总有一个力把它拉回来。
On the left, a mass bounces on a spring. In the middle, a dot travels around a circle. On the right, a graph traces a smooth wave.
看这三幅图：左边，物体在弹簧上上下振动； 中间，一个点绕圆周运动；右边，图像画出一条平滑的波。
<slow>They are three views of one single motion</slow>, and they stay perfectly in step.
它们是同一种运动的三种视角，而且始终完全同步。
Here is the rule that defines simple harmonic motion. The acceleration equals minus omega squared, times the displacement. Two ideas hide inside it.
这就是定义简谐运动的规则：加速度等于负的欧米伽平方，乘以位移。
First, the acceleration is proportional to the displacement: go twice as far from the centre, and the pull back is twice as strong.
里面藏着两个要点。 第一，加速度与位移成正比：离中心远一倍，回拉的力就强一倍。
Second, and this is the key, that minus sign.
第二，也是关键，那个负号。
It means the acceleration always points back toward the centre. Every single time.
它表示加速度始终指向中心，每一次都是。
That is what keeps the motion repeating forever.
这正是运动不断重复的原因。
Look at a mass on a spring, pulled above the equilibrium line and pushed below it.
看弹簧上的物体，被拉到平衡线上方，或被推到下方。
In each case the displacement arrow points away from the centre, but the acceleration arrow always points back toward equilibrium — opposite in sign to the displacement.
每一次，位移箭头都指向远离中心， 但加速度箭头始终指回平衡位置——与位移符号相反。
That is the whole SHM condition in one picture: acceleration proportional to displacement, and always restoring.
这就是简谐运动条件的一幅图：加速度与位移成正比，并且始终是恢复的。
If the arrows ever pointed the same way, the motion would run away, not oscillate.
如果箭头曾经指向同一方向，运动就会失控，而不是振动。
Many systems near a stable equilibrium do the same thing.
许多靠近稳定平衡的系统都会这样做。
A mass on a spring.
弹簧上的物体。
A pendulum on a small swing — like the pendulum clock that keeps time with simple harmonic motion.
小角度摆动的单摆—— 就像用简谐运动计时的摆钟。
A floating block pushed down and released.
被按下再放开的浮块。
The charge on a capacitor in an L C circuit, swinging between electric and magnetic form.
电容在电感电容电路中的电荷，在电场与磁场形式之间来回交换。
Even atoms in a solid, jiggling about their lattice points.
甚至固体中的原子，在晶格点附近抖动。
Different hardware, same defining equation: acceleration equals minus omega squared times displacement.
硬件不同，定义式相同： 加速度等于负的欧米伽平方乘以位移。
Before we go further, five words, and you must use them precisely.
在继续之前，先记住五个词，而且必须准确使用。
Displacement is how far the object is from the centre right now; it can be positive or negative.
位移是物体此刻离中心的距离，可正可负。
Amplitude is the largest displacement it ever reaches, and it is always positive.
振幅是它所能到达的最大位移，永远为正。
The period is the time for one full swing.
周期是完成一次完整振动所用的时间。
The frequency is how many swings happen each second.
频率是每秒振动的次数。
And the angular frequency, omega, ties them all together.
而角频率欧米伽把它们全部联系在一起。
Find omega first, and every other quantity follows.
先求出欧米伽，其余每个量都能随之得到。
Omega links three exam quantities.
欧米伽把三个考试量连在一起。
Omega equals two pi over the period, and also two pi times the frequency.
欧米伽等于二派除以周期，也等于二派乘以频率。
So the period is two pi over omega, and the frequency is omega over two pi.
所以周期等于二派除以欧米伽，频率等于欧米伽除以二派。
Given any one of omega, f, or T, you can find the other two.
只要知道欧米伽、频率或周期中的任意一个，就能求出另外两个。
Unit of omega: radians per second. Unit of frequency: hertz.
欧米伽的单位是弧度每秒，频率的单位是赫兹。
Always work in radians when omega is in play — degree mode will wreck every sine and cosine you touch.
凡是用到欧米伽，就用弧度——角度制会毁掉你碰到的每一个正弦和余弦。
Now watch how displacement, velocity, and acceleration change over time.
现在看位移、速度和加速度如何随时间变化。
Displacement follows a sine curve.
位移沿正弦曲线变化。
Velocity is a quarter of a cycle ahead; it is largest when the object races through the centre.
速度超前四分之一个周期；当物体高速冲过中心时最大。
Acceleration is the exact opposite of displacement, turned upside down, and biggest at the two ends.
加速度与位移完全相反， 上下颠倒，在两端最大。
<slow>Notice the pattern.</slow> Each step shifts the wave forward by a quarter of a cycle.
注意这个规律：每前进一步，波形就向前移动四分之一个周期。
Learn that pattern once, and these graphs become easy marks.
记住这个规律，这些图像就是轻松得分。
That quarter-cycle shift has a name: phase difference.
那个四分之一周期的错位有个名字：相位差。
Phase difference is the fraction of a cycle by which one oscillation leads or lags another, measured in radians.
相位差是一个振动超前或落后另一个 振动的周期分数，用弧度量度。
A quarter cycle apart is a phase difference of pi over two.
相差四分之一周期，相位差就是二分之派。
Velocity leads displacement by pi over two.
速度超前位移二分之派。
Acceleration is out of phase with displacement by pi — a full half cycle, one hundred and eighty degrees.
加速度与位移相位差为派——整整半个周期，也就是一百八十度。
If the particle starts at the centre moving positive, use sine: x equals x-nought sine omega t.
若粒子从中心沿正方向出发，用正弦：位移等于振幅乘以正弦欧米伽 t。
If it starts at the extreme, use cosine: x equals x-nought cosine omega t.
若从端点出发，用余弦：位移等于振幅乘以余弦欧米伽 t。
Choose the form that matches the start conditions.
按起始条件选形式。
Now plot acceleration against displacement — not against time.
现在画加速度对位移的图——不是对时间。
You get a straight line through the origin with a negative gradient.
你得到一条过原点、斜率为负的直线。
That gradient is minus omega squared.
这个斜率就是负的欧米伽平方。
So from the graph alone you can read omega: take the absolute value of the gradient, then square-root it.
所以单从图就能读出欧米伽：取斜率的绝对值，再开方。
Period follows as two pi over omega.
周期等于二派除以欧米伽。
This is a favourite exam pattern — they show the line, you extract omega and T.
这是常考题型——他们给出直线，你求出欧米伽和周期。
Remember: negative slope means restoring acceleration, the SHM signature.
记住：负斜率表示恢复加速度，那是简谐运动的标志。
On a displacement–time graph, simple harmonic motion is a smooth sine or cosine wave.
在位移–时间图上，简谐运动是平滑的正弦或余弦波。
Amplitude is the peak height, x-nought.
振幅是波峰高度，也就是最大位移。
Period is the time from crest to next crest, or any full repeat.
周期是从波峰到下一个波峰，或任意一次完整重复的时间。
Differentiate once and you get velocity: maximum speed is omega times the amplitude, as the particle races through equilibrium.
微分一次得到速度： 最大速率等于欧米伽乘以振幅，出现在粒子冲过平衡位置时。
Differentiate again and acceleration comes back as minus omega squared times x — the defining equation, recovered from the wave.
再微分一次， 加速度又变回负的欧米伽平方乘以位移——定义式从波形中重新出现。
Let's put the rule to work.
让我们运用这条规则。
A mass on a spring oscillates with an amplitude of five centimetres, at a frequency of two point five hertz. Find its largest acceleration.
一个弹簧上的物体做振动，振幅为五厘米，频率为二点五赫兹， 求它的最大加速度。
First, get the angular frequency, omega.
首先，求角频率欧米伽。
Omega is two pi times the frequency, two pi times two point five, which comes to fifteen point seven radians per second.
欧米伽等于二派乘以频率，即二派乘以二点五， 得到十五点七弧度每秒。
Next: when is the acceleration largest?
接着：加速度何时最大？
At the extremes, where its size equals omega squared times the amplitude.
在两端，那里它的大小等于欧米伽平方乘以振幅。
Put the numbers in — fifteen point seven squared, times zero point zero five — and you get about twelve metres per second squared.
代入数字——十五点七的平方，乘以零点零五——约等于十二米每二次方秒。
One more tool before we talk about energy.
在讨论能量之前，还有一个工具。
This equation gives the speed at any position, without needing the time at all.
这个公式给出任意位置的速度，而完全不需要时间。
At the centre, the object moves fastest. At the two ends, it stops, just for an instant, before turning back.
在中心，物体最快；在两端，它会停顿一瞬，然后折返。
And a changing speed means changing energy — so where does that energy go?
速度在变，能量也在变—— 那么能量去了哪里呢？
With no damping the total stays constant: this is conservation of energy.
没有阻尼时，总能量保持不变：这就是能量守恒。
Use it.
用起来。
Same oscillation: amplitude zero point zero five metres, omega fifteen point seven radians per second.
同样的振动：振幅零点零五米，欧米伽十五点七弧度每秒。
Find the speed when the displacement is zero point zero three metres — same oscillation, a new question.
求位移为零点零三米时的速率——同一振动，新问题。
Speed equals omega times the square root of amplitude squared minus displacement squared.
速率等于欧米伽乘以振幅平方减位移平方的平方根。
Put the numbers in: fifteen point seven times the square root of zero point zero five squared minus zero point zero three squared.
代入数字：十五点七乘以零点零五的平方减零点零三的平方的平方根。
That square root is zero point zero four, so the speed is about zero point six three metres per second.
那个平方根是零点零四，所以速率约为零点六三米每秒。
Both signs are allowed: the particle passes each position twice each cycle, once each way.
正负号都可以：粒子每个周期两次经过同一位置，方向相反。
In simple harmonic motion, energy is never lost — it only changes form.
在简谐运动中，能量从不损失——只会改变形式。
At the centre, the object moves fastest, so all its energy is kinetic.
在中心，物体最快，能量全是动能。
At the ends it stops, and all its energy is potential — stored in the spring, or in the height.
在两端，物体停下，能量全是势能——储存在弹簧里，或储存在高度中。
Watch the two bars trade back and forth. Kinetic becomes potential, potential becomes kinetic, but the total stays exactly the same.
看这两个能量条来回交换：动能变势能，势能变动能，但总能量始终不变。
Now plot energy against displacement.
现在画出能量对位移的图。
Kinetic energy is largest at the centre and zero at the ends — a downward curve.
动能在中心最大，在两端为零——一条向下的曲线。
Potential energy is the mirror image: zero at the centre, largest at the ends.
势能正好相反：在中心为零，在两端最大。
Add them together, and the total is a flat line — the same at every position.
把它们相加，总能量是一条水平线—— 在每个位置都相同。
That flat line is the whole idea: the energy is conserved.
这条水平线就是核心：能量守恒。
Write the total energy as a formula.
把总能量写成公式。
Using the maximum speed, omega times the amplitude, the total energy is one half m omega squared x-nought squared.
用最大速率——欧米伽乘以振幅——总能量等于二分之一乘以质量 乘以欧米伽平方乘以振幅的平方。
Two facts for the exam.
考试记住两点。
First, total energy is proportional to the square of the amplitude: double the amplitude and you store four times the energy.
第一，总能量正比于振幅的平方： 振幅翻倍，储存的能量就是四倍。
Second, it is also proportional to omega squared.
第二，它也正比于欧米伽的平方。
Kinetic energy at any position is one half m omega squared times open bracket amplitude squared minus x squared close bracket.
任意位置的动能是二分之一乘以质量乘以欧米伽平方，再乘以振幅平方减位移平方。
Potential energy is one half m omega squared x squared.
势能是二分之一乘以质量乘以欧米伽平方乘以位移的平方。
Add them: the x terms cancel, and the total is constant.
相加后位移项抵消，总能量恒定。
Let's use that.
我们来用一用。
The same two-hundred-gram mass, amplitude five centimetres, frequency two point five hertz. Find its greatest kinetic energy.
同样的二百克物体，振幅五厘米，频率二点五赫兹，求它的最大动能。
Pause here and try it yourself. Ready?
先暂停，自己试一试。
The object is fastest at the centre, where the speed is omega times the amplitude — fifteen point seven times zero point zero five, about zero point seven nine metres per second.
好了吗？ 物体在中心最快，那里的速度等于角频率乘以振幅—— 十五点七乘以零点零五，约为零点七九米每秒。
Then the kinetic energy is one half m v squared: one half, times zero point two, times zero point seven nine squared — about zero point zero six joules.
然后动能等于二分之一乘以质量乘以速度的平方： 二分之一，乘以零点二，乘以零点七九的平方——约等于零点零六焦耳。
Real oscillations don't last forever.
真实的振动不会永远持续。
Friction and air resistance drain the energy away — we call this damping.
摩擦和空气阻力把能量耗散掉——这叫做阻尼。
With light damping, the object still swings, but the amplitude shrinks a little each time, fading inside a decaying envelope.
在轻阻尼下，物体仍然摆动，但每次振幅都减小一点，逐渐衰减在一个包络线内。
With critical damping, it returns to the centre as fast as possible without overshooting — exactly what you want in a car suspension.
在临界阻尼下，它以最快的速度回到中心且不越过——这正是汽车悬架所需要的。
With heavy damping, it creeps back slowly, without swinging at all.
在过阻尼下，它缓慢地爬回，完全不摆动。
Look closely at light damping.
仔细看轻阻尼。
The displacement–time graph is still a wave, but each crest is smaller than the last.
位移–时间图仍是波形，但每个波峰都比前一个小。
A smooth decaying envelope bounds the peaks above and below.
一条平滑衰减的包络线上下包住波峰。
Energy is lost as heat through friction, drag, and air resistance, so the amplitude shrinks slowly over many cycles.
能量通过摩擦、阻力和空气阻力变成热而损失， 所以振幅在许多周期里缓慢减小。
The system still oscillates near its natural frequency — only the size dies away.
系统仍在固有频率附近振动——只是幅度在消亡。
A car's suspension is light-to-medium damped: bumps die out, but the ride stays smooth.
汽车悬架属于轻到中等阻尼：颠簸会消失，但乘坐仍然平稳。
Critical damping is the least damping that brings the system home without overshooting and without oscillating.
临界阻尼是能让系统回到平衡、既不越过也不振荡的最小阻尼。
It returns in the shortest time.
它用最短时间返回。
A galvanometer or analogue voltmeter is critically damped so the needle settles quickly on the reading.
检流计或模拟电压表用临界阻尼，好让指针快速停在读数上。
Heavy damping — overdamping — has even more resistance: the system still does not swing, but it creeps back more slowly than the critical case.
过阻尼——也叫重阻尼——阻力更大：系统仍不摆动，但比临界情况爬得更慢。
A door with a strong closer is heavily damped.
带强闭门器的门就是过阻尼。
On the graph: critical is the fast smooth return; heavy is the slow crawl.
图上：临界是快速平滑返回；过阻尼是缓慢爬回。
So far the oscillator was free, or dying under damping.
到目前为止，振子是自由的，或在阻尼下衰减。
Now drive it.
现在去驱动它。
A forced oscillation is driven by an outside periodic force at a driving frequency the experimenter chooses.
受迫振动由外部周期性力驱动，驱动频率由实验者选定。
The system then oscillates at that driving frequency — not at its own natural frequency.
于是系统按这个驱动频率振动——而不是按自己的固有频率。
A plucked guitar string rings at its resonant frequencies; a continuous driver can lock the motion to any chosen rate.
拨动的吉他弦在其共振频率上发声；持续的驱动可以把运动锁定在任意选定的速率。
Plot amplitude against driving frequency and you get a resonance curve with a peak.
把振幅对驱动频率作图，就得到带有峰值的共振曲线。
Now push the oscillator with a repeating force — a driver.
现在用一个反复的力去推动振子——这叫驱动力。
When the driving frequency is far from the object's natural frequency, not much happens.
当驱动频率远离物体的固有频率时， 几乎没什么变化。
But as the driver matches the natural frequency, the amplitude climbs higher and higher.
但当驱动频率与固有频率一致时，振幅越来越大。
This is resonance: the largest response, when you push in time with the natural rhythm.
这就是共振：当你顺着固有节奏去推时，产生最大的响应。
The resonance curve plots amplitude against driving frequency.
共振曲线把振幅对驱动频率作图。
It rises to a peak near the natural frequency and falls away on either side.
它在固有频率附近升到峰值，两侧再落下。
Damping shapes the peak.
阻尼决定峰的形状。
Lighter damping gives a sharper, higher peak — more amplitude, but only in a narrow band of frequencies.
更轻的阻尼给出更尖、更高的峰——振幅更大，但只在很窄的频带。
Heavier damping gives a broader, lower peak, shifted slightly toward lower frequency.
更重的阻尼给出更宽、更低的峰，并略向低频偏移。
At resonance the energy transfer from the driver is most efficient, which is why the amplitude is largest.
共振时驱动源的能量传递最有效，所以振幅最大。
Resonance is everywhere.
共振无处不在。
A swing pushed at the right rate builds a large amplitude.
秋千在合适的节拍下会被推得振幅很大。
A wine glass can shatter when a sound matches its natural ringing frequency.
酒杯在声音匹配其固有振铃频率时可能碎裂。
A building shaken by an earthquake whose frequency matches a natural frequency can suffer huge swings — so engineers design buildings so their natural frequencies avoid the main earthquake range.
建筑物被频率匹配其固有频率的地震摇动时，摆动会很大—— 所以工程师会设计建筑，使固有频率避开主要地震频段。
And in nineteen forty, wind pushed the Tacoma Narrows Bridge close to its natural frequency; with little damping the twisting grew until the deck ripped apart.
一九四零年，风把塔科马海峡大桥推到接近固有频率；阻尼很小，扭动越来越大，直到桥面撕裂。
Now we can explain the bridge.
现在我们可以解释那座桥了。
The wind gave the deck a small push on every cycle, and by chance it pushed at the bridge's own natural frequency.
风每个周期都给桥面一个小小的推力， 而它恰好与桥的固有频率一致。
Each push added energy, in perfect time — resonance.
每一次推动都在恰当的时刻注入能量——这就是共振。
The swaying grew until the bridge tore itself apart.
摇晃越来越大，直到桥把自己撕裂。
The same effect is all around us: it tunes a radio, it makes a wine glass sing, and engineers now design bridges and buildings to keep it away.
这种效应无处不在：它让收音机调谐，让酒杯发声； 如今工程师会特意设计桥梁和建筑来避开它。
Before you go, three marks students always lose.
结束之前，三个学生常失的分。
First, never drop the minus sign in the defining equation — the acceleration is directed back towards that point, always back to the centre.
第一，永远不要在定义式中丢掉负号—— 加速度始终指向中心。
Second, work in radians, not degrees, whenever you use omega.
第二，凡是用到角频率，就用弧度而不是角度。
Third, remember that energy depends on amplitude squared — double the amplitude, and you get four times the energy.
第三，记住能量正比于振幅的平方——振幅翻倍，能量就是四倍。
Get these right, and this topic is yours.
把这些做对，这个专题就是你的了。