Hey Henry,

嘿，亨利。

A while ago you did a video... called "What if the Earth were Hollow?" where you showed

不久前，你做了一個視頻......叫做 "如果地球是空心的呢？"在那裡你展示了

how long it would take to fall through the earth. I was simply wondering how that was

它要花多長時間才能穿過地球。我只是想知道那是怎麼回事

even calculated since the force of gravity would constantly be changing due to the growing

甚至可以計算出來，因為重力將不斷地變化，因為日益增長的

amount of mass above you.

你上面的品質數量。

Peter

彼得

Ok, so we've got a hole through the earth, and it goes from the north pole to the south

好的，所以我們有一個貫穿地球的洞，它從北極到南極。

pole – that way we don't have to worry about the Coriolis effect from the earth's spinning.

這樣我們就不必擔心地球自轉產生的科里奧利效應。

There's also no air in the hole, otherwise you'd reach terminal velocity pretty quickly

洞裡也沒有空氣，否則你會很快達到終點速度。

and your trip would be slow and boring. There are a few different ways to figure out how

而你的旅行將是緩慢而無聊的。有幾種不同的方法來弄清楚如何

long it'll take for you to reach the other side.

你要花多長時間才能到達另一邊。

One is to jump in, but it'll be faster – and probably a greater chance of survival – if

一個是跳進去，但會更快，而且可能有更大的生存機會，如果

we calculate using math & physics.

我們用數學和物理學進行計算。

First, more simplifications: assume the earth is perfectly spherical and has the same density

首先，更多的簡化：假設地球是完全球形的，具有相同的密度

everywhere. It turns out that the gravitational attraction from any spherically symmetric

到處都是。事實證明，來自任何球體對稱的引力

object is the same as if all its mass were concentrated at the center of that object

的品質與所有品質都集中在該物體中心的情況相同。

– the closer parts attract more than average, the far away parts attract less, but over

- 較近的部分比平均水平吸引更多，較遠的部分吸引較少，但超過了

the whole sphere it averages out. In a similar vein, if you're _inside_ a spherical shell,

在整個球體上，它是平均的。類似地，如果你在一個球殼的_內。

then the gravitational pulls from all the different parts cancel out and you experience

那麼來自所有不同部分的引力就會抵消，你就會體驗到

zero effect from the shell.

炮彈的效果為零。

This means that, inside the earth, any parts that are farther away from the center than

這意味著，在地球內部，任何遠離中心的部分都比

you are cancel out and have no effect – almost like they've been trimmed off and you're temporarily

你被取消了，沒有任何影響--幾乎就像它們被修剪掉了，而你是暫時的

on the surface of a smaller, shaved earth. Since we assumed the same density everywhere,

在一個較小的、被刨開的地球的表面上。由於我們假設各地的密度相同。

the shaved-earth's mass is simply proportional to its volume, which is proportional to its

刨花地球的品質與它的體積成正比，而體積又與它的

radius cubed. And because it's a sphere we get to pretend all that mass is actually concentrated

半徑的立方。因為它是一個球體，我們可以假裝所有的品質實際上都集中在一起。

at a single point in the middle.

在中間的一個點上。

So how much does the shaved-earth-point pull on you? Well, the gravitational attraction

那麼，剃光頭的地球點對你有多大的拉力？嗯，引力的吸引力

between two objects is proportional to their masses but inversely proportional to the distance

兩個物體之間的距離與它們的品質成正比，但與距離成反比。

between them, squared, so we have to divide the mass of the shaved-earth by the square

它們之間的平方，所以我們必須用刨花地球的品質除以其平方。

of the distance you are from the center – which is just the radius of the shaved earth. R

你離中心的距離--這只是刨地的半徑。R

cubed divided by r squared is r, so the force on you is simply F equals some constant stuff

立方除以r的平方就是r，所以你所受的力只是F等於一些常數。

times r, your distance from the center.

乘以r，你與中心的距離。

Essentially, as you fall the mass beneath you decreases, while the **average gravitational

從本質上講，當你下降時，你腳下的品質會減少，而**的平均引力

pull on you** from any bit of that mass increases, but the mass decreases **more than the average

從該品質的任何一點對你**的拉力增加，但品質的減少**超過了平均水平

pull** of gravity increases.

重力的拉力**增加。

So as you approach the earth's center, you go faster and faster but the force pulling

是以，當你接近地心時，你的速度越來越快，但拉動你的力量卻越來越大。

you towards the middle gets smaller and smaller. Exactly in the middle you experience zero

你朝向中間的時候會變得越來越小。恰恰是在中間，你的體驗是零

net force because the earth is pulling you equally in all directions, though since you're

淨力，因為地球在各個方向上對你的拉扯是相等的，不過由於你是

going so fast you'll continue to speed towards the other side, gradually slowed by the now

速度如此之快，你將繼續向另一邊加速，逐漸被現在的

increasing force pulling you back towards the middle. F equals some constant stuff times

越來越大的力把你拉回到中間。F等於一些常數的東西乘以

distance.

距離。

The exact same equation – some constant stuff times a distance – also describes

完全相同的方程--一些常量的東西乘以一個距離--也描述了

a mass on a spring, or simple pendulum, or a cat in a parabola. And from studying _those_

一個彈簧上的品質，或簡單的擺，或拋物線上的貓。而從研究_那些_

equations we know that **the time taken by the moving object – whatever it is –

我們知道，**運動物體--不管它是什麼--所花費的時間

to go from** one side to the other has a simple formula: pi times the square root of the mass

從**的一邊到另一邊有一個簡單的公式：π乘以品質的平方根

divided by the "constant stuff". In the case of falling through the earth, your mass cancels

除以 "不變的東西"。在通過地球下落的情況下，你的品質抵消了

out of the equation so we just need to put in numbers for the density of the earth and

所以我們只需要把地球的密度數字輸入到方程中。

the gravitational constant to get the answer - 42 minutes to fall through the earth.

通過對引力常數的計算，可以得到答案--42分鐘後落入地球。

This turns out to be exactly the same as the time it takes to fall _around_ the earth to

事實證明，這與繞地球落下的時間完全相同。

the other side, and it's the number you'll find commonly mentioned on the internet. Even

另一邊，這是你會發現在互聯網上經常提到的數字。甚至

more surprising, the radius of the earth didn't factor into the time calculation – it predicts

更令人驚訝的是，地球的半徑並沒有被納入時間的計算中--它預測了

you'll take 42 minutes to fall through or orbit around to the other side of ANY sphere

你將需要42分鐘才能穿過或繞過任何球體的另一端

with the same density as the earth.

具有與地球相同的密度。

But the earth isn't exactly the same density throughout – we know from seismology that

但地球的密度並不是完全相同的，我們從地震學中知道

the earth's core is much denser than its mantle and crust. So as you begin to fall, most of

地球的核心比地幔和地殼的密度大得多。是以，當你開始下降時，大部分的

the mass is still below you, pulling, so the pull of gravity doesn't decrease as much as

品質仍然在你的下面，拉著你，所以重力的拉力並沒有減少得那麼多。

our simple model predicted. In fact, the force is actually pretty constant until about halfway

我們的簡單模型預測的。事實上，這個力實際上是相當恆定的，直到大約一半的時候

to the center, at which point it starts quickly decreasing as more and more of the earth is

到中心，在這一點上，它開始迅速減少，因為地球上越來越多的地方是

"above" you.

"高於 "你。

The calculations here are a bit more annoying because we have to piece together two different

這裡的計算有點煩人，因為我們必須把兩個不同的

parts – the falling with constant acceleration part, which is easy, and the falling with

部分--以恆定加速度下降的部分，這很容易，而以恆定加速度下降的

decreasing gravity proportional to your radius part, which is the same thing we did before,

減少重力與你的半徑部分成正比，這與我們之前所做的相同。

except now you're starting out halfway to the middle of the earth with a speed of 17

除了現在你在地球中部的半路上開始，速度為17

thousand miles per hour, instead of on the surface with no speed. Once our mathemagical

每小時一千英里，而不是在表面上沒有速度。一旦我們的數學

dust settles, we combine the two parts and multiply by two to get the total time back

塵埃落定後，我們將兩部分結合起來，乘以2，得到總的時間。

to the surface on the other side: 37 minutes.

到另一側的水面：37分鐘。

Of course, this is still just an approximation – slightly more realistic than before, but

當然，這仍然只是一個近似值--比以前稍微現實一些，但

far from perfect. If you carefully piece together the time for a falling-through-the-earth trip

遠非完美。如果你仔細拼湊出落地旅行的時間

based on a more detailed density profile of the earth, like maybe the Preliminary Reference

基於更詳細的地球密度剖面，也許像《初步參考》那樣

Earth Model, you can can be even more precise – 38 minutes and 6 seconds from pole to

地球模型，你可以更精確的 - 38分鐘和6秒從極地到地球。

pole.

杆。

But either way, if instead of calculating you jumped into the hole at the start of this

但無論如何，如果你不去計算，而是在這一開始就跳進洞裡的話

video, you still have a long ways to go before reaching the other side of the earth. Safe travels!

視頻，在到達地球的另一端之前，你還有很長的路要走。一路順風!