Name: 簡諧運動：物理閃電學習 #16 (Simple Harmonic Motion: Crash Course Physics #16)
Uploaded: 2026-01-10T02:47:24.000Z
Duration: 9 min 11 s

In June 2001, officials in London unveiled a striking new feat of engineering.

倫敦在 2001 年 6 月時，揭開了一項令人驚豔的全新工程。

The Millennium Bridge, a pedestrian bridge spanning the River Thames.

千禧橋，一座橫跨泰晤士河的人行橋。

It promised to be very useful, and it was cool to look at.

But it had to be closed almost immediately, because when people used the bridge, it swayed back and forth noticeably due to the force of their footsteps.

但啟用後不久就關閉了，因為大家走在橋上時，腳步的力量會讓橋明顯地來回擺動。

Undeterred, people kept using the bridge, but

儘管如此，人們還是繼續使用這座橋，但是

as they walked, they began to lean into the swaying to keep themselves from falling over.

當他們行走時，開始順著擺動的力道傾斜，以免自己跌倒。

Eventually, the motion of the bridge became so severe that the bridge took on the shape of a giant S, essentially a horizontal wave.

最後，橋的運動變得非常劇烈，橋身呈現一個巨大的 S 形，基本上就是一個水準的波浪。

The bridge had to be closed, and the engineers took nearly two years to fix the problem.

這座橋被迫關閉，工程師花了將近兩年才解決這個問題。

So what was wrong with the Millennium Bridge?

那麼，千禧橋到底出了什麼問題呢？

And why didn't the engineers foresee the problem?

那工程師們為什麼沒有預料到這個問題呢？

The physics that caused the swaying of the Millennium Bridge has to do with oscillations, or back and forth motion.

造成千禧橋搖晃的物理原理，跟震盪有關，也就是來回的運動。

Or more specifically, it has to do with simple harmonic motion, where oscillations follow a particular consistent pattern.

更具體來說，這跟簡諧運動有關，也就是震盪會遵循一個特定的規律模式。

But before we had the Millennium Bridge as a real life example, physicists often described simple harmonic motion in terms of a ball attached to a horizontal spring lying on a table.

不過，在我們有了千禧橋這個真實世界的例子之前，物理學家常常用一個球連接到放在桌上的水準彈簧來描述簡諧運動。

While it's lying there at rest, it's in equilibrium.

當它靜止不動時，就處於平衡點。

And when you move the ball so that it stretches the spring, then let go, the ball keeps moving back and forth forever.

當你移動球讓彈簧伸長，然後放手，球就會永遠來回移動。

So in a frictionless world, that back and forth motion caused by the force of the spring is simple harmonic motion.

所以，在沒有摩擦力的世界裡，由彈簧力造成的來回運動就是簡諧運動。

Now, we want to know two things about this oscillating ball.

現在，我們想知道關於這個震盪的球的兩件事。

What kinds of energy does it have, and what's its maximum velocity?

它擁有什麼樣的能量，以及它的最大速度是多少？

To better understand what's happening to the ball, let's start with its energy.

為了更瞭解球的狀況，我們先來看看它的能量。

As the ball compresses and stretches the spring, both kinetic energy and potential energy come into play.

當球壓縮和拉伸彈簧時，動能和位能都會出現。

And as the ball moves, there are two points, the turning points, where it's not moving.

而且當球移動時，有兩個點，也就是轉折點，球在那裡是靜止不動的。

One point is where the spring is compressed all the way, and

其中一個點是彈簧被壓縮到最緊的時候，

the other is where it's stretched all the way.

另一個點則是彈簧被拉伸到最長的時候。

And the distance between either of these two points and the equilibrium point is called the amplitude.

而這兩個點到平衡點之間的距離，就叫做振幅。

At those two turning points, the ball won't have any kinetic energy since it isn't moving.

在那兩個轉折點，球不會有任何動能，因為它沒有在移動。

Instead, all of the ball's energy will be potential energy from the spring, half of the spring constant times the amplitude squared.

取而代之的是，球的全部能量都會是來自彈簧的位能，也就是二分之一的彈簧常數乘以振幅的平方。

Now, as the ball moves towards the middle, its kinetic energy starts to increase because it's moving faster and faster.

現在，當球往中間移動時，它的動能開始增加，因為它的速度越來越快。

And at the same time, its potential energy decreases, keeping its total energy the same and exactly in the middle of the ball's motion at the equilibrium point, its potential energy goes down to zero.

同時，它的位能會減少，總能量保持不變，而在球運動的平衡點，位能會降到零。

The ball is back where it started, so the spring is no longer pulling on it.

球回到了它開始的地方，所以彈簧不再對它施力。

Its kinetic energy, on the other hand, has reached its maximum, which means that at that point, the total energy of the ball will be equal to half of its mass times its maximum velocity squared.

另一方面，它的動能已經達到最大值，這表示在那一點，球的總能量會等於二分之一的質量乘以最大速度的平方。

Now, we have two equations for the total energy in this oscillating spring, which we can combine into one equation.

現在，我們有了這個震盪彈簧總能量的兩個方程式，我們可以把它們合併成一個方程式。

And if we use algebra to move around its variables, we can start to answer the second question we had about the ball.

如果我們用代數來整理它的變數，我們就可以開始回答關於這個球的第二個問題了。

We wanted to know the ball's maximum velocity, and

this equation tells us that it's equal to the amplitude times the square root of the spring constant divided by its mass.

這個方程式告訴我們，它等於振幅乘以彈簧常數除以其質量的平方根。

So we've answered our two questions about the ball on the spring.

所以我們已經回答了關於彈簧上球的兩個問題。

We know about its energy, and we have an equation for its maximum velocity.

我們瞭解了它的能量，也得到了它最大速度的方程式。

But there's a lot more going on with this ball than just its energy and velocity.

但這個球的運動不只跟它的能量和速度有關。

It also has properties like period, a frequency, and an angular velocity.

它還有週期、頻率和角速度等性質。

而且，它的位置會隨著時間改變。

You might recognize those terms because we already talked about them in our episode on uniform circular motion.

你可能對這些名詞很熟悉，因為我們在講等速圓周運動時已經討論過了。

Simple harmonic motion is actually a lot like uniform circular motion, mathematically speaking.

其實，簡諧運動在數學上跟等速圓周運動很像。

You can see this for yourself if you compare the ball's motion on the spring to an object in uniform circular motion, say a marble moving along a ring at a constant speed.

你可以自己比較看看，把彈簧上的球的運動，跟一個在做等速圓周運動的物體做對比，比如一顆在固定速度下沿著圓環移動的彈珠。

Okay, I admit it might seem like kind of a weird comparison at first.

好啦，我承認一開始聽起來有點奇怪。

For one thing, the ball on the spring is moving in one dimension, while a marble moving along a circular path is in two dimensions.

其中一個原因，彈簧上的球是在一個維度上移動，而沿著圓形路徑移動的彈珠則是在兩個維度上。

But what if you take that ring and look at it from the side?

但如果你把那個圓環從側面看呢？

The marble keeps moving along its circular path, but

彈珠沿著它的圓形路徑持續移動，但是

to you, it looks like it's just moving back and forth along a straight line.

看起來就像是在一條直線上來回移動。

it looks like this marble is stopping momentarily as it changes direction and moving faster as it gets closer to the middle, which is exactly the same way the ball was moving on the spring.

這個彈珠看起來在改變方向時會暫停一下，然後越靠近中間速度越快，這跟彈簧上的球的運動方式完全一樣。

Now, let's take this comparison one step further.

接下來，我們把這個比較再往前推進一步。

Let's assume that the radius of the ring is the same as the amplitude of the ball's motion on the spring, and

我們假設圓環的半徑跟彈簧上球的振幅一樣大，而且

that the marble's constant speed along the ring is equal to the maximum speed of the ball on the spring.

彈珠沿著圓環移動的固定速度，等於彈簧上球的最大速度。

In that case, if you did the math, you'd find that the equation for the marble's velocity, when you look at it edge on, is exactly the same as the equation that described the velocity of the ball on the spring.

在這種情況下，如果你做數學計算，你會發現從側面看彈珠的速度方程式，跟描述彈簧上球的速度的方程式是完全一樣的。

So let's recall what we know about uniform circular motion to see what it can tell us about simple harmonic motion.

所以，讓我們回顧一下我們對等速圓周運動的瞭解，看看它能告訴我們什麼關於簡諧運動的知識。

We know that the time it takes for the marble to move around the ring once is called the period.

我們知道彈珠繞著圓環轉一圈所需的時間，就叫做週期。

We also know that the period will be equal to the circumference of the ring divided by the marble's speed.

我們也知道週期會等於圓環的周長除以彈珠的速度。

And the radius of the circle is the same as the ball's amplitude on the spring.

而且圓環的半徑，跟彈簧上球的振幅是一樣的。

So its circumference will be equal to 2 times PI times the amplitude.

所以圓環的周長就會等於 2 乘以 PI 乘以振幅。

This means that the period will be equal to 2 times PI times the amplitude divided by the marble speed, which again, is the same as the ball's maximum speed as it moves on the spring.

這表示週期會等於 2 乘以 PI 乘以振幅，再除以彈珠的速度，這又跟彈簧上球的最大速度是一樣的。

And we can simplify that equation, since we know that the maximum speed of the ball is equal to the amplitude times the square root of the spring constant divided by the mass.

我們可以簡化這個方程式，因為我們知道球的最大速度等於振幅乘以彈簧常數除以質量的平方根。

So the period of the marble's motion around the ring is equal to 2 PI times the square root of m over k.

所以彈珠繞著圓環運動的週期，會等於 2 PI 乘以 m 除以 k 的平方根。

Now, we've also talked about the frequency of uniform circular motion.

現在，我們也討論過等速圓周運動的頻率。

It's the number of revolutions the marble makes around the ring every second.

頻率是彈珠每秒繞著圓環轉動的圈數。

And it's equal to one divided by the period.

In this case, the frequency will also be equal to 1 over 2 PI times the square root of k over m.

這種情況下，頻率也會等於 1 除以 2 PI 乘以 k 除以 m 的平方根。

And that'll apply to the ball on the spring too, because the rules are the same.

這也適用於彈簧上的球，因為規則是一樣的。

Finally, there's angular velocity to consider in uniform circular motion.

最後，在等速圓周運動裡，我們還要考慮角速度。

We've described it as the number of radians per second that the marble covers as it moves around the ring.

我們把它描述成彈珠繞著圓環移動時，每秒掃過的徑度量。

An angular velocity is just equal to the frequency times 2pi, which means that in the case of the ball on the spring, it's equal to the square root of k over m.

角速度就等於頻率乘以 2 PI，這表示在彈簧上的球的情況下，它就等於 k 除以 m 的平方根。

So now, with the help of our knowledge about circular motion, we can understand the period, frequency and angular velocity of the ball's simple harmonic motion as it oscillates on the spring.

所以現在，藉由我們對圓周運動的瞭解，我們可以理解彈簧上球的簡諧運動在震盪時的週期、頻率和角速度。

How does the ball's position change over time?

球的位置會隨著時間如何變化呢？

為了找出答案，我們必須分析。

Analyze the marble's motion along the ring again.

再次分析彈珠沿著圓環的運動。

And the answer will involve some trigonometry.

But it's not particularly complicated trig.

At any given point along its path, the marble will be at a certain angle to the right hand side of the ring.

在軌跡上的任何一點，彈珠相對於圓環的右側都會有一個特定的角度。

And the cosine of that angle will be equal to its horizontal distance from the center of the ring divided by the ring's radius.

而那個角度的餘弦值，就等於它離圓環中心的水準距離除以圓環的半徑。

We already know that the radius of the ring is the same as the amplitude of the ball's motion along the spring.

我們已經知道圓環的半徑，就等於彈珠沿著彈簧運動的振幅。

And if you turn the ring so it looks like a line again, you can see that the marble's horizontal distance from the center of the ring is the same as the ball's distance from the equilibrium point.

如果你把圓環轉回來，讓它看起來像一條線，你就會發現彈珠離圓環中心的水準距離，就等於球離平衡點的距離。

So the cosine of theta is equal to the ball's position divided by its amplitude.

所以 theta 的餘弦值，就等於球的位置除以它的振幅。

In other words, the ball's position is equal to the amplitude times the cosine of the angle.

換句話說，球的位置就等於振幅乘以角度的餘弦值。

In the same way that distance is equal to velocity multiplied by time, the angle is equal to the angular velocity multiplied by time.

就像距離等於速度乘以時間一樣，角度等於角速度乘以時間。

So we can write the equation for the position of the ball as x equals a cosine omega t.

所以我們可以把球的位置方程式寫成 x 等於 a 乘以 cosine omega t。

And when you graph this equation, something interesting happens.

當你畫出這個方程式的圖形時，會發生一件有趣的事。

We'll be talking a lot more about waves in our next three episodes, but

在接下來的三集裡，我們會更深入地探討波，但

for now, it's helpful just to see the connection here.

現在，能看到這裡的關聯就很有幫助了。

For an object in simple harmonic motion, the graph of its position versus time is a wave, which is why the swaying of the Millennium Bridge looked like a wave.

對於做簡諧運動的物體來說，它的位置對時間的圖形就是一個波，這也是為什麼千禧大橋的搖晃看起來像一個波。

Speaking of the bridge, we can now better understand what happened to it.

說到這座橋，我們現在可以更好地理解它發生了什麼事。

The bridge's shimmy was the result of oscillation, but it was made worse by another resonance.

這座橋的搖晃是震盪的結果，但另一個共振讓情況變得更糟。

Resonance can increase the amplitude of an oscillation by applying force at just the right frequency, Kind of like how you can get a kid on a swing set to swing higher by pushing at just the right moment.

共振可以透過在恰當的頻率施力，來增加震盪的振幅，就像你透過在恰當的時機推，能讓鞦韆上的小孩盪得更高一樣。

The engineers of the Millennium Bridge were reminded of that the hard way.

千禧大橋的工程師們用慘痛的教訓記取了這一點。

When pedestrians on the bridge started to lean into its swaying, they created resonance.

當橋上的行人開始隨著橋的搖晃而傾斜時，他們就造成了共振。